Universität Innsbruck...SIAM J. CONTROL OPTIM. c 2006 Society for Industrial and Applied Mathematics Vol. 45, No. 4, pp. 1467–1507 STABILITY AND STABILIZATION OF MULTIDIMENSIONAL

SIAM J. CONTROL OPTIM. c© 2006 Society for Industrial and Applied MathematicsVol. 45, No. 4, pp. 1467–1507

STABILITY AND STABILIZATION OF MULTIDIMENSIONALINPUT/OUTPUT SYSTEMS∗

ULRICH OBERST†

Abstract. In this paper we discuss stability and stabilization of continuous and discrete multi-dimensional input/output (IO) behaviors (of dimension r) which are described by linear systems ofcomplex partial differential (resp., difference) equations with constant coefficients, where the signalsare taken from various function spaces, in particular from those of polynomial-exponential functions.Stability is defined with respect to a disjoint decomposition of the r-dimensional complex space intoa stable and an unstable region, with the standard stable region in the one-dimensional continuouscase being the set of complex numbers with negative real part. A rational function is called stableif it has no poles in the unstable region. An IO behavior is called stable if the characteristic varietyof its autonomous part has no points in the unstable region. This is equivalent to the stability ofits transfer matrix and an additional condition. The system is called stabilizable if there is a com-pensator IO system such that the output feedback system is well-posed and stable. We characterizestability and stabilizability and construct all stabilizing compensators of a stabilizable IO system(parametrization). The theorems and proofs are new but essentially inspired and influenced by andrelated to the stabilization theorems concerning multidimensional IO maps as developed, for instance,by Bose, Guiver, Shankar, Sule, Xu, Lin, Ying, Zerz, and Quadrat and, of course, the seminal papersof Vidyasagar, Youla, and others in the one-dimensional case. In contrast to the existing literature,the theorems and proofs of this paper do not need or employ the so-called fractional representationapproach, i.e., various matrix fraction descriptions of the transfer matrix, thus avoiding the oftenlengthy matrix computations and seeming to be of interest even for one-dimensional systems (at leastto the author). An important mathematical tool, new in systems theory, is Gabriel’s localizationtheory which, only in the case of ideal-convex (Shankar, Sule) unstable regions, coincides with theusual one. Algorithmic tests for stability, stabilizability, and ideal-convexity, and the algorithmicconstruction of stabilizing compensators, are addressed but still encounter many difficulties; see inparticular the open problems listed by Xu et al.

Key words. stability, stabilization, multidimensional system, behavior, stable transfer matrix

AMS subject classifications. 93D15, 93D25, 93C20, 93C35

DOI. 10.1137/050639004

1. Introduction. Stabilization theory is a part of control theory and usuallyinvolves the following ingredients [7, p. 60].

1. Stability : Select the class of admissible systems and define and characterizethe stable systems in this class.

2. Stabilizability : Determine which admissible systems can be stabilized by out-put feedback.

3. Stabilization: Construct a stabilizing compensator for a given stabilizablesystem.

4. Parametrization: Classify or construct all stabilizing compensators for a givenstabilizable system.

In this paper we discuss these problems for continuous and discrete multidimen-sional input/output (IO) behaviors, which are described by linear systems of complexpartial differential equations on R

r (resp., difference equations on Nr) with constant

∗Received by the editors August 26, 2005; accepted for publication (in revised form) May 12,2006; published electronically October 16, 2006.

http://www.siam.org/journals/sicon/45-4/63900.html†Institut fur Mathematik, Universitat Innsbruck, Technikerstraße 25, A-6020 Innsbruck, Austria

([email protected]).

1467

1468 ULRICH OBERST

coefficients, where the signals are taken from various function spaces—in particu-lar from those of polynomial-exponential functions. Polderman and Willems in [15,sec. 10.8] and Rocha [19] suggest renouncing the IO structure and output feedbackin favor of more general behavior interconnections. The first stabilization results formultidimensional systems in this generality are due to Shankar [21], [22].

In contrast to our approach, most multidimensional stabilization papers use acommutative integral domain S of “SISO-stable plants” and describe the admissiblesystems, often called plants, by a transfer operator or IO map, which is a matrixwith coefficients in the quotient field of S; cf., for instance, [26], [6], [23], [25], [31],[32], [7], [8], [16], [17], [18]. This approach to stabilization theory is originally dueto Desoer, Kucera, Vidyasagar, Youla, and their coworkers. These systems are calledstructurally [6] or internally [17] stable if their transfer matrix has entries in S. Intheir recent paper [29], Wood, Sule, and Rogers treat stability and causality, but notstabilization of continuous multidimensional IO systems.

An IO behavior B gives rise to its autonomous part B0, its transfer matrix H,and the largest controllable subbehavior Bcont which, in turn, has the autonomouspart B0

cont. The entries of the transfer matrix are complex rational functions in rindeterminates sρ, i.e., contained in the quotient field C(s) of the polynomial algebraA := C[s] = C[s1, . . . , sr]. In general, the transfer matrix does not act on arbitraryinputs as an operator or IO map, and it is an important task to identify those inputson which it does. Associated with these behaviors is the complex variety sing(B)of rank singularities and the characteristic varieties char(B0) ⊇ char(B0

cont) of theautonomous subbehaviors, the latter coinciding with the variety of poles of H. Inthe one-dimensional theory, the elements of char(B0) are called the poles, modes,characteristic values, or natural frequencies of the system.

Stability and stabilization of an IO system are defined with respect to a disjointdecomposition C

r = Λ1 � Λ2 of the complex space into a stable region Λ1 and anunstable region Λ2, with the standard continuous (resp., discrete) cases being

Λ2 = C+r, C+ := {z ∈ C; �(z) ≥ 0} or Λ2 = C+ × iRr−1 ([29]),

resp., Λ2 = {z ∈ C; |z| ≥ 1}r or Λ2 = {z ∈ C; |z| ≥ 1} × (S1)r−1.(1)

A rational function is called stable if it has no poles in the unstable region. The ringof all stable rational functions or SISO-stable plants is the quotient ring AT ⊆ C(s)with T := {t ∈ A; ∀λ ∈ Λ2 : t(λ) = 0} [25]. The discrete IO maps in the literature areusually assumed to be causal or proper and are considered rational functions in theindeterminates s−1

ρ ; then the set Λ2 from (1) is replaced with the closed unit polydisc[6], [7]. Properness is not assumed in the present paper because it is rather restrictivefor partial differential equations, but properness and the ensuing BIBO stability willbe discussed in the paper [20].

An IO system B is called stable if the characteristic variety of its autonomous partis contained in the stable region or, equivalently, if all polynomial-exponential trajec-tories in B0 are stable, i.e., involve exponents in the stable region only. In [29] this iscalled a characteristic variety (CV) condition and used to define stable autonomoussystems. Stability of B in this sense is equivalent to the stability of the transfer ma-trix and an additional condition (see Theorem/Definition 5.1 and Remark 5.2). Inparticular, a one-dimensional IO system is stable if and only if its autonomous part isasymptotically stable or, equivalently, if its transfer matrix is stable and its singularvariety is contained in the stable region. One of the reviewers points out that thesetwo equivalent descriptions of the stability of one-dimensional IO systems should be

STABILITY AND STABILIZATION 1469

called time domain (resp., frequency domain) stability. Multidimensional stable IOsystems are externally or IO stable in the sense that the transfer matrix acts as anoperator on interesting classes of inputs and generates outputs of the same type asshown in Theorem/Definition 5.3 and Theorem 5.4; the latter applies results on par-tial differential equations from [4] as do Theorems 6.4 and 7.2 in [29] which, however,hold in special cases only. Theorem 7.6.2 of [15, p. 265], for instance, treats theconnection between asymptotic stability and bounded input/bounded output (BIBO)stability of one-dimensional IO systems as done already by Kalman in his fundamen-tal work. The paper [29] contains the interesting idea that stable systems shouldgenerate stable outputs from stable inputs and initial conditions, with a necessaryrequirement that the initial value problem be defined and uniquely solvable. For dis-crete multidimensional systems this is the case for which we give a partial answer andpose Open Problem 5.13, whose study is also worthwhile for continuous systems andcertain function spaces.

An IO system B is called stabilizable if there is an IO system B′ such that thefeedback system (in (20), (21)) of B and B′ is well-posed [26] and stable; then B′

is called a stabilizing compensator. In Theorems 4.4 and 5.8 we characterize sta-bilizability of B and construct one stabilizing compensator, whereas Theorems 2.14and 4.6 describe the parametrization or construction of all stabilizing compensatorsof a stabilizable IO behavior. The famous prototype of such a parametrization is thatof Kucera, Youla, Bongiorno, and Jabr and is detailed by Vidyasagar in [26, Chap. 5].The theorems on the stabilization of general multidimensional IO systems and theirproofs are new but essentially influenced and inspired by and related to the resultson the stabilization of IO maps in the references given above.

The proofs employ localization, after the work of Gabriel, as a new mathematicaltool in systems theory which is described in Stenstrom’s book [24] and in section 3 ofthis paper. At no time do the results and proofs need or employ the so-called fractionalrepresentation approach (i.e., matrix fraction descriptions of the transfer matrix ofvarious kinds and the, sometimes long [26], [17], [18], ensuing matrix computations);thus they seem simpler and of interest even in the one-dimensional case (at least tothe author). The localization technique also avoids the difficulties in [29] with the lackof ideal-convexity [23] of the unstable regions Λ2. Such a region is called ideal-convexif

V (a) ∩ Λ2 = ∅ ⇒ a ∩ T = ∅ ∀ ideals a ⊂ C[s], where

V (a) := {λ ∈ Cr; ∀t ∈ a : t(λ) = 0}

(2)

denotes the algebraic variety of a. Ideal-convexity is characterized by the coincidenceof Gabriel localization with the standard localization functor M �→ MT on A-modulesM (see Theorem/Definition 5.6).

Algorithmic problems are addressed in Remark/Open Problem 5.10. The algo-rithmic test of stability, stabilizability, and ideal-convexity and the algorithmic con-struction of one or all stabilizing compensators still encounter many difficulties; seein particular the open problems in [7] and [30] which, however, address these difficul-ties only for the closed unit polydisc of arbitrary dimension as region of instability.Solutions for the closed unit polydisc are known in interesting special cases [6], [7],[8].

Sections 2, 3, and 4 of this paper consider abstract IO systems whose signalspaces are injective cogenerators over a factorial Noetherian integral domain as in [11,Chap. 7] and use, in particular, Matlis’ theory of injective modules over Noetherian

1470 ULRICH OBERST

rings as in [12]. This generality makes the proofs more transparent and may possiblybe used for other types of systems too. As preparation for the stable and stabilizablesystems in section 4, section 2 discusses trivial and trivializable systems and theconstruction of all trivializing compensators. An IO behavior is called trivial if itsautonomous part is zero or, equivalently by Theorem/Definition 2.5, if it is controllableand its transfer matrix is polynomial. In section 5 the results of sections 2, 3, and 4are specialized to multidimensional systems proper as described above.

Due to the large number of papers on stabilization theory, we list only thosereferences which are actually used. But the author is fully conscious of many otherimportant contributions and contributors, such as, for instance, Bisiacco, Fornasini,Marchesini, and Valcher of the Padovian school.

2. Triviality and trivialization by feedback. Let A denote a commutativeNoetherian integral domain with its quotient field K = quot(A), and let F denotean injective cogenerator which is used as a signal space with its scalar multiplication◦. In this section we consider F-systems or F-behaviors as introduced and studiedin [11], in particular [11, Chap. 7, p. 139]. Refer to the first pages of [28] or to [27] fora newer, more elegant introduction to multidimensional behavioral systems theory.It was shown that many cases of interest for systems theory can be developed inthis abstract setting. For instance, the continuous case of systems governed by linearsystems of partial differential equations with constant coefficients uses the data

A := C[s] = C[s1, . . . , sr], F � y = y(z), sρ ◦ y = ∂y/∂zρ,

z := (z1, . . . , zr) ∈ Rr, λ = (λ1, . . . , λr) ∈ C

r, λ • z := λ1z1 + · · · + λrzr,

F := C∞(Rr,C) or F := D′(Rr) or

F := C∞(Rr,C)lf := {y ∈ C∞(Rr,C); [C[s] ◦ y : C] < ∞}= D′(Rr)lf = ⊕λ∈Cr C[z] exp(λ • z),

(3)

whereas multidimensional discrete or r-dimensional systems theory applies the poly-nomial algebra A as in (3) and the signal space

F := CN

r

= C[[z]] = C[[z1, . . . , zr]] � y = (yμ)μ∈Nr =∑μ∈Nr

yμzμ

with (sν ◦ y)μ := yμ+ν , μ, ν ∈ Nr.

(4)

The corresponding objects over the real field R instead of the complex field C arelikewise admissible. Recall that an A-module F is an injective cogenerator if thecontravariant duality functor,

D := HomA(−,F) : (ModA)op → ModA, M �→ HomA(M,F),(5)

preserves and reflects exact sequences, where ModA is the category of A-modules.In [11] we used a large injective cogenerator F , but this is unnecessary, as was observedin [12]. The column vectors in F l are suggestively called trajectories also in theabstract module situation.

A matrix R ∈ Ak×l gives rise to the row submodule

U := A1×kR =

k∑i=1

ARi− ⊂ F := A1×l,(6)


the factor module

M := A1×l/U =

l∑j=1

Aδj , δj := (0, . . . , 0,j

1, 0, . . . , 0) ∈ A1×l,

and the behavior

B := U⊥

:= {w = (wj)j=1,...,l ∈ F l; ∀f = (f1, . . . , fl) ∈ U : f ◦ w = f1 ◦ w1 + · · · + fl ◦ wl = 0}= {w = (wj)j=1,...,l ∈ F l; R ◦ w = 0} =

ident.HomA(M,F), w = (δj �→ wj).

Since A is Noetherian, every submodule U ⊂ A1×l arises in this fashion. Like U⊥ wedefine, for every submodule B of F l, the orthogonal submodule

B⊥ := {f = (f1, . . . , fl) ∈ A1×l; ∀w ∈ B : f ◦ w = 0} ⊂ A1×l(7)

of all linear equations which are satisfied by all trajectories in B. If B := U⊥ is abehavior, the relation B⊥⊥ = U⊥⊥⊥ = U⊥ = B holds since (−)⊥ is a Galois corre-spondence, whereas the identity U = B⊥ = U⊥⊥ is a consequence of the cogeneratorproperty of F [11, Cor. 2.47].

The matrix R with M and B as in (6) also gives rise to its transfer space (= signalflow space in [11, Thm./Def. 2.91])

B := {w ∈ Kl; Rw = 0} =ident.

HomA(M,K) =ident.

