Common Lyapunov Functions

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Common Lyapunov functions for families of

commuting nonlinear systems

∗

Linh Vu and Daniel Liberzon

Coordinated Science Laboratory

Univ. of Illinois at Urbana-Champaign

Urbana, IL 61801, U.S.A.

Email: {linhvu,[email protected]}

August 21, 2007

Abstract

We present constructions of a local and global common Lyapunov function for a finite family of pairwise commuting globally asymptotically stable nonlinear systems. The constructions are based onan iterative procedure, which at each step invokes a converse Lyapunov theorem for one of the individualsystems. Our results extend a previously available one which relies on exponential stability of the vectorfields.

Keywords: common Lyapunov functions, switched nonlinear systems, commutativity, uniform asymp-totic stability.

1 Introduction

A family of dynamical systems and a switching signal which specifies the active system at each time giverise to a switched system. Switched systems are common in situations where system behavior can hardlybe described by a single ordinary differential equation, for instance, when a physical system exhibits severalmodes or when there are several controllers and a switching among them. As is well known, it is possibleto have unstable trajectories when switching among globally asymptotically stable (GAS) systems (see,e.g., [4, 9]). It is then interesting to study uniform asymptotic stability of switched systems with respectto switching signals, which is the property that the switched system state goes to zero asymptoticallyregardless of what a switching sequence is [9]. If this property holds for all initial conditions, we haveglobal uniform asymptotic stability (GUAS).

Stability of switched systems under arbitrary switching has been the subject of a number of studies,and several classes of switched systems that possess the GUAS property have been identified [1, 2, 8,16, 17, 19, 7]. In particular, it is known that a switched system generated by a finite family of GASpairwise commuting subsystems is GUAS. When the subsystems are linear, it is easy to prove this fact bymanipulating matrix exponentials. When the subsystems are nonlinear, which is the subject of our studyhere, the GUAS property has been proved by using comparison functions in [11].

∗This work was supported by NSF ECS-0134115 CAR, NSF ECS-0114725, and DARPA/AFOSR MURI F49620-02-1-0325grants.

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The converse Lyapunov theorem for switched systems asserts the existence of a common Lyapunovfunction when the switched system is GUAS [12]. A Lyapunov function is of theoretical interest and alsouseful for p erturbation analysis. For a GUAS switched system generated by locally exponentially stablesystems, a construction of a common Lyapunov function is studied in [15, 3]. The general construction of acommon Lyapunov function for GUAS switched nonlinear systems presented in [12] is a consequence of the

converse Lyapunov theorem for robust stability of nonlinear systems [10]. Although these constructionscan be applied to a family of pairwise commuting systems, they are too general for our setting since theydo not utilize commutativity. The alternative constructions proposed here involve handling the individualsystems sequentially rather than simultaneously, resulting in more constructive procedures (as comparedwith, e.g., [3, 12]). Further, one of our constructions also gives a bound on the gradient of the Lyapunovfunction and thus allows us to infer about stability of the switched system under perturbations (which isnot possible with the approach of [12]).

For a finite family of pairwise commuting systems, there is an iterative procedure for constructing acommon Lyapunov function which employs Lyapunov functions of individual systems. The procedure wasfirst proposed for a family of linear systems in [17] and later applied to a family of exponentially stable non-linear systems in [18]. In this paper, we provide more general constructions of common Lyapunov functions

for a finite family of pairwise commuting GAS—but not necessarily locally exponentially stable—nonlinearsystems. We achieve this by basing the iterative procedure on general converse Lyapunov theorems forGAS nonlinear systems.

There are primarily two ways of constructing a converse Lyapunov function for a GAS nonlinear system.One is the integral construction due to Massera [14], the other is Kurzweil’s construction [6]. UtilizingGUAS property, we describe an integral construction of a common Lyapunov function for a family of pairwise commuting GAS systems on a bounded region around the origin. This Lyapunov function is usedto derive a result on stability of the corresponding switched system under perturbations. We then useKurzweil’s method to obtain a common Lyapunov function which is valid on the whole state space. Thelatter construction actually does not rely on GUAS of the switched system. As with non-switched systems,a smoothing procedure can be used to achieve arbitrary smoothness of the Lyapunov functions.

2 Background

2.1 Notations and definitions

Recall that a continuous function V : D ⊆ Rn → [0, ∞) is positive definite if V (x) = 0 ⇔ x = 0. A

continuous function α : [0, a) → [0, ∞) is of class K if it is increasing and α(0) = 0. If a = ∞ andα(r) → ∞ as r → ∞, we say that α belongs to class K∞. A continuous function β : [0, a)× [0, ∞) → [0, ∞)is of class KL if for each fixed t, β (r, t) is of class K and for each fixed r, β (r, s) is decreasing with respectto s and lims→∞ β (r, s) = 0.

For a nonlinear system

x(t) = f (x(t)) (1)

where x(t) ∈ Rn is the state vector and f : Rn → R

n is a locally Lipschitz vector field, denote by φ(t, ξ )the solution with initial condition x(0) = ξ ∈ R

n. If for each ξ , the solution is defined for all t ∈ [0, ∞),the system is forward complete . If the solution is defined for all t ∈ (−∞, 0], the system is backward complete . The system (1) is complete if it is both backward and forward complete. The system is globally asymptotically stable (GAS) if there exists a class KL function β such that |φ(t, ξ )| ≤ β (|ξ |, t) ∀ξ ∈ Rn, ∀t ≥

2



0, where | · | denotes the Euclidean norm. We denote by Br the open ball of radius r centered at the origin,Br := {x ∈ Rn : |x| < r}.