HomK(K ⊗A M,K) ⊂ Kl(8)

and, more precisely, to the contravariant exact functor HomA(−,K) on finitely gen-

erated A-modules or to the covariant exact functor B �→ B on behaviors. Standardlinear algebra over the field K can be applied to the K-space B. In particular, itdetermines the number

rank(B) := rank(M) := [K ⊗A M : K] = [B : K] = l − rank(R),(9)

where [F : A] denotes the dimension of a free A-module F .Let w =

(yu

)∈ Fp+m, l = p + m, be a decomposition of the trajectories w into

two components y and u, possibly after a permutation of the components wj of w,and let R = (P,−Q) ∈ Ak×(p+m) be the corresponding decomposition of the matrixR such that B := {w =

(yu

)∈ Fp+m; P ◦ y = Q ◦ u}. The matrix P gives rise to the

module M0 := A1×p/A1×kP and the behavior B0 := {y ∈ Fp; P ◦ y = 0}.Result 2.1 (IO structure and transfer matrix; see [11, Thms. 2.69, 2.94]). The

following assertions are equivalent:1. rank(P ) = rank(R) = p.2. The projection

proj := (0 idm)◦ : B =

{w =

(y

u

)∈ Kp+m; P y = Qu

}→ Km,

(y

u

)�→ u,(10)

is a K-isomorphism, i.e., for each u ∈ Km the equation P y = Qu is uniquely solvablefor y ∈ Kp.

3. The A-module sequence

0 −→ A1×m inj:=◦(0 idm)−→ Mproj:=◦(idp

0 )−→ M0 −→ 0,

η �→ (0, η), (ξ, η) �→ ξ(11)

1472 ULRICH OBERST

is exact, and the module M0 is a torsion module or rank(M0) = 0.4. The dual sequence of behaviors

0 −→ B0(idp

0 )◦−→ B proj:=(0 idm)◦−→ Fm −→ 0,

y �→(y0

),(yu

)�→ u

(12)

is exact, and the behavior B0 is autonomous or rank(B0) = 0.If these equivalent conditions are satisfied, the behavior is called an IO behavior

with the IO structure(yu

), the input u, and the output y. The rank of B is m. There

is a unique matrix H ∈ Kp×m, the transfer matrix of B, such that PH = Q. Theinverse of the isomorphism (10) has the graph form u �→

(Huu

), and for every input

u ∈ Fm there is a trajectory(yu

)∈ B.

If B is an IO behavior as in the preceding result, the A-sequence

A1×k ◦(P,−Q)−→ A1×(p+m)◦( H

idm)

−→ K1×m(13)

is a complex, i.e., (P,−Q)(

Hidm

)= PH −Q = 0, and induces an A-epimorphism

M = A1×(p+m)/A1×k(P,−Q)( Hidm

)−→ M := im

(◦(

H

idm

))= A1×pH + A1×m,

(ξ, η) �→ ξH + η.

(14)

The A-submodule M of K1×m is a lattice. Lattices play a decisive part in Quadrat’streatment of stabilization [17, Eq. (38) and Thm. 3], [18] (see also Quadrat’s earlierpapers quoted there).

Theorem and Definition 2.2 (controllable behaviors and realization). Themodule M is torsion free if and only if the sequence (13) is exact or

B⊥ = A1×k(P,−Q) = {(ξ, η) ∈ A1×(p+m); ξH + η = 0}.(15)

If this is the case, the behavior B is called controllable and is indeed the unique con-trollable IO behavior with transfer matrix H or, in other words, the unique controllablerealization of H, and moreover,

A1×kP = {ξ ∈ A1×p; ξH ∈ A1×m}.(16)

Remark 2.3. As shown by Willems and Rocha for discrete two-dimensional sys-tems and by Pillai and Shankar [14] for continuous multidimensional ones, the termcontrollable is justified by the concatenability of trajectories in controllable systems. Amodule is torsion free if and only if it can be embedded into a free module and hence,by duality, the system B is controllable if and only if there is a system epimorphismφ : Fm → B, which Pommaret (resp., Willems) calls a parametrization (resp., animage representation) of B. The first reviewer suggests calling a controllable systemB torsion free or, more generally, to systematically use the attribute of the systemmodule M also for the system itself. We will stick to the term controllable since it isused by most researchers. If d is a common denominator of the entries of H, i.e., if0 = d ∈ A and dH ∈ Ap×m, the multiplication with d is an isomorphism on K, andtherefore (15) can also be expressed as

A1×k(P,−Q) = ker

(A1×(p+m)

◦( dHd idm

)−→ A1×m

).


Proof. The theorem is a reformulation of [11, Thm. 7.24]. We give a slightlysimpler and more direct proof. If (13) is exact, then (14) is an isomorphism, and henceM is torsion free. If, conversely, M is torsion free, we obtain the monomorphisms

M = A1×(p+m)/A1×kR → K ⊗A M = K1×(p+m)/K1×kR◦( H

idm)

∼= K1×m,

(ξ, η) �→ 1 ⊗ (ξ, η) = (ξ, η) �→ ξH + η,

(17)

which imply A1×kR = {(ξ, η) ∈ A1×(p+m); ξH + η = 0} and the exactness of (13).The first map in (17) is a monomorphism since its kernel is the torsion submodule ofM and thus zero. The second isomorphism follows from

(P,−Q) = P (idp,−H) and rank(P ) = p; hence K1×k(P,−Q) = K1×p(idp,−H),

and the exactness of

0 −→ K1×p ◦(idp,−H)−→ K1×(p+m)◦( H

idm)

−→ K1×m −→ 0

or K1×(p+m)/K1×p(idp,−H)◦( H

idm)

∼= K1×m.

Concerning equality (16), the identity PH = Q ∈ Ak×m implies

A1×kP ⊆ {ξ ∈ A1×p; ξH ∈ A1×m}.

If, conversely, η := −ξH ∈ A1×m and thus ξH + η = 0, the monomorphism (17)implies (ξ,−η) = ζ(P,−Q) for some ζ ∈ A1×k and hence ξ = ζP ∈ A1×kP .

Definition 2.4. A behavior B is called free (resp., projective) if its module Mhas this property, and is then controllable. Free behaviors were characterized in [11,Thm. 7.53] and were called strongly controllable by Rocha in her thesis.

With these preparations we can now draw the following simple conclusion which,however, is basic for the present paper.

Theorem and Definition 2.5 (trivial IO behaviors). For an IO behavior B asin Result 2.1, the following conditions are equivalent:

1. The autonomous behavior B0 is zero or M0 = 0 or A1×kP = A1×p; i.e., therows of P generate the full free module A1×p.

2. The behavior B is controllable and H ∈ Ap×m.If these equivalent conditions are satisfied, the IO behavior B is called trivial and thelinear maps

◦(0 idm) : A1×m ∼= M, η �→ (0, η), and proj := (0 idm)◦ : B ∼= Fm,

(y

u

)�→ u,(18)

are isomorphisms; in particular M and B are free. In accordance with the literature,we reserve the term stable to a more general situation in sections 4 and 5.

Proof. ⇒: The exact sequences (11) and (12) and M0 = 0 imply the isomor-phisms (18) and hence the freeness and controllability of B. The equality

A1×p = A1×kP = {ξ ∈ A1×p; ξH ∈ A1×m} implies A1×pH ⊆ A1×m;

hence H ∈ Ap×m.

1474 ULRICH OBERST

� � � �

� �

�

�

��

�

B1 B2

u2u1

y1 y2

+ +

Fig. 2.1.

⇐: The controllability of B implies equality (16) and then, with H ∈ Ap×m, alsoA1×p = A1×kP .

In what follows we consider a feedback system given by the following block dia-gram [11, Chap. 8] as in Figure 2.1.

The two IO subbehaviors Bi, i = 1, 2, of Fp+m are given as

B1 :=

{(y1

u1

)∈ Fp+m; P1 ◦ y1 = Q1 ◦ u1

}, P1H1 = Q1, P1 ∈ Ak1×p,

B2 :=

{(u2

y2

)∈ Fp+m; P2 ◦ y2 = Q2 ◦ u2

}, P2H2 = Q2, P2 ∈ Ak2×m.

(19)

Let l := p + m. The feedback behavior B := feedback(B1,B2) is the behavior

B :=

{(y

u

)∈ F2l; y =

(y1

y2

), u =

(u2

u1

)∈ Fp+m satisfy (21)

}(20)

with

P1 ◦ y1 = Q1 ◦ (u1 + y2), P2 ◦ y2 = Q2 ◦ (u2 + y1).(21)

Lemma 2.6. The map

B = feedback(B1,B2) → B1 × B2,

(y

u

)�→

((y1

u1 + y2

),

(y2

u2 + y1

))is a behavior isomorphism with the inverse map

((y1

u1

),(u2

y2

))�→

(yv

), with v1 := u1 −y2

and v2 := u2 − y1. In particular, B is controllable (resp., projective) if and only ifboth Bi have these properties. If B is trivial, then both Bi are projective.

Proof. It is obvious that the indicated map is indeed the inverse map. Recallthat a trivial behavior is free and that projective modules are precisely the directsummands of free ones.

From Result 2.1 we know that the transfer space of B1 is

B1 =

{(y1

u1

)∈ Kp+m; y1 = H1u1

}and likewise for B2. Therefore, the equations of the transfer space of B accordingto (8) are

y1 = H1(u1 + y2), y2 = H2(u2 + y1)

⇔ (idp −H1H2)y1 = H1u1 + H1H2u2, y2 = H2(u2 + y1).(22)


Theorem 2.7. The feedback behavior B = feedback(B1,B2) is an IO behaviorwith input u =

(u2

u1

)∈ Fp+m if and only if one or, equivalently, all of the matrices

idp −H1H2, idm −H2H1,

(idp −H1

−H2 idm

)are invertible, i.e., contained in Gl•(K). If this is the case, the autonomous part B0

and the transfer matrix of B are given as

B0 =

{y =

(y1

y2

)∈ Fp+m;

(P1 −Q1

−Q2 P2

)◦ y = 0

}= B1 ∩ B2,

H :=

(idp −H1

−H2 idm

)−1 (0 H1

H2 0

).

Moreover,

B⊥1 ⊕ B⊥

2 = (B0)⊥.

Vidyasagar then calls the feedback system well-posed [26, p. 100]. Rocha calls B0

a regular interconnection of B1 and B2 [19, Def. 1]. Direct sum decompositions andregular interconnections of systems have also been treated in [1] and [33] and in paperson the stabilization of IO maps; see, for instance, [17], [18].

Proof. It is obvious that (22) are uniquely solvable for(y1

y2

)∈ Kp+m for given(

u2

u1

)∈ Kp+m if and only if idp −H1H2 ∈ Glp(K). The assertion then follows from

Result 2.1. The simultaneous invertibility of these matrices is standard and followstrivially by elementary row and column operations. According to Result 2.1, theequations of B0 follow from (21) of B by setting the input u =

(u2

u1

)to zero. The

expression for H is implied by the equations(P1 −Q1

−Q2 P2

)H =

(P1 00 P2

)(idp −H1

−H2 idm

)H

=

(0 Q1

Q2 0

)=

(P1 00 P2

)(0 H1

H2 0

)by cancelling the matrix

(P1 00 P2

)of rank l = p + m on the left.

The sum (B0)⊥ = B⊥1 + B⊥

2 follows from B0 = B1 ∩ B2 by duality. Moreover,

rank((B0)⊥) = p + m = rank(B⊥1 ) + rank(B⊥

2 ) implies B⊥1 ∩ B⊥

2 = 0;

hence (B0)⊥ = B⊥1 ⊕ B⊥

2 .

Remark 2.8. If the feedback behavior of the preceding theorem is well-posed, thematrices

G :=

(0 H1

H2 0

)and (id−G)−1(id−G) = id in K(p+m)×(p+m)(23)

imply

H = (id−G)−1G and (id−G)−1 = id +H,

and hence that (id−G)−1 and H share many properties, in particular that H belongsto A(p+m)×(p+m) if and only if (id−G)−1 does [26, Chap. 5, Lem. 9].

1476 ULRICH OBERST

Corollary and Definition 2.9. If the feedback behavior of the preceding the-orem is well-posed, then B is trivial, i.e., B0 = 0, if and only if

B⊥1 ⊕ B⊥

2 = A1×l or, by duality, B1 ⊕ B2 = Fp+m.

If this holds, the modules Uj = B⊥j , j = 1, 2, are complementary direct summands of

A1×l and the modules Mj = A1×l/Uj and the behaviors Bj are projective. We saythat the IO behavior B2 trivializes (instead of stabilizes) B1.

In the following we assume that the IO behavior B1 is given and construct alltrivializing IO behaviors B2. The set of all these B2 is parametrized by an opensubset (in the Zariski topology) of a finitely generated polynomial module. Thisparametrization generalizes the important Youla–Kucera parametrization of the one-dimensional stabilization theory. According to the preceding theorem, we make thenecessary assumption that the behavior B1 is projective. We use the exact sequence

F1 := A1×k1d0:=◦R1−→ F0 := A1×l can−→ M1 := A1×l/U1 → 0

with l := p + m, R1 = (P1,−Q1), U1 := im(d0) = A1×k1R1.(24)

The following remark establishes the well-known one-to-one correspondence betweenidempotent endomorphisms e = e2 of a module and direct sum decompositions. If

e = e2 ∈ HomA(F0, F0)

is an idempotent or a projection, then

F0 = im(e) ⊕ ker(e) � x = e(x) + (x− e(x)),

x = e(x) ⇔ x ∈ im(e), ker(e) = im(id−e).

Conversely, any direct sum decomposition

F0 = V1 ⊕ V2 � x = x1 + x2 =: e(x) + (x− e(x))

gives rise to the projection e = e2 ∈ HomA(F0, F0). The map

{e ∈ HomA(F0, F0); e = e2} → {F0 = V1 ⊕ V2},e �→ F0 = im(e) ⊕ ker(e)

is bijective. By restricting the preceding bijection to decompositions F0 = U1 ⊕ U2

with the fixed U1 from above we obtain the following.Corollary 2.10. For the data from (24) the following map is bijective:

{e ∈ HomA(F0, F0); e = e2, im(e) = U1} → {F0 = U1 ⊕ U2},e �→ F0 = U1 ⊕ ker(e), ker(e) = im(idl −e).

The preceding decompositions can also be described by means of homomorphismsg ∈ HomA(F0, F1) ∼= Al×k1 , as was already shown more generally in [9, Lem. 6.2,p. 88]. This is important for constructive purposes in particular.

Lemma 2.11. The following conditions are equivalent for a submodule U1 of F0

and its factor module M1 = F0/U1 = cok(d0):1. The module M1 is projective.2. U1 is a direct summand of F0. Then U1 is also projective and there is an

idempotent e = e2 ∈ HomA(F0, F0) such that U1 = im(e).


3. There is a homomorphism g ∈ HomA(F0, F1) such that d0gd0 = d0. Thene := d0g is an idempotent with U1 = im(d0) = im(e).

Proof. The equivalence of 1 and 2 is standard since the projectivity of M1 isequivalent with the splitting of the exact sequence (24).

2 ⇒ 3: Let e be an idempotent with image im(e) = U1. Then the map e : F0 →U1 = im(e) is surjective. Since U1 is projective and the map d0 : F1 → U1 is surjective,there is a linear section

s : U1 → F1 such that idU1 = d0s. Define g := se : F0e−→ U1

s→ F1. Thene(x) = d0s(e(x)) ∈ U1 for x ∈ F0; hence e = d0se = d0g and

d0gd0 = d0sed0 = ed0 = d0 since e(d0(x)) = d0(x) for d0(x) ∈ U1 = im(e).

3 ⇒ 2: Assume d0gd0 = d0 and define e := d0g. Then e ∈ HomA(F0, F0) isan idempotent since e2 = (d0gd0)g = d0g = e and, moreover, im(e) = im(d0g) ⊆im(d0) = U1. From d0 = d0gd0 = ed0 we infer U1 = im(d0) ⊆ im(e) in the samefashion; hence U1 = im(e).

Theorem 2.12. Assume that M1 = A1×l/U1 is projective and that g1 ∈ HomA(F0, F1)and e1 := d0g1 with im(e1) = U1 are constructed according to the preceding lemma.Then the map

ϕ : {h ∈ HomA(F0, F1); d0hd0 = 0}/{h ∈ HomA(F0, F1); d0h = 0}→ {e ∈ HomA(F0, F0); e = e2, U1 = im(e)}, h �→ ϕ(h) := e1 + d0h,

is bijective. It furnishes a parametrization of the set of direct complements U2 = ker(e)with F0 = U1 ⊕U2 by the finitely generated A-module on the left. If A is a field, thenthe right side is an affine open subset of the projective Grassmann variety of all m-dimensional subspaces of A1×(p+m).

Proof. 1. The equation d0gd0 = d0 is an inhomogeneous linear equation for g inthe A-module HomA(F0, F1) and g1 is one solution. Hence

{g ∈ HomA(F0, F1); d0gd0 = d0} = g1 + {h ∈ HomA(F0, F1); d0hd0 = 0}.

2. The map is well defined: If

d0hd0 = 0, then g2 := g1 + h satisfies d0g2d0 = d0, and hencee2 := d0g2 = d0g1 + d0h = e1 + d0h

is an idempotent with im(e2) = U1 according to the preceding lemma. If h2 and h3

are homogeneous solutions such that

h2 = h3 or d0(h2 − h3) = 0, then e1 + d0h2 = e1 + d0h3.

3. It is obvious that the map ϕ is injective, and it is surjective by the precedinglemma.

We reformulate the preceding theorem in matrix terms. We identify

Am×n = HomA(A1×m, A1×n), X = ◦X = (ξ �→ ξX),

and emphasize that (◦X)(◦Y ) = ◦(Y X).Theorem 2.13. Let B1 ⊆ Fp+m be an IO behavior with the transfer matrix H1

and the data

U1 = B⊥1 = A1×k1R1, M1 = A1×l/B⊥

1 ,

R1 = (P1,−Q1) ∈ Ak1×(p+m), rank(P1) = p, P1H1 = Q1, H1 ∈ Kp×m.

1478 ULRICH OBERST

1. The module M1 or the behavior B1 is projective if and only if there is a matrix

G1 ∈ Al×k1 with R1G1R1 = R1 or, equivalently, with (idp,−H1)G1P1 = idp.(25)

Then E1 := G1R1 = E21 ∈ Al×l is an idempotent matrix with B⊥

1 = A1×lE1.This algorithm is related to the algorithms of [33]; its idea goes back at least to

MacLane [9] as explained above.2. Assume that B1 is projective with the data from 1. Then the map

ϕ : {X ∈ Al×k1 ; R1XR1 = 0}/{X ∈ Al×k1 ; XR1 = 0}→ {E ∈ Al×l; E = E2, B⊥

1 = A1×lE}, X �→ ϕ(X) := E1 + XR1,(26)

is bijective. Moreover,

R1XR1 = 0 if and only if (idp,−H1)XP1 = 0 if and only if (t idp,−tH1)XP1 = 0 and

XR1 = 0 if and only if XP1 = 0,

(27)

where 0 = t ∈ A is a common denominator of the entries of H1, i.e., tH1 ∈ Ap×m.The direct complement behavior B2 of B1 in Fp+m, which is constructed by means ofthe idempotent E = E1 + XR1, is

B2 := {w ∈ Fp+m; (idl −E1 −XR1) ◦ w = 0} = U⊥2 , B1 ⊕ B2 = Fp+m,

where U2 := ker(◦(E1 + XR1)) = A1×l(idl −E1 −XR1).(28)

3. Since B⊥1 = A1×lE1, the bijection (26) holds if R1 is replaced with E1.

4. If U1 is even free, and if the rows of R1 are a basis of U1, i.e., if R1 =(P1,−Q1) ∈ Ap×(p+m) and det(P1) = 0, then the situation of (26) simplifies to

G1 ∈ Al×p, R1G1 = idp, E1 = G1R1, and

ϕ : {X ∈ Al×k; R1X = 0} ∼= {E ∈ Al×l; E = E2, B⊥1 = A1×lE},

X �→ E1 + XR1.

(29)

Proof. The equivalence of (25) and (27) follows from

R1 = (P1,−Q1) = P1(idp,−H1) and rank(P1) = rank(idp,−H1) = p,

which imply that P1 (resp., (idp,−H1)) can be cancelled as a left (resp., right) fac-tor.

Of course, each direct complement behavior

B2 =

{w2 =

(u2

y2

)∈ Fp+m; (idl −E) ◦ w2 = 0

}with

rank(idl −E) = l − rank(E) = l − p = m

of the preceding theorem admits an IO structure, but not necessarily that with u2 asinputs, as needed for the feedback construction. To enforce this additional propertywe proceed as follows. Let X1, . . . , Xn be a system of A-generators of the submodule

{X ∈ Al×k1 ; R1XR1 = 0( ⇔ R1XP1 = 0)}.(30)


The same matrices generate {X ∈ Kl×k1 ; (idp,−H1)XP1 = 0} as a K-vector space.In the relevant examples the Xi can be computed by means of Grobner bases. Thelinear map θ = (θi)i=1,...,n �→

∑ni=1 θiXi induces the A-isomorphism

ϕ1 : A1×n/V ∼= {X ∈ Al×k1 ; R1XP1 = 0}/{X ∈ Al×k1 ; XP1 = 0},

where ϕ1(θ) :=

n∑i=1

θiXi and V :=

{θ ∈ A1×n;

n∑i=1

θiXiP1 = 0

}.

(31)

Again, in the relevant examples the submodule V and its factor module A1×n/V canbe computed by means of Grobner bases. In addition we consider the polynomial map

d : Kl×m → K( lm), Y �→ d(Y ) = (dα(Y ))α,(32)

where the dα(Y ) are the(lm

)m × m minors of the matrix Y . The function d is

polynomial in the entries of Y and d(Y ) = 0 if and only if rank(Y ) = m. With thedata from above, we obtain the induced map

{E ∈ Al×l; E = E2, B⊥1 = A1×lE} → A( l

m), E �→ d

((idl −E)

(0

idm

)).

If

idl −E =: (−Q2, P2) ∈ Al×(p+m), hence (idl −E)

(0

idm

)= P2,

the behavior

B2 :=

{w2 =

(u2

y2

)∈ Fp+m; (idl −E) ◦ w2 = 0 or P2 ◦ y2 = Q2 ◦ u2

}is an IO behavior with input u2 if and only if rank(P2) = m. Composing the mapsjust constructed, we obtain the map

Φ : K1×n → K( lm), θ �→ d

((idl −E)

(0

idm

)),

with E := E1 + XR1, X :=

n∑i=1

θiXi,

(33)

which is polynomial in the components of θ; i.e., its components are contained inK[θ]. Summing up we obtain the final parametrization theorem.

Theorem 2.14 (parametrization of trivializing behaviors). Let B1 = {w1 =(y1

u1

)∈ Fp+m} be a projective behavior as in Theorem 2.13, and assume that A is

infinite. Then, with the data introduced above, the map

θ �→ B2 :=

{w2 =

(u2

y2

)∈ Fp+m; (idl −E) ◦ w2 = 0

},

where E := E1 + XR1, X :=

n∑i=1

θiXi,

is a bijection from the nonempty set{θ ∈ A1×n/V ; Φ(θ) = d

((idl −E)

(0

idm

))= 0

}

1480 ULRICH OBERST

onto the set of all IO behaviors B2 ⊆ Fp+m, which trivialize B1. This signifies thatthe feedback system B := feedback(B1,B2) is well-posed and trivial. In particular,such behaviors exist.

Proof. It remains only to show that such behaviors exist. The matrix P1 of rankp has a left inverse

S1 ∈ Kp×k1 with S1P1 = idp.(34)

Define

E :=

(idp

0

)(idp,−H1) =

(idp −H1

0 0

)∈ Kl×l and G :=

(idp

0

)S1 ∈ Kl×k1 ;

hence idl −E =

(0 H1

0 idm

), (idl −E)

(0

idm

)=

(H1

idm

),

rank

((idl −E)

(0

idm

))= m, (idp,−H1)GP1 = idp, and E := GR1.

Thus X := G −G1 ∈ Kl×k1 satisfies

(idp,−H1)XP1 = 0, E = E1 + XR1 and rank

((idl −E)

(0

idm

))= m;

in particular, X is of the form

X =n∑

i=1

θiXi with θ ∈ K1×n and Φ(θ) = 0.

Since the components of Φ are polynomials in K[θ] and since A is an infinite subsetof K, there is also a parameter θ′ ∈ A1×n with nonzero Φ(θ′), which induces anIO behavior B2 such that the feedback system feedback(B1,B2) is well-posed andtrivial.

Remark 2.15. We remark that {θ ∈ K1×n; Φ(θ) = 0} is a nonempty open subsetof K1×n with respect to the Zariski topology and is therefore dense. This signifiesthat, generically, the behaviors B2 with Fp+m = B1 ⊕ B2 are IO behaviors with thedesired input u2 ∈ Fp.

3. Localization. The assumptions of the preceding section remain in force.Moreover, we assume that the ring A is factorial. We use Matlis’ structure the-ory of injective modules over Noetherian rings and refer to [10, pp. 145–150] and [12,sec. 2], where this theory was used in the system-theoretic context. Moreover, we useGabriel’s theory of localization as detailed in the book [24, Chap. IX].

Let Spec(A) (resp., Max(A)) denote the set of prime (resp., maximal) ideals of A.A prime ideal p is associated with an A-module M if and only if there is an x ∈ M suchthat p = ann(x) = {a ∈ A; ax = 0} or, in other words, that A/p is a submodule of Mup to isomorphism. Let Ass(M) ⊂ Spec(A) denote the set of prime ideals associatedwith M . A module M is p-coprimary if Ass(M) consists exactly of one prime idealp; then

a : M → M, x �→ ax is

{injective

locally nilpotent⇔

{a /∈ p

a ∈ p.(35)


Local or almost nilpotency of a on M means that for all x ∈ M there is an index msuch that amx = 0.

The injective module F is a cogenerator if and only if it contains all simplemodules A/m, m ∈ Max(A), up to isomorphism, i.e., if Max(A) ⊂ Ass(F). Eachmodule M has an injective envelope E(M) ⊃ M which is unique up to noncanonicalisomorphism. This signifies that E(M) is injective and that V ∩ M = 0 for eachnonzero submodule V of E(M). Since A is Noetherian, a direct sum or coproductof modules is injective if and only if each direct summand has this property. Eachinjective module E admits a direct sum decomposition into directly indecomposableinjective modules, and this decomposition is unique up to an automorphism of E.An indecomposable injective module E is coprimary, and if p is its unique associatedprime ideal, then E = E(A/p). The map

Spec(A) → {E; E indecomposable injective}/isomorphism,

p �→ E(A/p), Ass(E(A/p)) = {p}(36)

is a bijection of the prime spectrum Spec(A) onto the set of indecomposable injectivesup to isomorphism. Since E(A/p) is injective and indecomposable, the injective map

s : E(A/p) → E(A/p), x �→ sx, s ∈ A \ p,

is even bijective, and therefore E(A/p) is an Ap-module, where

Ap :={a

s; a ∈ A, s ∈ A \ p

}⊂ K = quot(A)

is the local ring of the prime ideal p with its unique maximal ideal pp = App. ForM ∈ ModA, the adjointness isomorphism

HomA(M,E(A/p)) ∼= HomAp(Mp, E(A/p)) = HomA(Mp, E(A/p)),

Mp := Ap ⊗A M ={x

s; x ∈ M, s ∈ A \ p

}∈ ModAp

(37)

holds. Together with N = Np for N ∈ ModApthis shows that the duality functor

HomAp(−, E(A/p)) on ModAp

is exact, and hence E(A/p) is an injective Ap-module.Since it contains the unique simple Ap-module (A/p)p = Ap/pp, the module E(A/p)is the unique minimal injective cogenerator of the category ModAp

to which the con-siderations of section 2 are applicable. In particular,

Mp = 0 ⇔ HomA(M,E(A/p)) = 0.(38)

Let

F := ⊕i∈I Fi, Ass(Fi) = {pi}, P := Ass(F) = {pi; i ∈ I}(39)

denote a direct sum decomposition of the injective cogenerator F into indecomposableinjectives Fi and let

P := Ass(F) = P1 � P2, P1,P2 = ∅,(40)

be a disjoint decomposition of Ass(F). All subsequent objects depend on the choice ofF and of the decompositions (39) and (40).

1482 ULRICH OBERST

In particular, the latter imply direct sum decompositions

F = ⊕p∈P F(p) = F1 ⊕ F2 with F(p) := ⊕i∈I, pi=p Fi∼= E(A/p)(μ(p)),

μ(p) := |{i ∈ I; pi = p}|, Fj := ⊕p∈PjF(p), j = 1, 2,

(41)

where E(A/p)(μ(p)) denotes the direct sum of μ(p) copies of E(A/p). All mod-ules in these direct sum decompositions are injective. Like E(A/p), the moduleF(p) ∼= E(A/p)(μ(p)) is an injective cogenerator of the category of Ap-modules. Thedecompositions (41) induce corresponding decompositions of F behaviors. Let

R ∈ Ak×l, U := A1×kR, M := A1×l/U,

B := U⊥ =ident.

HomA(M,F) = {w ∈ F l; R ◦ w = 0}.(42)

Since M is finitely generated, the functor HomA(M,−) preserves direct sums, andtherefore (41) induces direct decompositions

B = ⊕i∈I Bi ⊂ F l = ⊕i∈I F li ,

Bi := B ∩ F li =

ident.HomA(M,Fi) =

ident.{w ∈ F l

i ; R ◦ w = 0},

B = ⊕p∈P B(p) = B1 ⊕ B2,

B(p) := B ∩ F(p)l = HomA(M,F(p)) = {w ∈ F(p)l; R ◦ w = 0}∼= HomAp

(Mp, E(A/p))(μ(p)),

Bj := B ∩ F lj = HomA(M,Fj) = {w ∈ F l

j ; R ◦ w = 0}, j = 1, 2.

(43)

We use the following suggestive system-theoretic terminology. The elements of F(resp., F1) are called signals (resp., stable signals). If y = y1 + y2 ∈ F = F1 ⊕ F2 isa signal, then y1 is called its stable part and y2 its steady state.

Example 3.1. We indicate the data for the important complex continuous case.The algebra A is the complex polynomial algebra C[s] = C[s1, . . . , sr]. The map

Cr → Max(C[s]),

λ = (λ1, . . . , λr) �→ mλ :=

r∑ρ=1

C[s](sρ − λρ) = {t ∈ C[s]; t(λ) = 0}

is bijective. As an injective cogenerator we take the module

F := D′(Rr)lf := {y ∈ D′(Rr); [C[s] ◦ y : C] < ∞} = ⊕λ∈Cr C[z] exp(λ • z),

z = (z1, . . . , zr) ∈ Rr, λ • z := λ1z1 + · · · + λrzr

(44)

of locally finite distributions or polynomial-exponential functions [12, Thm. 6.6]. Then

Ass(F) = Max(C[s]) = {mλ; λ ∈ Cr},

F(λ) := F(mλ) = C[z] exp(λ • z) ∼= E(C[s]/mλ);(45)

i.e., F is the unique minimal injective cogenerator of the category of C[s]-modules.We choose a disjoint decomposition

Cr = Λ1 � Λ2 and Ass(F) = Max(C[s]) = P1 � P2, Pj := {mλ; λ ∈ Λj}.(46)


Standard choices for continuous stability theory are

Λ2 := C+r, where C+ := {w ∈ C; �(w) > 0}, C+ := {w ∈ C; �(w) ≥ 0},

or [29, Def. 3.1] Λ2 := C+ × iRr−1.(47)

A discrete analogue is Λ2 := {w ∈ C; |w| ≥ 1}r; see the end of section 5.The modules Fj , j = 1, 2, are Fj = ⊕λ∈Λj

C[z] exp(λ • z). For r = 1 and timez = z1 the two sets Λ2 from (47) coincide and yield

F = ⊕λ∈C C[z] exp(λz) = F1 ⊕ F2 = ⊕(λ)<0 C[z] exp(λz) ⊕ ⊕(λ)≥0 C[z] exp(λz).

Then F1 = {y ∈ F ; limz→∞ y(z) = 0}. Hence

limz→∞

y1(z) = 0fory = y1 + y2 ∈ F = F1 ⊕ F2.

For practical engineering applications the stable part y1 of y is negligible and thesignal y coincides with its steady state y2. This justifies the suggestive terminologyintroduced above.

We go on with the general situation. The set

T := ∩p2∈P2(A \ p2) ⊂ A(48)

is a multiplicative submonoid of A and saturated; i.e., each divisor of an element inT belongs to T . The set T gives rise to its quotient ring

AT :={a

t; a ∈ A, t ∈ T

}⊂ Ap ⊂ quot(A), p ∈ P2, and (AT )pT

= Ap.(49)

Lemma 3.2. AT = ∩p∈P2 Ap ⊂ quot(A).Proof. The inclusion ⊆ follows from (49). For the reverse inclusion we use that

A is factorial. Let x = as ∈ quot(A) with relatively prime a, s be contained in all

Ap, p ∈ P2, hencea

s=

ap

sp

, sp ∈ A \ p, p ∈ P2.

Then

aps = spa, gcd(a, s) = 1 ⇒ ∀p ∈ P2 : s | sp

⇒ ∀p ∈ P2 : s ∈ A \ p ⇒ s ∈ ∩p∈P2(A \ p) = T and x =

a

s∈ AT .

In the context of Gabriel’s localization theory the set P2 ⊂ Ass(F) and its associ-ated injective module F2 give rise to a full localizing or Serre subcategory or hereditarytorsion class C of ModA and a Gabriel topology T [24, Thm. VI.5.1], where

C := {C ∈ ModA; HomA(C,F2) = 0}= {C ∈ ModA; ∀p ∈ P2 : HomA(C,E(A/p)) = 0}

= {C ∈ ModA; ∀p ∈ P2 : Cp = 0} and

T := {a ⊆ A; A/a ∈ C}.

(50)

Since F2 is injective, the class C of T-torsion modules is obviously closed under takingsubmodules, factor modules, extensions, and direct sums; in particular

C = {C ∈ ModA; ∀x ∈ C : ann(x) = {a ∈ A; ax = 0} ∈ T}(51)

1484 ULRICH OBERST

and T is directed downward. Moreover, C is stable, i.e., closed under taking injectiveenvelopes [24, Props. VI.7.1 and VII.4.5]. If M is a finitely generated A-module anda an ideal of A, then

M ∈ C, resp., a ∈ T

⇔ ∀p ∈ P2 ∃sp ∈ A \ p such that spM = 0, resp., sp ∈ a ∩ (A \ p).(52)

The largest submodule of M in C is called the T-torsion radical of M and denotedby tT(M). If tT(M) = 0, the module is called T-torsion free. Due to (52) a T-torsionmodule is a torsion module in the usual sense, and hence a torsion free module is T-torsion free. Every subcategory C with the indicated properties arises from a suitableinjective module F2; in other words, there are no other localization functors thanthose described below.

An A-module M also induces its quotient module

MT ={x

t; x ∈ M, t ∈ T

}= AT ⊗A M.

Since Mp = (MT )pT∀p ∈ P2, we see that any module with MT = 0 is a T-torsion

module.An A-module N is called T-closed if

N ∼= HomA(a, N), x �→ (a �→ ax) ∀a ∈ T.(53)

The full additive subcategory of ModA of all T-closed submodules is denoted byModA,T. According to [24, pp. 195–200; 213–216] the inclusion functor inj : ModA,T ⊂ModA has an exact left adjoint functor

ModA → ModA,T, M �→ MT, with the functorial adjunction morphism

ηM : M → MT, i.e., HomA(MT, N) ∼= HomA(M,N), g �→ gηM ,(54)

for M ∈ ModA and N ∈ ModA,T, and is given by the directed colimit [24, Prop. IX.1.7]

MT = colima∈T

HomA(a,M) = {α : a → M ; a ∈ T}/ ∼� [α : a → M ], where

(α : a → M) ∼ (β : b → M) ⇔ ∃c ∈ T with c ⊆ a ∩ b, α | c = β | c, and

ηM : M → MT, x �→ [A → M, a �→ ax].

(55)

Moreover [24, IX.1.2,1.3],

tT(M) = ker(ηM ) and M ∈ C ⇔ M = tT(M) ⇔ MT = 0.(56)

The functor M �→ MT is called localization with respect to P2, C, or T. A moduleN is T-closed if and only if ηN is an isomorphism [24, Prop. IX.1.8]; in particularMT = (MT)T. The category ModA,T is itself abelian with exact directed colimits;the localization functor is exact, but the inclusion functor inj is left exact only [24,p. 214]. This means that the kernels of an A-linear map between T-closed modulesin ModA (resp., ModA,T) are the same, but that an epimorphism in ModA,T is notsurjective in general. Indeed, if f : N1 → N2 is an A-linear map between T-closedmodules and C := cokA(f) := N1/f(N2) is its cokernel in ModA, the exact sequence

N1f−→ N2

can−→ C → 0 in ModA induces the exact sequence

N1 = (N1)T

f−→ N2 = (N2)T

can−→ CT → 0 in ModA,T,(57)


and hence f is an epimorphism in the category ModA,T if and only if C is a T-torsionmodule.

Lemma 3.3.

1. The modules F2 = ⊕p∈P2 F(p) and ⊕p∈P2 E(A/p) are injective cogeneratorsof the category ModA,T; in particular, as already indicated,

HomA(M,F2) ∼= HomA(MT,F2) = 0 ⇔ MT = 0 ⇔ M ∈ C.(58)

2. These modules are also injective AT -modules. Only in important special casesare they AT -cogenerators; see Theorem/Definition 5.6 of section 5.

3. The module FT∼= ⊕i{Fi; pi ∩ T = ∅} is A- and AT -injective.

Proof.1. This is shown in [24, Prop. X.1.9].2. Since these AT -modules are injective A-modules, the injectivity as AT -

modules follows from the identity

HomA(N,F2) = HomAT(N,F2), N = NT ∈ ModAT

.

3. This follows from

E(A/p)T =

{E(A/p) if p ∩ T = ∅0 if p ∩ T = ∅

.

Since A is Noetherian and hence the functor HomA(a,−) in (53) preserves di-rect sums, a direct sum of A-modules is T-closed if and only if all summands havethis property. This implies in particular that the injection functor from ModA,T toModA also preserves direct sums and that every projective AT -module is containedin ModA,T since AT = AT is T-closed (see Lemma 3.4).

The module MT is always an AT -module, i.e., the multiplication with t ∈ T isbijective [24, p. 196], but the functors M �→ MT and M �→ MT coincide only inexceptional cases [24, Prop. XI.3.4], for instance, for the principal ideal ring C[s1]in one-dimensional systems theory. Again in Theorem/Definition 5.6 we characterizethis exceptional situation in the most important case and show its relation to ideal-convexity in the sense of [29]. The difficulties of [29] with the possible absence ofideal-convexity are avoided in our theory by using MT instead of the standard MT .But see the following lemma.

Lemma 3.4. AT = AT ⊂ K := quot(A).Proof. For a ∈ T, the map

HomA(a, A) → K, α �→ xα := α(a)/a, with 0 = a ∈ a,(59)

is a well-defined A-monomorphism. Indeed, if

0 = a1, a2 ∈ a, then α(a1a2) = a1α(a2) = a2α(a1); henceα(a2)/a2 = α(a1)/a1 and α(a2) = a2α(a1)/a1.

Since directed colimits are exact, the injections (59) induce an injection

AT = colima∈T

HomA(a, A) =ident.

{xα = α(a)/a; 0 = a ∈ aα→ A} ⊂ K.

But according to (52) and Lemma 3.2,

∀p ∈ P2 ∃ap ∈ a ∩ (A \ p) = ∅; hence xα = α(ap)/ap ∈ ∩p∈P2Ap = AT ,

1486 ULRICH OBERST

and thus AT ⊆ AT . If, conversely, x = a/t ∈ AT , then

∀p ∈ P2 : t ∈ At ∩ (A \ p); hence

At ∈ T, α : At → A, t �→ a, and x = a/t = α(t)/t = xα ∈ AT.

As in the preceding lemma, we define for every A-module M and prime idealp ∈ P2 the canonical A-linear map

canp : MT → Mp, [α : a → M ] �→ α(sp)/sp, with sp ∈ a ∩ (A \ p),(60)

independent of the choice of sp.Lemma 3.5. The map

can : MT →∏

p∈P2

Mp, [α] �→ (α(sp)/sp)p∈P2 ,

is a functorial monomorphism.Proof. Only the injectivity has to be shown. Assume therefore that

∀p ∈ P2 : α(sp)/sp = 0, i.e., ∃tp ∈ A \ p such that

tpα(sp) = α(tpsp) = 0.

The ideal a :=∑

p∈P2Atpsp belongs to T since it contains the tpsp and, by construc-

tion, α | a = 0, and hence [α] = 0 in MT.A further functorial homorphism is the map

can : MT → MT,x

t�→ [α : At → M, t �→ x],(61)

which, in general, is neither injective nor surjective. The kernel and cokernel of thecanonical map M → MT are modules C with CT = 0 and hence CT = 0, and arethus T-torsion modules. The same holds for the canonical map ηM : M → MT [24,Lemmas IX.1.2, IX.1.5]. Application of the exact functors (−)T and (−)p thus yieldscanonical isomorphisms

MT∼= (MT )T and Mp

∼= (MT )p∼= (MT)p for p ∈ P2,

K ⊗A M ∼= K ⊗ApMp

∼= K ⊗A MT∼= K ⊗A MT;

hence rank(M) = rank(MT ) = rank(MT) = rank(Mp) = [K ⊗A M : K].

(62)

For the data from (42), the inclusion U ⊂ A1×l induces the inclusions

UT = ATU ⊂ UT ⊂ A1×lT = A1×l

T.(63)

Lemma and Definition 3.6. For the data from (42), the submodule Ust :=UT ∩A1×l of A1×l is the largest one such that

(Ust)T = UT or (A1×l/Ust)T = MT.

Moreover,

Ust = ∩p∈P2 Up ∩A1×l = {x ∈ A1×l; ∀p ∈ P2 ∃sp ∈ A \ p with spx ∈ U},UT = (Ust)T = ATUst and K ⊗A U = K ⊗A Ust ⊂ K1×l.


The last equality shows that the behavior Bst := U⊥st is in the same transfer class, or

has the same controllable subbehavior, as B.Proof. The exactness of (−)T implies

(Ust)T = (UT ∩A1×l)T = (UT)T ∩A1×lT

= UT ∩A1×lT = UT.

If UT = VT, then V ⊂ VT ∩A1×l = UT ∩A1×l = Ust.

Finally,

UT = AT (UT ∩A1×l) = ATUst = (Ust)T .

The last equations follow from the fact that UT is an AT -submodule of A1×lT and that

V = AT (V ∩A1×l) for each AT -submodule of A1×lT .

The injection from Lemma 3.5 and the exact sequence

0 → Uinj−→ A1×l can−→ M → 0

induce the exact sequence in ModA,

0 → UT

inj−→ A1×lT

can−→∏

p∈P2

A1×lp /Up,

and this implies

Ust = UT ∩A1×l = ∩p∈P2 Up ∩A1×l.

Remark 3.7. If the matrix R with U = A1×kR is given it is, in general, a difficultproblem to compute a matrix

Rst ∈ Ak1×l such that Ust = A1×k1Rst and hence UT = A1×k1

T Rst.

4. Abstract stability and stabilization. The assumptions and notation ofthe preceding section remain in force. We apply the results of the preceding sectionsto stability questions, first in the abstract situation and then, in the next section, tomultidimensional systems governed by partial differential or difference equations.

Theorem and Definition 4.1 (equal steady state). 1. Let

B := U⊥ = {w ∈ F l; R ◦ w = 0} and B′ := U ′⊥ = {w ∈ F l; R′ ◦ w = 0}

be two behaviors as in (42) with their corresponding decompositions (43) and thesubmodules Ust (resp., U ′

st) according to Lemma/Definition 3.6. Then

B2 := B ∩ F l2 = B′

2 ⇔ UT = U ′T ⇔ Ust = U ′

st ⇔ MT = M ′T.

Then we say that the behaviors B and B′ have the same steady state.2. The behavior

Bst := U⊥st = {w ∈ F l; Rst ◦ w = 0} with Ust = A1×k1Rst

is the smallest one with the same steady state as B.Proof. 1. Replacing B with B + B′, we assume without loss of generality that

B′ ⊆ B or U ⊆ U ′. Let f : U ⊂ U ′ (resp., g : M → M ′) be the canonical injection

1488 ULRICH OBERST

(resp., surjection). Application of the exact functor (−)T furnishes the commutativediagram with exact rows in ModA,T

0 → UT ⊂ A1×lT

canT−→ MT → 0↓ fT ↓ id ↓ gT

0 → U ′T ⊂ A1×l

T

can′T−→ M ′

T → 0,

where fT : UT ⊂ U ′T and gT is the canonical epimorphism. Hence UT = U ′

T if andonly if gT : MT → M ′

T is an isomorphism. Moreover,

Hom(g,F2) =ident.

Hom(gT,F2) =ident.

inj :

B′2 =

ident.HomA(M ′,F2) =

ident.HomA(M ′

T,F2)

→ B2 =ident.

HomA(M,F2) =ident.

HomA(MT,F2).

By Lemma 3.3 F2 is an injective cogenerator in ModA,T and hence

B′2 = B2 ⇔ inj = Hom(gT,F2) is an isomorphism

⇔ gT : MT → M ′T is an isomorphism ⇔ MT = M ′

T.

2. This is a reformulation of Lemma/Definition 3.6.

In what follows we assume that an IO system B ⊂ F l is given with the data

l = p + m, R = (P,−Q) ∈ Ak×(p+m), rank(P ) = rank(R) = p, PH = Q,

U := A1×kR ⊂ A1×l, U0 := A1×kP ⊂ A1×p,

M := A1×l/U, M0 = A1×p/U0,

B := U⊥ =

{w =

(y

u

)∈ Fp+m; P ◦ y = Q ◦ u

},

B0 := {y ∈ Fp; P ◦ y = 0}.

(64)

Theorem and Definition 4.2 (stable systems). For the data from (64) thefollowing assertions are equivalent:

1. B0 ⊆ Fp1 , i.e., B0

2 = 0 or B0(p) = 0 ∀p ∈ P2. In other words, all trajectoriesof the autonomous part B0 of B are stable. Recall that the signals y ∈ F1 aresuggestively called stable.

2. M0T = 0, i.e., for all p ∈ P2 M0

p = 0 or A1×kp P = A1×p

p or, in the terminologyof Theorem/Definition 2.5, the Ap IO behavior{(

y

u

)∈ F(p)p+m; P ◦ y = Q ◦ u

}with the Ap-cogenerator F(p)

is trivial.3. For all p ∈ P2 the module Mp is torsion free, and H ∈ Ap×m

p .

4. The module MT is torsion free, and H ∈ Ap×mT .

If these conditions are satisfied, the IO system B is said to be stable with respect to


the decompositions (39) and (40), and then the A-linear map

A1×m → M, η �→ (0, η), induces the isomorphisms

A1×mT

∼= MT and

proj = (0 idm)◦ : B2 → Fm2 ,

(y2

u2

)�→ u2, with the inverse u2 �→

(H ◦ u2

u2

);

hence B := B1 ⊕{(

H ◦ u2

u2

); u2 ∈ Fm

2

}.

(65)

Here ◦ denotes the scalar multiplication of F2 as an AT -module. In other words,the module MT is AT -free of dimension m. For every input u2 ∈ Fm

2 ,(H◦u2

u2

)is the

unique trajectory in B ∩ F l2 with input u2. We remark that H acts on Fm

2 as anoperator although no analogue of the customary properness of H is assumed.

Proof. 1 ⇔ 2: This follows from

B0(p) ∼= HomA(M0,F(p)) ∼= HomAp(M0

p ,F(p))

and the cogenerator property of F(p) as an Ap-module.2 ⇔ 3: This is a special case of Theorem/Definition 2.5.3 ⇔ 4: The property of H follows from AT = ∩p∈P2 Ap according to Lemma 3.2.

If all Mp, p ∈ P2, are torsion free, Lemma 3.5 implies the same property for MT.Conversely, if MT is torsion free, then so are all Mp

∼= (MT)p, p ∈ P2, accordingto (62).

The condition M0T = 0 and the application of the exact functors

(−)T and HomA(−,F2) ∼= HomA((−)T,F2)

to the exact IO sequence

0 → A1×m ◦(0 idm)−→ Mcan−→ M0 → 0

furnishes the isomorphisms (65).Recall that F2 is T-closed and an AT -module in particular; hence a ◦ y is defined

for a ∈ AT and y ∈ F2. The equation PH = Q of matrices with coefficients in AT

hence implies P ◦ (H ◦ u2) = Q ◦ u2 for u2 ∈ Fm2 and therefore the asserted inverse

map of the projection proj = (0 idm)◦ : B2∼= Fm

2 .Theorem and Definition 4.3 (stabilizable systems). An IO system as in (64)

is called stabilizable with respect to the decompositions (39) and (40) if there is

another IO system B′ ={(

u′

y′

)∈ Fp+m; P ′ ◦ y′ = Q′ ◦ u′} such that the feedback

system B′′ := feedback(B,B′) according to (20) and 21 is well-posed and stable in thesense of Theorems 2.7 and 4.2. If this is the case, then

A1×lT = UT ⊕ U ′

T and Fp+m2 = B2 ⊕ B′

2,(66)

and B′ is called a stabilizing compensator of B.Proof. According to Theorem 2.7 the direct sum decomposition

(B′′0)⊥ = B⊥ ⊕ B′⊥ holds and implies (B′′0)⊥T = B⊥T ⊕ B′⊥

T .

The stability of B′′ signifies that B′′02 = 0 or (B′′0)⊥T = A1×l

T , and hence we have theassertion.

1490 ULRICH OBERST

Direct decompositions as in (66) appear in almost all stabilization theories, oftendisguised as Bezout identities for matrices. The next theorem contains a test forstabilizability of the IO system B and the construction of a stabilizing compensator.It is just a reformulation of Theorem 2.13.

Let

Ust = A1×k1Rst, Rst = (Pst,−Qst) ∈ Ak1×(p+m)(67)

be the largest submodule of

A1×l with UT = (Ust)T = A1×k1

T Rst; hence

A1×kp R = Up = (Ust)p = A1×k1

p Rst ∀p ∈ P2

(68)

according to (62) and Lemma/Definition 3.6. The use of Rst is necessary since epi-morphisms in ModA,T are not surjective, and hence UT is not generated over AT bythe rows of R in general. Since U ⊂ Ust, there is a matrix X such that

R = XRst; hence P = XPst and p = rank(P ) ≤ rank(Pst) ≤ rank(Rst).Moreover, p = rank(R) = rank(U) = rank(Ust) = rank(Rst), and therefore

p = rank(Pst) = rank(Rst).

Thus the behavior

Bst := U⊥st =

{(y

u

); Pst ◦ y = Qst ◦ u

}⊂ B(69)

is an IO behavior with the same IO structure and steady state as B. Due to K⊗AU =K⊗AUst from Lemma/Definition 3.6, B and Bst have also the same transfer matrix H.

Theorem 4.4 (characterization of stabilizable behaviors). The following asser-tions are equivalent for the IO behavior (64):

1. The behavior B is stabilizable.2. UT = A1×k1

T Rst is a direct summand of A1×lT .

3. There is a matrix

G1 ∈ Al×k1

T such that Rst = RstG1Rst, and then

E1 := G1Rst = E21 ∈ Al×l

T with UT = A1×lT E1 = UT.

4. With G1 and E1 from item 3 there is a matrix

X ∈ Al×k1

T with RstXRst = 0 such that E := E1 + XRst = E2 ∈ Al×l,

UT = A1×lT E and rank

((idl −E)

(0

idm

))= m.

If these conditions are satisfied, choose E ∈ Al×lT according to item 4. Let t ∈ T be 1

or the greatest common divisor of the denominators of E. Define

R′ := (−Q′P ′) := t(idl −E) ∈ Al×(p+m); hence

rank(R′) = l − rank(E) = l − p = m = rank(P ′) and

B′ :=

{(u′

y′

)∈ Fp+m; P ′ ◦ y′ = Q′ ◦ u′

}.


Then B′ is a stabilizing compensator of B, i.e., B′ is an IO behavior with input u′,and the feedback system B′′ = feedback(B,B′) is well-posed and stable.

Proof. 1 ⇒ 2: This follows from Theorem 4.3.2 ⇔ 3: This is an application of Theorem 2.13(1) to the AT -module A1×l

T /UT.3 ⇒ 4: This implication follows from Theorem 2.14.4 ⇒ 1: Construct R′ and B′ as indicated. The equalities rank(R′) = rank(P ′) =

m show that B′ is an IO system with the desired IO structure. By construction,

UT = A1×k1

T Rst = A1×lT E, p = rank(U) = rank(E),

U ′ := B′⊥ = A1×lt(idl −E) ⇒ U ′T = A1×l

T t(idl −E) = A1×lT (idl −E),

U ∩ U ′ ⊆ UT ∩ U ′T = 0 ⇒ A1×(k+l)

(P −Q

−Q′ P ′

)= U ⊕ U ′.

The last equation shows that B′′ is well-posed with

(B′′0)⊥ = U ⊕ U ′ and (B′′0)⊥T = UT ⊕ U ′T = A1×l

T E ⊕A1×lT (idl −E) = A1×l

T .

The last equation signifies the stability of B′′.Remark 4.5. Condition 2 of the preceding theorem is equivalent to the AT -

projectivity of A1×lT /UT, and then this module coincides with MT. This does not

imply that UT is a direct summand of A1×lT or that MT is AT -projective. If, however,

this is the case, then condition 2 follows and UT = UT, MT = MT. We remark that,in general, AT is not projective in ModA,T since epimorphisms are not surjective inthis category. Therefore condition 2 has been expressed without using the notion ofprojectivity.

Theorem 4.6 (parametrization of stabilizing controllers). Assume that B from(64) is stabilizable and that E1 is the matrix from item 3 of the preceding theorem.The steady state parts B′

2 ⊂ F l2 of the stabilizing compensators B′ of B are uniquely

determined by the idempotent matrices

E = E2 ∈ Al×lT with rank

((idl −E)

(0

idm

))= m,

B⊥T = A1×l

T E, B′⊥T = A1×l

T (idl −E),

and these are parametrized by the bijection of Theorem 2.14 applied to the directsummand B⊥

T of A1×lT or the projective AT -module A1×l

T /B⊥T .

5. Continuous and discrete stability and stabilization. We specialize theresults of the preceding section to complex partial differential equations on R

r � z;i.e., we assume A := C[s] = C[s1, . . . , sr], the data from (64), and Example 3.1 withan arbitrary decomposition

Cr = Λ1 � Λ2, F := D′(Rr)lf , F(λ) := C[z] exp(λ • z) ∼= E(C[s]/mλ),

Ass(F) = Max(C[s]) = P1 � P2, Pj := {mλ; λ ∈ Λj},F = ⊕λ∈Cr F(λ) = F1 ⊕ F2, F1 := ⊕λ∈Λ1 F(λ), F2 := ⊕λ∈Λ2 F(λ).

(70)

Standard examples for such decompositions are those from (47).The main goal of this section is to show that for these data the stable sys-

tems according to Theorem 4.2 have various stability properties known from theone-dimensional theory, and that therefore the stability terminology and the abstract

1492 ULRICH OBERST

theorems of the preceding section are justified. At the end of this section we explainthe necessary modifications for the discrete case.

For any polynomial ideal a ⊂ C[s] we define its algebraic variety as usual as

V (a) := {λ ∈ Cr; ∀t ∈ a : t(λ) = 0},

in particular V (t) := V (C[s]t) = {λ ∈ Cr; t(λ) = 0}.

(71)

The maximal ideals of A = C[s] are the

mλ := {t ∈ C[s]; t(λ) = 0 or λ ∈ V (t)}, λ ∈ Cr; hence

λ ∈ V (a) ⇔ a ⊆ mλ,

and the multiplicative monoid T from (48) is

T := ∩λ∈Λ2(A \ mλ) = {t ∈ A; ∀λ ∈ Λ2 : t(λ) = 0}= {t ∈ A; V (t) ⊂ Λ1}; hence λ ∈ Λ2 ⇒ mλ ∩ T = ∅.

(72)

The quotient rings Ap, p ∈ P2, and AT are

A(λ) := Amλ=

{a

t∈ C(s); t(λ) = 0

},

AT ={a

t∈ C(s); ∀λ ∈ Λ2 : t(λ) = 0

}.

(73)

Quotient rings AT of this type are customarily used in stabilization theory as ringsof SISO-stable plants [26], [25]. From (70) and (43) we further obtain direct decom-positions of B and likewise of B0:

B := ⊕λ∈Cr B(λ) = B1 ⊕ B2, where

B1 := B ∩ F l1, B2 := B ∩ F l

2, B(λ) := B ∩ F(λ)l

=ident.

HomC[s](M,F(λ)) =ident.

HomA(λ)(Mmλ,F(λ)).

(74)

For the system B from (64) its variety sing(B) of rank singularities (resp., the char-acteristic variety char(M0) = char(B0)) of its autonomous part are [11, Thm. 7.69,Cor. 7.71, Rem. 7.72, Cor. 7.78], [28, Thm. 4.4]

sing(B) := {λ ∈ Cr; Mmλ

is not free} = {λ ∈ Cr; rank(R(λ)) < rank(R) = p},

char(B0) := char(M0) := sing(B0) = {λ ∈ Cr; M0

mλ= 0}

= {λ ∈ Cr; rank(P (λ)) < rank(P ) = p} = V (ann(M0)),

where ann(M0) := {t ∈ A; tM0 = 0}.

(75)

We remark that the characteristic variety of a nonautonomous system B coincideswith C

r since ann(M) = 0. The elements of char(B0) are called the modes or polesor characteristic values of B. The decomposition (74) implies

B0 = ⊕λ∈char(B0) B(λ).(76)

We further need the controllable subbehavior Bcont of B, its autonomous part B0cont,

and their dual modules Mcont (resp., M0cont). There are the exact sequences [28,

diagram (5.1)]

0 → t(M) → M0 can−→ M0cont → 0 and hence, ∀λ ∈ Λ2,

0 → t(M)mλ= t(Mmλ

) → M0mλ

can−→ M0cont,mλ

→ 0,(77)


where t(M) is the torsion submodule of M .These exact sequences and (16) directly imply [28, Thm. 5.3, Cor. 5.4]

char(B0cont) = set of poles of H ⊂ char(B0) = char(B0

cont) ∪ char(t(M)),

where char(t(M)) = {λ ∈ C; t(M)mλ= 0}.

(78)

The modes or poles of B in char(B0cont) are called controllable and coincide with the

poles of H. In the trivial one-dimensional example s21 ◦ y = s1 ◦ u we have

t(M) ∼= A/As1 = 0 and char(B0) = char(B0cont) = char(t(M)) = {0}.

The following simple theorem characterizes stable behaviors and is related to theprincipal Theorem 7.2 from [29] which has, however, been shown only for the cases p =k = 1 or r ≤ 2 or for regions Λ2 which are ideal-convex (see Theorem/Definition 5.6and Example 5.7 below).

Theorem and Definition 5.1 (Λ2-stability). For the behavior B from (64)and the decomposition (70) the following statements are equivalent:

1. B is stable in the sense of Theorem/Definition 4.2, i.e., B0 ⊂ Fp1 or M0

mλ= 0

∀λ ∈ Λ2.2. char(B0) ⊂ Λ1 or char(B0) ∩ Λ2 = ∅.3. M0

cont,mλ= t(Mmλ

) = 0 ∀λ ∈ Λ2 or, in other words,(i) char(B0

cont) ∩ Λ2 = ∅.(ii) For all λ ∈ Λ2 the module Mmλ

is torsion free.4. (i) The transfer matrix H is stable; i.e., by definition, H ∈ Ap×m

T .(ii) For all λ ∈ Λ2 the module Mmλ

is torsion free or the F(λ) behaviorB(λ) :=

{(yu

)∈ F(λ)p+m; P ◦ y = Q ◦ u

}is controllable.

Then B is called Λ2-stable.Proof. 1 ⇔ 2 according to definition (75).1, 2 ⇔ 3 follows from the exactness of the sequences in (77) and the equivalence

1 ⇔ 2 applied to B0cont instead of B0.

1 ⇔ 4 is a special case of Theorem/Definition 4.2, 1 ⇔ 4.Remarks 5.2.

1. Once the data from (70) have been chosen we call the region Λ1, the poly-nomials in T , the rational functions in AT , and the polynomial-exponentialfunctions in F1 stable and Λ2 the unstable region or domain of instability [25,p. 1692]. Stable rational functions are those without poles in the unstableregion. Of course, not all unstable regions Λ2 are of practical importance.

2. The preceding theorem shows that stability of B in the sense of this paper ischaracterized both by properties 1 and 2 of its autonomous part, and by thestability of the transfer matrix H together with 4(ii), interpreted as externalstability due to the following theorems.

3. Trivial systems satisfying B0 = 0 are stable for each Λ2.4. A one-dimensional IO system (r = 1) is stable with respect to Λ1 := {z ∈ C;

�(z) < 0} if and only if its autonomous part is asymptotically stable or,equivalently, if its transfer matrix is stable and its singular variety is alsocontained in Λ1. In this case, condition 4(ii) is equivalent to

sing(B) ∩ Λ2 = ∅ or rank(R(λ)) = rank(R) ∀λ ∈ Λ2

since torsion freeness and freeness of Mmλcoincide in dimension 1. Recall

that B is controllable if and only if sing(B) = ∅. Lemma 17 from [26, p. 103]is a special case of the implication 4 ⇒ 1 of Theorem/Definition 5.1.

1494 ULRICH OBERST

5. In most papers on multidimensional stabilization the systems are discrete anddescribed by a transfer operator or IO map H, and stability signifies just thestability of H [26], [6], [23], [25], [31], [32], [7], [8], [17], [18], so our stabilitynotion is stronger. We remark, however, that for multidimensional systemsand nonproper H this transfer matrix cannot, in general, be considered asan operator on the usual function spaces, and therefore properness of H isusually required in the literature. We do not assume this since properness israther restrictive for multidimensional systems in contrast to one-dimensionalsystems. For instance, none of the standard equations

P ◦ y = u, P = s1 + s2, s1 − s22, s

21 + / − s2

2

has a proper transfer function.6. If the first independent variable z1 is distinguished as time and if Λ2 =

C+ × iRr−1, Wood, Sule, and Rogers [29, Def. 3.1] call the condition

char(B0) ∩ Λ2 = ∅ (resp., char(B0cont) ∩ Λ2 = ∅)

the CV condition for the autonomous behavior B0 (resp., B0cont) and consider

it as the stability condition for autonomous behaviors [29, p. 1500].7. For the unstable regions Λ2 from (47) and r > 1 the system B0 ⊂ Fp

1 contains,in general, many polynomial-exponential functions which are not stable in anaive sense. This is unavoidable due to the interplay of conditions 2 and 4(i)of the theorem. Indeed, with increasing Λ1 the sets of stable polynomial-exponential functions, polynomials, and rational functions also grow; hencethe generally desired existence of sufficiently many stable rational functionsimplies the same for the stable polynomial-exponential functions. For in-stance, for Λ1 = ∅ a stable system is trivial, whereas for Λ1 = C

r each systemis stable.

8. Consider Λ2 := {(λ1, λ2) ∈ C2; �(λ1),�(λ2) ≥ 0} and the simple system

(s1 + 1) ◦ y = u

with the characteristic variety

char(B0) = {−1} × C and B0 = ⊕λ2∈C C[z2] exp(−z1) exp(λ2z2).

This is obviously Λ2-stable and B0 contains the functions f(z) = exp(−z1)g(z2)with g(z2) := exp(λ2z2), �(λ2) > 0, which grow exponentially with increasingz2 and are not stable in a naive sense. However, they share this property withtheir initial part f(0, z2) = g(z2) and such initial conditions are not permit-ted according to [29, pp. 1499–1500] and the idea that a stable system shouldgenerate stable outputs for stable inputs and initial conditions. This requiresthat the initial value problem can be formulated and uniquely solved. OpenProblem 5.13 below addresses this problem for discrete systems.

We are going to show next that a stable system is also IO stable in the sense thatinputs of various types generate outputs of the same type. Since the used signals arenot necessarily bounded, we do not use the acronym BIBO.

The modules F(λ) = C[z] exp(λ • z) = E(A/mλ) (resp., F2) are A(λ)-modules(resp., AT -modules). The module F(λ) has the canonical increasing filtration of


finite-dimensional A(λ)-submodules [12, Eq. 1.17, Thm. 1.25, Thm. 6.6]

F(λ) = ∪∞k=0 F(λ)k with

F(λ)k := {y ∈ D′(Rr); mk+1λ ◦ y = 0} = C[z]≤k exp(λ • z), where

C[z]≤k := {f ∈ C[z]; deg(f) ≤ k}(79)

and deg denotes the total degree of a polynomial in C[z1, . . . , zr].Theorem and Definition 5.3 (IO stability).1. Let t ∈ A \mλ or t(λ) = 0 and y ∈ F(λ)k. Then t is invertible in the C finite-

dimensional local ring A/mk+1λ or At+ m

k+1λ = A. Via Grobner bases one constructs

t1 ∈ A such that

t1t ≡ 1(mk+1λ ) or t1t = 1 ∈ A/mk+1

λ .

Then the differential operator t◦ : F(λ)k → F(λ)k is bijective and its inverse is thedifferential operator t1◦ : F(λ)k → F(λ)k. Therefore the scalar multiplication withat ∈ A(λ) = C[s]mλ

on F(λ)k is given by the differential operator

a

t◦ : F(λ)k → F(λ)k, y �→ (at1) ◦ y.

2. If H ∈ Ap×mT is a stable rational matrix and u ∈ Fm

2 = ⊕λ∈Λ2C[z]m exp(λ • z)

is an input of the form

u =∑λ∈Λ2

fλ(z) exp(λ • z) with fλ ∈ C[z]m≤k ∀λ ∈ Λ2

and fλ = 0 for almost all λ, then the corresponding output y := H ◦ u has the sameform

y =∑λ∈Λ2

gλ(z) exp(λ • z) with gλ ∈ C[z]p≤k ∀λ ∈ Λ2

and can be computed with the algorithm from part 1. This property of the operator H◦ :Fm

2 → Fp2 is suggestively called its IO stability with respect to the decomposition (70).

3. Item 2 is applicable to any stable system B according to Theorem/Definition5.1 with its stable transfer matrix H; in particular B2 := B ∩ F l

2 ={(

H◦uu

); u ∈ Fm

2

}is its steady state part.

Proof. Item 1 follows from

tt1 + a = 1, a ∈ mk+1λ , m

k+1λ ◦ y = 0 ⇒ t1 ◦ t ◦ y = y.

Items 2 and 3 are direct consequences of 1 and Theorem/Definition 4.2.The next IO stability theorems are just reformulations of the results [4, Thm. on

p. 16 and Thm. 2 on p. 24]. These results also play an essential part in the stabilitypaper [29, subsections 5, 6, 7]. As in [4, pp. 10–12] let S (⊃ C∞

0 (Rr)) denote the spaceof rapidly decreasing C∞ functions and S ′ (⊂ D′(Rr)) its topological dual space oftemperate distributions. Further we consider the space O (⊃ S) of slowly increasingC∞ functions f for which there is an index m > 0 and C > 0 such that

|f (μ)(z)| ≤ C(1 + |z|)m ∀ z ∈ Rr, μ ∈ N

r,

1496 ULRICH OBERST

and finally the space O′ (⊂ S ′) of rapidly decreasing distributions. In the paper [29]the first variable t := z1 is distinguished as time. In this case the spaces

Rr+ := {z ∈ R

r; t = z1 ≥ 0} and

F3,+ := {y ∈ F3; supp(y) ⊂ Rr+}, F3 = S,S ′,O,O′,

play an important part in [4] and [29]. In the next theorem we use the data from(70)–(79).

Theorem 5.4 (IO stability).1. Let F3 be one of the spaces S,S ′,O,O′. If Λ2 ⊃ iRr, then F3 is an AT -

module. In particular, if H ∈ Ap×mT is a rational matrix and u ∈ Fm

3 is an input,then y := H ◦ u ∈ Fp

3 is an output of the same type. If B is stable, then according toTheorem/Definition 5.1 this holds.

2. If the variable t := z1 is distinguished and if

Λ2 ⊃ C+ × iRr−1, C+ := {λ ∈ C; �(λ) ≥ 0}

as in [29], then the statement of item 1 applies to the spaces S+ and (O′)+.If, in addition, B0 is time-autonomous in the sense of [29, Def. 2.1, Thm. 2.2],

then

Fm3

∼= B ∩ ((D′)p+ × Fm3 ), u �→

(H ◦ u

u

), F3 = S+, (O′)+.(80)

3. If in the situation of item 2

Λ2 ⊃ C+ × iRr−1, C+ := {λ ∈ C; �(λ) > 0},

then the statement of item 1 applies to the space (S ′)+.Proof. 1. Since iRr ⊂ Λ2, each t ∈ T has no zero in iRr. According to [4, Thm. on

p. 16] this is necessary and sufficient for t◦ : F3 → F3 to be bijective, and this in turnimplies that F3 is an AT -module with the scalar multiplication

a

t◦ u =: y with t ◦ y = a ◦ u.

2, 3. These assertions follow from [4, Thm. 2, p. 22] in the same fashion. Itremains to show the surjectivity of (80). Assume that

u ∈ Fm3 , y := H ◦ u, y1 ∈ (D′)p+, and

(y1

u

)∈ B, i.e., P ◦ y1 = Q ◦ u. Then

y1 − y ∈ (D′)p+ and P ◦ (y1 − y) = 0.

The time-autonomy and [29, Thm. 2.2] imply y1 = y = H ◦ u and hence the asser-tion.

Remarks 5.5.

1. The preceding theorem suggests to require iRr ⊂ Λ2 for the decomposi-tions (70) in context with stability questions of partial differential equations.The standard set

Λ2 := {λ ∈ Cr; ∀ρ = 1, . . . , r : �(λρ) ≥ 0} = C+

r

satisfies the assumptions of items 1–3.


2. The condition(H◦uu

)∈ B∩Fp+m

3 implies in particular that the input u can befreely chosen in Fm

3 and therefore condition 1 of Theorem 7.2 of [29] which,however, was shown only for p = m = 1 or r ≤ 2. The isomorphism (80)is the IO stability of B according to [29, Def. 7.1]. Theorem/Definition 5.1and Theorem 5.4 show that stability in the sense of the present paper forΛ2 = C+ × iRr−1 implies the various stability notions of that paper.

The next theorem characterizes those sets Λ2 ⊂ Cr for which the localization

functor M �→ MT from section 3 is perfect, i.e., isomorphic to M �→ MT . It turns outthat this coincides with ideal-convexity of Λ2 in the sense of [23, Def. on p. 25] or [29,Def. 5.4]. Recall from (62) that MT = 0 implies MT = (MT )T = 0, in particular {a;a ∩ T = ∅} ⊆ T.

Theorem and Definition 5.6 (perfect localization and ideal-convexity). Forthe decomposition (70) and A := C[s] the following assertions are equivalent:

1. An A-module is T -torsion if and only if it is T-torsion, i.e.,

MT = 0 ⇔ MT = 0 ⇔(50)

∀λ ∈ Λ2 : Mmλ= 0.

2. T = {a; a ideal of A, T ∩ a = 0}.3. For each M ∈ ModA the canonical map (61) MT = AT ⊗A M → MT is an

isomorphism.4. ModAT

= ModA,T.5. The set Λ2 is ideal-convex according to [29, Def. 5.4]; i.e., for each ideal a

of A the implication

V (a) ∩ Λ2 = ∅ ⇒ a ∩ T = ∅ or ∃t ∈ a with V (t) ∩ Λ2 = ∅

holds.6. The ideals (mλ)T ⊂ AT , λ ∈ Λ2, are the only maximal ideals of AT .7. The module F2 from (70) is a cogenerator in ModAT

. Recall from Lemma 3.3that F2 is an injective AT -module and an injective cogenerator in ModA,T.

Stenstrom [24, Chap. XI, Prop. 3.4] talks about perfect localization in this context.Proof. The equivalence of items 1–4 follows from [24, Chap. XI, Prop. 3.4].2 ⇔ 5 follows from the equivalences

V (a) ∩ Λ2 = ∅ ⇔ ∀λ ∈ Λ2 : a ⊂ mλ or (A/a)mλ= 0 ⇔

(50)a ∈ T.

The equivalence of items 5 and 6 is a consequence of [23, Prop. 3.1.19], but we givethe short proof for completeness.

5 ⇒ 6: The maximal ideals of AT are exactly the ideals n = mT , where m isan ideal of A maximal with respect to m ∩ T = ∅. Such an m is always prime, butnot necessarily a maximal ideal of A [10, Thm. 4.1]. Recall that mλ ∩ T = ∅ forλ ∈ Λ2. Consider such an n = mT . If V (m)∩Λ2 = ∅, condition 5 implies m∩T = ∅, acontradiction. Hence V (m)∩Λ2 = ∅ and therefore there is a λ ∈ Λ2 such that m ⊆ mλ

and mλ ∩ T = ∅; hence m = mλ by the maximality of m.6 ⇒ 5: Assume indirectly that a ∩ T = ∅ and let m be maximal with a ⊂ m and

m ∩ T = ∅. Then mT is a maximal ideal of AT and therefore by 6 of the form

mT = (mλ)T , λ ∈ Λ2 ⇒ a ⊂ m = mλ ⇒ λ ∈ V (a) ∩ Λ2.

4 ⇒ 7: The module F2 = ⊕λ∈Λ2 E(A/mλ) is an injective cogenerator in ModA,T

and hence by 4 in ModAT.

1498 ULRICH OBERST

7 ⇒ 6: Let n = mT be a maximal ideal of AT as before. The primeness of m

implies

A/m ⊂ (A/m)T = AT /mT = AT /n.

Since F2 is an injective AT -cogenerator it contains the simple AT -module AT /n andthus A/m; hence

m ∈ AssA(F2) = {mλ; λ ∈ Λ2} and m = mλ for some λ ∈ Λ2.

Examples 5.7.

1. If Λ2 ⊂ Cr is ideal-convex, then

Λ2 := {λ ∈ Cr; ∀t ∈ T : t(λ) = 0} or Λ1 = C

r \ Λ2 = ∪t∈T V (t),(81)

i.e., Λ1 is a union of hypersurfaces. Equation (81) is false for

Λ2 := {λ ∈ C2; �(λ1) ≥ 0 or �(λ2) ≥ 0},

Λ1 = {λ ∈ Cr; �(λ1) < 0,�(λ2) < 0}, T = C \ {0}, ∪t∈T V (t) = ∅.

2. In dimension 1 each Λ2 is ideal-convex.3. If r = 2, (81) characterizes ideal-convexity. In particular, the noncompact

regions C2+ and C+ × iR from Theorem 5.4 are ideal-convex, whereas for

r > 2 this is an open question [29, Thm. 5.5].4. Each polynomially convex compact subset Λ2 of C

r is ideal-convex [23, Prop.3.1.20, Rem. on p. 28]. Polynomial convexity signifies that

Λ2 =

{λ ∈ C

r; ∀ f ∈ C[s] : |f(λ)| ≤ supz∈Λ2

|f(z)|}.

In particular, the closed unit polydisc Ur, U := {z ∈ C; |z| ≤ 1}, is ideal-

convex [23, Prop. 3.1.20, Rem. on p. 28].5. Each compact cuboid Λ2 of the form

Λ2 := {λ ∈ Cr; ∀ρ = 1, . . . , r : aρ ≤ �(λρ) ≤ bρ, cρ ≤ �(λρ) ≤ dρ},

where aρ ≤ bρ, cρ ≤ dρ in R, ρ = 1, . . . , r,(82)

is ideal-convex. The proof is similar to that of the preceding item in [23] anduses the cohomological theorems A and B concerning coherent modules onStein spaces for these cuboids and the approximation of holomorphic func-tions by polynomials on such Λ2. The proof is omitted.

The proofs of the last two examples use the compactness of K in different placesessentially, in particular for the approximation of analytic functions on Λ2 by poly-nomials. It seems difficult to derive constructive algorithms from these proofs; seeRemark/Open Problem 5.10 below. The theorems A and B, however, are valid inmuch more generality and can possibly be used to prove ideal-convexity also for cer-tain noncompact subsets Λ2 of C

r.Theorem 5.8 (stabilizability and ideal-convexity).1. If B is stabilizable with respect to (70), then

sing(B) ∩ Λ2 = ∅ or rank(R(λ)) = rank(R) ∀λ ∈ Λ2.


2. If Λ2 is ideal-convex, the following assertions are equivalent:(a) B is stabilizable.(b) MT is a projective AT -module.

(c) i. The AT -lattice M := A1×(p+m)T

(H

idm

)= A1×p

T H + A1×mT ⊂ C(s)1×m

is projective.ii. MT is torsion free; i.e., B2 := B ∩ Fp+m

2 is a controllable ATF2

behavior.(d) sing(B) ∩ Λ2 = ∅ or rank(R(λ)) = rank(R) ∀λ ∈ Λ2.

Proof. 1. According to Theorem 4.4, UT is a direct summand of A1×lT . Hence for

all λ2 ∈ Λ2 also Umλ= (UT)mλ,T

is a direct summand of (AT )1×lmλ,T

= A1×lmλ

. Therefore

Mmλ= A1×l

mλ/Umλ

is projective and thus free, and λ2 is not contained in sing(B)according (75).

2. (a) ⇔ (b): Due to the assumed ideal-convexity, we have

UT = UT, MT = MT = A1×lT /UT . Hence

B is stabilizable ⇔Thm. 4.4

UT is a direct summand ⇔ MT is projective.

(b) ⇔ (c): The assumed torsion freeness of MT implies the exact sequence (13)

A1×kT

◦(P,−Q)−→ A1×(p+m)T

◦( Hidm

)−→ C(s)1×m, hence

MT = A1×(p+m)T /A1×k

T (P,−Q) ∼= A1×(p+m)T

(H

idm

)= M.

Hence MT is AT -projective if and only if the lattice M has this property.

(a) ⇒ (d) follows from 1.

(d) ⇒ (b): Condition (d) implies that for all λ ∈ Λ2 Mmλ= (MT )mλ,T

is projec-tive. Since, by Theorem/Definition 5.6(6), the mλ,T , λ ∈ Λ2, are all maximal idealsof AT , the projectivity of MT follows from [10, Thm. 7.12].

Remarks 5.9.

1. For general, not ideal-convex unstable regions, stabilizability is characterizedin Theorem/Definition 4.3 and Theorem 4.4.

2. For r = 1 and Λ2 := {λ ∈ C; �(λ) ≥ 0} condition 2(d) coincides withcondition 4(ii) from Theorem/Definition 5.1; see item 4 of Remark 5.2. Theequivalence (a) ⇔ (d) is Theorem 5.2.30 of [15], where the stabilizability of Bis, however, defined by asymptotic controllability to zero of its trajectories.

3. The condition 2(c)i of the preceding theorem characterizes internally stabi-lizable systems according to Quadrat [17, Thm. 3, Eq. (38)] with AT as aring of SISO-stable plants.

Remarks and open problems 5.10 (algorithmic questions). The assumptionsare those from (64) and (70)–(75). It turns out that basic algorithmic problems cannotcurrently be solved. Refer to [7] and to [30] where the history, the state of the art,and open problems concerning algorithms in multidimensional stability theory aredescribed.

1. According to Theorem/Definition 5.1, the stability of B is checked via

char(B0) ⊂ Λ1 or char(B0) ∩ Λ2 = ∅, where

char(B0) = V (a) and a := ann(M0) = {f ∈ C[s]; fM0 = 0},

1500 ULRICH OBERST

and a can be computed from the matrix P ∈ C[s]k×p via Grobner bases.Recall from Theorem/Definition 5.6 that

V (a) ∩ Λ2 = ∅ ⇔ a ∩ T = ∅ ∀ ideals a

if and only if Λ2 is ideal-convex. If this is not the case, it is the analogue ofthe open problem 3 from [7, p. 73] for Λ2 instead of the closed unit polydiscto decide whether V (a) ∩ Λ2 = ∅. No algorithm is presently known. If Λ2 isideal-convex, then to decide a ∩ T = ∅ and to actually construct a t ∈ a ∩ Tis the analogue of the open problem 4 from [7, p. 73] or problem 1 from [30].For compact, polynomially convex sets Λ2, such as the closed unit polydisc,this requires a constructive version of the proof of Example 5.7(4), and eventhis seems hard to obtain. If

[M0 : C] < ∞ ⇔ a is Krull–zero-dimensional ⇔ V (a) = char(B0) is finite,

then the finite variety V (a) can be computed via Grobner bases, and V (a) ∩Λ2 = ∅ can be decided (compare [7, p. 73, Rem.]).

2. Theorem 4.4 is used to test stabilizability and to construct a stabilizing com-pensator. Presently the computation of a matrix Rst ∈ Akst×l such that

Ust = {x ∈ A1×l; ∀λ ∈ Λ2 ∃sλ ∈ A with

sλ(λ) = 0 and sλx ∈ U = A1×kR}= A1×kstRst and hence UT = A1×kst

T Rst

is unsolved. If, however, Λ2 is ideal-convex and MT = MT for all A-modulesM , then UT = UT = A1×k

T R and the construction of Rst is superfluous. SeeTheorem 5.14 for a partial result on the construction of Rst. If, in general, amatrix Rst ∈ Akst×l with UT = A1×kst

T Rst is known, one computes the moduleL of all solutions (G′

1, t1) ∈ Al×kst ×A of the polynomial linear system

RstG′1Rst − t1Rst = 0 ⇔ if t1 = 0 : Rst

G′1

t1Rst = Rst(83)

via Grobner bases and obtains a system of generators of the ideal

a := {t1 ∈ A; ∃(G′1, t1) ∈ L}.(84)

According to Theorem 4.4(3), the system is stabilizable if and only if a∩T = ∅.This is again the open problem quoted in item 1. If

t1 ∈ a ∩ T and (G′1, t1) ∈ L, then G1 :=

G′1

t1∈ Al×kst

T

is a matrix according to Theorem 4.4(3), and

E1 := G1Rst = E21 ∈ Al×l

T , UT = A1×lT E1.

Generically, but not always, this matrix already satisfies the rank conditionrank((idl −E1)

(0

idm

)) = m and can be used to construct a stabilizing compen-

sator of B according to Theorem 4.4.


3. All stabilizing compensators or idempotent matrices

E = E2 ∈ Al×lT with UT = A1×l

T E and rank

((idl −E)

(0

idm

))= m

are constructed according to Theorems 2.14 and 4.6 in the form

E = E1 + XRst, X =

n∑i=1

ΘiXi, Θi ∈ AT ,

where the Xi generate the solution module of the polynomial linear systemRstX

′Rst = 0, X ′ ∈ Al×kst . Generically, these E satisfy the rank condi-tion rank((idl −E)

(0

idm

)) = m and can be used to construct all stabilizing

compensators of B.In the remainder of this section we indicate the necessary modifications of the

preceding theory for the discrete case of complex partial difference equations and thelocally finite elements of the A-module C

Nr

[12, Thm. 1.25, Cor. 1.26] with the leftshift action

(sμ ◦ y)(ν) := y(μ + ν), y ∈ CN

r

, μ, ν ∈ Nr.

For

α ∈ C, k, i ∈ N, λ ∈ Cr, μ, ν ∈ N

r define eα,k ∈ CN, eλ,μ ∈ C

Nr

by

eα,k(i) :=

{(ik

)αi−k if α = 0

δi,k if α = 0, eλ,μ(ν) :=

r∏ρ=1

eλρ,μρ(νρ). Then

(s− λ)ν ◦ eλ,μ =

{eλ,μ−ν if μ ∈ ν + N

r

0 otherwiseand

F := CN

r

lf := {y ∈ CN

r

; [C[s] ◦ y : C] < ∞} = ⊕λ∈Cr, μ∈Nr Ceλ,μ.

(85)

Again F is the minimal injective cogenerator in ModA; indeed

F(λ) := ⊕μ∈Nr Ceλ,μ = E(A/mλ) ∀λ ∈ Cr.(86)

The Borel isomorphism

CN

r

= C[[z]] ∼= C[[z]], y =∑μ∈Nr

y(μ)zμ �→∑μ∈Nr

y(μ)zμ

μ!

induces the C[s] isomorphism

F(λ) = ⊕μ∈Nr Ceλ,μ ∼= C[z] exp(λ • z), eλ,μ �→ zμ

μ!exp(λ • z).(87)

The decomposition Cr = Λ1 � Λ2 with its implied data from (70) is again arbitrary,

but, of course, interesting choices in the discrete case are different from those in thecontinuous case. Equations (70) to (77) remain valid, and so do Theorems/Definitions5.1 and 5.3 if F(λ)k from (79) is replaced with

F(λ)k := {y ∈ CN

r

; mk+1λ ◦ y = 0} = ⊕μ∈Nr, |μ|≤k Ceλ,μ, |μ| := μ1 + · · · + μr.(88)

1502 ULRICH OBERST

A standard example for Λ2 in the discrete case, in particular for r = 1, is

Λ2 := {w ∈ C; |w| ≥ 1}r, but not Ur

with U := {w ∈ C; |w| ≤ 1},(89)

where Ur

is the closed unit polydisc [7, p. 60], and for simplicity we assume this Λ2 inwhat follows. This seeming contrast to [7] and most other papers in this field comesfrom the fact that these papers do not consider IO systems, but systems defined by aproper or causal IO map H [6], [32, sec. 3.2]. To explain this connection we identify

zρ := s−1ρ , A := C[s] ⊂ C(s) = C(z) = quot(A), C[z] ⊂ C

Nr

= C[[z]]. Let

n :=C[z] 〈z1, . . . , zr〉 := {g ∈ C[z]; g(0) = 0}(90)

be the distinguished maximal ideal of C[z]. Then [6, Def. 3.16], [11, sec. 6, Thm. 60]

C[z]n = {n(z)/d(z); n, d ∈ C[z], d(0) = 0} = quot(A) ∩ C[[z]](91)

is the ring of proper or causal rational functions where the term “causal” is due tothe following property. The power series ring C

Nr

= C[[z]] is a module over itself viaconvolution, also denoted by ◦, and therefore also a module over the ring of properrational functions. In particular [11, sec. 6, Cor. 45], if the transfer matrix H of B isproper, any input u ∈ (CN

r

)m = C[[z]]m gives rise to an output

y := H ◦ u ∈ C[[z]]p with P ◦ y = P ◦ (H ◦ u) = (PH) ◦ u = Q ◦ u or

(y

u

)∈ B.

As noted before, properness is a rather restrictive property in the multidimensionalsituation and therefore we do not assume it in the present paper. For instance, thetransfer functions (s1 + s2)

−1 = z1z2z1+z2

of (s1 + s2) ◦ y = u and of other standardpartial difference or differential equations are not proper.

With the data from (89) and the terminology from [6, Def. 3.47], [7, p. 60], wesee that

S :=

{h =

n(z)

d(z)∈ C(z) = C(s); ∀w ∈ U

r: d(w) = 0

}⊂ C[z]n(92)

is the ring of structurally stable rational functions, whereas the denominators d(z) arecalled structurally stable polynomials.

Lemma 5.11. S ⊆ AT ∩ C[z]n = AT ∩ C[[z]].

Proof. Let h = n(z)d(z) ∈ S be structurally stable with relatively prime n and d.

Choose ν ∈ Nr such sνn, sνd ∈ C[s]; hence

h =n(z)

d(z)=

sνn(z)

sνd(z)=

p(s)

q(s)with relatively prime p, q ∈ C[s].

The factoriality of C[s] implies the existence of f(s) ∈ C[s] such that

q(s)f(s) = sνd(z) ∈ C[s], hence q(λ)f(λ) = λνd(w) = 0 ∀λ ∈ Λ2 or |λρ| ≥ 1, ρ = 1, . . . , r, and w := (λ−1

1 , . . . , λ−1r ) ∈ U

r,

where h ∈ S implies d(w) = 0. Hence q ∈ T and h ∈ AT as asserted.


Remarks 5.12.

1. Thus structural stability of h implies properness and stability, i.e., h ∈ AT , inthe sense of this paper. Equality holds in the preceding lemma if and only ifan irreducible polynomial d(z) ∈ C[z] is already structurally stable if d(0) = 0

and if it has no zero w in Ur

with wρ = 0 ∀ρ = 1, . . . , r. For r = 1 this is thecase; in general it is unknown (to the author). This is another example of thedifficulty of characterizing and manipulating the multiplicatively closed setsT of polynomials which have no zero in a given subset Λ2 of C

r.2. The standard one-dimensional continuous (resp., discrete) stabilization the-

ory for systems with a given proper transfer operator uses the Euclidean ringS = AT ∩ C[[z]] [26, Chap. 5] instead of AT here with

Λ2 := {λ ∈ C; �(λ) ≥ 0}, resp., Λ2 := {λ ∈ C; |λ| ≥ 1}.

A counterpart to Theorem 5.4 is the BIBO stability of structurally stable transfermatrices [6, sec. 3.4], i.e., the result that the space

l∞ :=

{u =

∑μ∈Nr

uμzμ ∈ C

Nr

= C[[z]]; ∃M > 0 ∀μ ∈ Nr : |uμ| ≤ M

}

of bounded multisequences is an S-submodule of CN

r

. This is “well-known” andfollows from the fact that a structurally stable rational function is a convergent powerseries in an open polydisc containing U

r.

For general stable discrete systems, we pose the following open problem. ConsiderΛ2 from (89), an arbitrary discrete IO system (64) with signals from C[[z]] = C

Nr

.Let Γ ⊂ {1, . . . , p} × N

r denote the associated canonical initial region [13, Eq. (14)]with respect to a chosen term order on {1, . . . , p} × N

r. Then the canonical Cauchyproblem [13, Thms. 5 and 8]

P ◦ y = Q ◦ u, u ∈ C[[z]]m, y =

( ∑μ∈Nr

yi,μzμ

)i≤p

∈ C[[z]]p,

y | Γ =

(∑μ

{yi,μzμ; (i, μ) ∈ Γ})

i≤p

= x ∈ CΓ ⊂ C[[z]]p

(93)

has a unique solution y for given input u and initial data x, and this can be computedusing Grobner bases. The space C〈z〉 of locally convergent power series is [13, Eq.(54)]

C〈z〉 =

{u =

∑μ∈Nr

uμzμ ∈ C[[z]]; ∃M > 0 ∃R ∈ R

r>0 ∀μ ∈ N

r :

|uμ| ≤ MRμ

}=

{u =

∑μ∈Nr

uμzμ ∈ C[[z]]; ∃R ∈ R

r>0 :

∑μ∈Nr

|uμ|Rμ < ∞}

and l∞, l1 :=

{u =

∑μ∈Nr

uμzμ ∈ C[[z]];

∑μ∈Nr

|uμ| < ∞}

⊂ C〈z〉

1504 ULRICH OBERST

are C[s]-submodules of C〈z〉. The unique solvability of the Cauchy problem (93) alsoholds for C〈z〉 instead of C[[z]] [13, Cor. 25], especially

B0 = {y ∈ C〈z〉p; P ◦ y = 0}

∼={x ∈ C

Γ; ∀i = 1, . . . , p :∑μ

{xi,μzμ; (i, μ) ∈ Γ} ∈ C〈z〉

}, y �→ y | Γ.

In particular, if u and x have components in l∞ or l1, the unique power series solutiony satisfies

|yi,μ| ≤ MRμ, resp.,∑μ∈Nr

|yi,μ|Rμ < ∞ for some R ∈ Rr>0 and M > 0.(94)

If all components Rρ are smaller (resp., greater) or equal to 1, then y ∈ (l∞)p (resp.,y ∈ (l1)p). Now assume that the IO system B is Λ2-stable for Λ2 from (89), i.e., thatconditions 2–4 of Theorem/Definition 5.1 are satisfied (the function space differs!!).With the idea from [29, pp. 1499–1500] that stability of a system should imply thatstable inputs and initial conditions generate stable outputs, we pose the following.

Open problem 5.13. Consider Λ2 := {λ ∈ Cr; ∀ρ = 1, . . . , r : |λρ| ≥ 1},

i = ∞, 1, and a Λ2-stable IO system B as above with input u ∈ (li)m, initial datax ∈ (li)p, and unique output y ∈ C〈z〉p. When is y ∈ (li)p, i.e., R = (1, . . . , 1)in (94) or, in other words, when is the stable system BIBO- or l∞-IO-stable (resp.,l1-IO-stable)?

This is always true for one-dimensional discrete IO systems—the properness of thetransfer matrix is not required. Compare [15, Thm. 7.6.2] for a continuous analogueunder the assumption that the transfer matrix is proper.

The question can also be asked for other natural unstable regions and is alsointeresting and reasonable in the continuous case of partial differential equations forthose spaces of analytic functions for which the Cauchy problem is uniquely solvable.In particular this holds for the space of entire functions of exponential type [13,Thm. 26] which, by means of the Borel isomorphism, is isomorphic to C〈z〉 with theshift action, and for the space C〈z〉, but now with the action by partial differentiationand under the assumption that the term order is graded [13, Thm. 29].

Consider, however, the trivial (in the sense of Theorem/Definition 2.5) and there-fore stable one-dimensional system y = s1 ◦ u with an empty initial condition andpolynomial transfer function H = s1. The analytic function u := exp(iz2

1) is boundedon R, but the output y = u′ = 2iz1u is not, so BIBO stability does not hold. Like-wise, u(z1) := (1 + z2

1)−1 exp(iz21) is in L1(R), but its derivative is not. So additional

assumptions, for instance properness, have to be made in the analogue of the openproblem for the general continuous case.

The following theorem without a detailed proof improves Theorem 5.8 and item 2of Remarks/Open Problems 5.10 and was added during the revision process. Thedata are those from (64) and (70)–(78). According to Theorem/Definition 2.2 weconstruct the unique controllable realization Bcont of H via

Ucont := B⊥cont := ker

(A1×(p+m)

◦( Hidm

)−→ K1×m

), hence

M/t(M) ∼= Mcont := A1×(p+m)/Ucont,

t(M) = Ucont/U ⊂ M = A1×(p+m)/U,

a := annA(t(M)).

(95)


The submodule Ucont and the annihilator a of the torsion module t(M) = Ucont/Uare computed via Buchberger’s algorithm. For the Gabriel localization UT, Lemma/Definition 3.6 implies

UT ⊂ UT = A1×(p+m)T ∩ ∩λ∈Λ2 Umλ

⊂ A1×(p+m)T .(96)

Generators of UT are needed for the application of the algorithm in Remarks/OpenProblems 5.10(2), and in general are hard to obtain. The next theorem gives a partialsolution.

Theorem 5.14. We use data as just introduced.(1) All localized modules Mmλ

, λ ∈ Λ2, are torsion free if and only if V (a)∩Λ2 = ∅.If this is the case, then UT = Ucont,T , and B is stabilizable if and only if Mcont,T is aprojective AT -module.

(2) The module MT is torsion free if and only if a ∩ T = ∅. If this is the case,then

UT = Ucont,T = UT and MT∼= M := A1×p

T H + A1×mT ⊂ C(s)1×m,

and B is stabilizable if and only if the lattice M is a projective AT -module. Accordingto Quadrat [17, Cor. 3] this characterizes the stabilizability of the transfer matrix Hin the usual sense.

6. Conclusion. In contrast to the literature on the stabilization of discrete mul-tidimensional IO maps, the present paper developed the stabilization theory for con-tinuous or discrete multidimensional IO behaviors. An important technical tool in thiscontext was the generalized localization theory due to Gabriel. Algorithmic problemswere addressed, but only partially solved. For a complete solution one needs con-structive solutions for the following problems from algebraic and analytic geometry.Let Λ2 be an arbitrary subset of the r-dimensional complex space and consider

T := {t ∈ C[s1, . . . , sr]; ∀λ ∈ Λ2 : t(λ) = 0} =: {stable polynomials},nonzero polynomials f and ideals a in C[s].

Open algorithmic problems 6.1 (compare [7], [30]).1. Decide f ∈ T .2. Decide V (a) ∩ Λ2 = ∅.3. Decide a ∩ T = ∅ and construct t ∈ a ∩ T .4. The implication a ∩ T = ∅ ⇒ V (a) ∩ Λ2 = ∅ is obvious. The reverse

implication is true for all ideals a if and only if a is ideal-convex. Decideideal-convexity constructively and construct t ∈ a ∩ T if V (a) ∩ Λ2 = ∅.

5. Of course, the preceding tasks have to be solved only for suitable Λ2, forinstance for the typical continuous (resp., discrete) cases

Λ2 := {z ∈ C; �(z) ≥ 0}r, resp., Λ2 = {z ∈ C; |z| ≥ 1}r.

As shown in Remarks/Open Problems 5.10 and in Theorem 5.14 a specialstabilization problem gives rise only to few ideals a for which the precedingtasks have to be solved. The paper [8] treats these problems for the closed unitpolydisc and a special class of ideals.

6. If a is Krull–zero-dimensional or V (a) is finite, then this variety can becomputed via Buchberger’s algorithm and the constructive tasks can in generalbe solved.

1506 ULRICH OBERST

At the recent workshops D2 and D3 of the Grobner semester in Linz, Austria(May 8–19, 2006), S. Tsarev (TU Berlin) pointed out the very interesting and highlyuseful fact that some of the preceding open problems can actually be constructivelysolved by means of the Tarski–Seidenberg theorem for real semi-algebraic geometryand cylindrical algebraic decomposition.

Acknowledgments. The author thanks the two referees for their reviewing workand critical comments which have been incorporated into the revised version of thepaper.

REFERENCES

[1] M. Bisiacco and M. E. Valcher, Two-dimensional behavior decompositions with finite-dimensional intersection: A complete characterization, Multidimens. Systems Signal Pro-cess., 16 (2005), pp. 335–354.

[2] V. D. Blondel and A. Megretski, eds., Unsolved Problems in Mathematical Systems andControl Theory, Princeton University Press, Princeton, NJ, 2004.

[3] N. K. Bose, ed., Multidimensional Systems Theory, D. Reidel, Dordrecht, 1985.[4] Y. V. Egorov and M. A. Shubin, eds., Partial Differential Equations III, Springer-Verlag,

Berlin, 1991.[5] K. Galkowski and J. Wood, eds., Multidimensional Signals, Circuits, and Systems, Taylor

and Francis, London, 2001.[6] J. P. Guiver and N. K. Bose, Causal and weakly causal 2-D filters with application in stabi-

lization, in Multidimensional Systems Theory, N. K. Bose, ed., D. Reidel, Dordrecht, 1985,pp. 52–100.

[7] Z. Lin, Output feedback stabilizability and stabilization of linear n-D systems, in Multidimen-sional Signals, Circuits, and Systems, K. Galkowski and J. Wood, eds., Taylor and Francis,London, 2001, pp. 59–76.

[8] Z. Lin, J. Lam, K. Galkowski, and S. Xu, A constructive approach to stabilizability andstabilization of a class of nD systems, Multidimens. Systems Signal Process., 12 (2001),pp. 329–343.

[9] S. MacLane, Homology, Springer-Verlag, Berlin, 1967.[10] H. Matsumura, Commutative Ring Theory, Cambridge University Press, Cambridge, UK,

1986.[11] U. Oberst, Multidimensional constant linear systems, Acta Appl. Math., 20 (1990), pp. 1–175.[12] U. Oberst, Variations on the fundamental principle for linear systems of partial differential

and difference equations with constant coefficients, Appl. Algebra Engrg. Comm. Comput.,6 (1995), pp. 211–243.

[13] U. Oberst and F. Pauer, The constructive solution of linear systems of partial difference anddifferential equations with constant coefficients, Multidimens. Systems Signal Process., 12(2001), pp. 253–308.

[14] H. K. Pillai and S. Shankar, A behavioral approach to control of distributed systems, SIAMJ. Control Optim., 37 (1998), pp. 388–408.

[15] J. W. Polderman and J. C. Willems, Introduction to Mathematical Systems Theory,Springer-Verlag, New York, 1998.

[16] A. Quadrat, An algebraic interpretation to the operator-theoretic approach to stabilizability.Part I: SISO systems, Acta Appl. Math., 88 (2005), pp. 1–45.

[17] A. Quadrat, A lattice approach to analysis and synthesis problems, Math. Control SignalsSystems, 18 (2006), pp. 147–186.

[18] A. Quadrat, On a generalization of the Youla–Kucera parametrization. Part II: The latticeapproach to MIMO systems, Math. Control Signals Systems, to appear.

[19] P. Rocha, Feedback control of multidimensional behaviors, Systems Control Lett., 45 (2002),pp. 207–215.

[20] M. Scheicher and U. Oberst, Proper Stabilization of Multidimensional Input/Output Sys-tems, in preparation.

[21] S. Shankar, Can one control the vibrations of a drum?, Multidimens. Systems Signal Process.,11 (2000), pp. 67–81.

[22] S. Shankar, The lattice structure of behaviors, SIAM J. Control Optim., 39 (2001), pp. 1817–1832.

[23] S. Shankar and V. R. Sule, Algebraic geometric aspects of feedback stabilization, SIAM J.Control Optim., 30 (1992), pp. 11–30.


[24] B. Stenstrom, Rings of Quotients, Springer-Verlag, Berlin, 1975.[25] V. R. Sule, Feedback stabilization over commutative rings: The matrix case, SIAM J. Control

Optim., 32 (1994), pp. 1675–1695.[26] M. Vidyasagar, Control Systems Synthesis: A Factorization Approach, The MIT Press, Cam-

bridge, MA, 1985.[27] J. Wood, Modules and behaviors in nD systems theory, Multidimens. Systems Signal Process.,

11 (2000), pp. 11–48.[28] J. Wood, U. Oberst, E. Rogers, and D. H. Owens, A behavioral approach to the pole struc-

ture of one-dimensional and multidimensional linear systems, SIAM J. Control Optim.,38 (2000), pp. 627–661.

[29] J. Wood, V. R. Sule, and E. Rogers, Causal and stable input/output structures on multidi-mensional behaviors, SIAM J. Control Optim., 43 (2005), pp. 1493–1520.

[30] L. Xu, Z. Lin, J.-Q. Ying, O. Saito, and Y. Anazawa, Open problems in control of lin-ear discrete multidimensional systems, in Unsolved Problems in Mathematical Systemsand Control Theory, V. D. Blondel and A. Megretski, eds., Princeton University Press,Princeton, NJ, 2004, pp. 221–228.

[31] J. Q. Ying, On the strong stabilizability of MIMO n-dimensional linear systems, SIAM J.Control Optim., 38 (1999), pp. 313–335.

[32] E. Zerz, Topics in Multidimensional Linear Systems Theory, Lecture Notes in Control andInform. Sci. 256, Springer-Verlag, London, 2000.

[33] E. Zerz and V. Lomadze, A constructive solution to interconnection and decomposition prob-lems with multidimensional behaviors, SIAM J. Control Optim., 40 (2001), pp. 1072–1086.

Documents

Universität Innsbruck...SIAM J. CONTROL OPTIM. c 2006 Society for Industrial and Applied Mathematics Vol. 45, No. 4, pp. 1467–1507 STABILITY AND STABILIZATION OF MULTIDIMENSIONAL