Consider a family of dynamical systems

x(t) = f p(x(t)), p ∈ P (2)

where f p :Rn

→Rn

, p ∈ P are locally Lipschitz vector fields parameterized by a finite index set P :={1, . . . , m} for some positive integer m. This gives rise to a switched system

x(t) = f σ(t)(x(t)) (3)

where σ : [0, ∞) → P is a piecewise constant switching signal . Such a function σ has a finite numberof discontinuities, which we call switching times , on every bounded time interval, and takes a constantvalue on every interval between two consecutive switching times. We denote by S the set of all admissibleswitching signals for the switched system (3). With the assumption of locally Lipschitz vector fields,piecewise constant switching signals, and no impulse effects, a unique solution of (3) for each σ ∈ S andeach initial condition exists and is continuous and piecewise differentiable. We write ψσ(t, ξ ) for the solutionof the switched system (3) starting at ξ at time 0 for a particular switching signal σ ∈ S and we writeφ p(t, ξ ) for solutions of the individual nonlinear systems indexed by p ∈ P . Sometimes, we will write φt

p(ξ )

instead of φ p(t, ξ ) for the ease of writing compositions of flows. We denote by φtI I (ξ ) the composition of

flows over a finite index set I ,

φtI I (ξ ) := φt1

k1◦ · · · ◦ φtm

km(ξ ) (4)

where k1, . . . , km are elements of I in the increasing order and t1, . . . , tm are the corresponding times.When we write t I ≥ 0 (resp. > 0, < 0, ≤ 0, = 0), it is equivalent to writing ti ≥ 0 ∀i ∈ I (resp. > 0, < 0,≤ 0, = 0). We write s I = (t + τ ) I meaning that si = ti + τ i ∀i ∈ I . Accordingly, t I ≥ τ I (resp. >, <, ≤,=) means ti ≥ τ i ∀i ∈ I (resp. >, <, ≤, =).

We now briefly review some stability concepts for switched systems (see, e.g., [7]). The switched system(3) is globally uniformly asymptotically stable (GUAS) if there exists a class KL function β such that forevery switching signal σ ∈ S , we have

|ψσ(t, ξ )| ≤ β (|ξ |, t) ∀ ξ ∈ Rn, t ≥ 0. (5)

It is easy to see that a necessary condition for GUAS is that all individual systems of (2) are GAS. Asufficient (as well as necessary) condition for GUAS is the existence of a common Lyapunov function forthe family of systems (2), which is a positive definite, continuously differentiable function V : Rn → [0, ∞)such that

Lf pV (ξ ) := ∇V (ξ )f p(ξ ) ≤ −W (ξ ) ∀ ξ ∈ Rn, p ∈ P (6)

where ∇V is the gradient (row vector) of V and W is some positive definite function. A weaker versiononly requires V to be continuous, positive definite, locally Lipschitz away from zero and satisfying

Df pV (ξ ) := lim

ε→0+

1

ε

[V (φ p(ε, ξ )) − V (ξ )] ≤ −W (ξ ) ∀ ξ ∈ Rn \ {0}, p ∈ P (7)

for some positive definite W , where Df pV (ξ ) is the directional derivative of V along the solution of x(t) =f p(x(t)). When V is continuously differentiable, Df pV becomes Lf pV .

We say that the family of systems (2) is pairwise commuting if

φti ◦ φs

j(ξ ) = φs j ◦ φt

i(ξ ) ∀ i, j ∈ P , ξ ∈ Rn, t ,s ≥ 0. (8)

If the systems are complete, it is easy to see that the property (8) is automatically extended to t, s ∈ R. If the vector fields are continuously differentiable, pairwise commutativity is equivalent to the Lie bracket of every two vector fields being zero.

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2.2 Converse Lyapunov theorems for nonlinear systems

In this section, we quickly review converse Lyapunov theorems for non-switched nonlinear systems (see,e.g., [5]) on which our subsequent results will be built. Basically, there are two constructions of converseLyapunov functions for GAS nonlinear systems. One is due to Massera [14], the other is due to Kurzweil[6]. Massera’s construction, sometimes referred to as the integral construction , seeks a converse Lyapunovfunction of the form

V (ξ ) =

∞0

G(|φ(t, ξ )|)dt (9)

with G : [0, ∞) → [0, ∞) being chosen by the following lemma, known as Massera’s lemma .

Lemma 1 Let g : [0, ∞) → [0, ∞) be a continuous and decreasing function with g(t) → 0 as t → ∞. Let h : [0, ∞) → (0, ∞) be a continuous and nondecreasing function. There exists a function G : [0, ∞) →[0, ∞) such that:

• G and its derivative G′ are class K functions defined on [0, ∞).

• There exist positive real numbers c1 and c2 such that for all continuous functions u : R → [0, ∞)satisfying 0 ≤ u(t) ≤ g(t) ∀t ≥ 0, we have

∞0

G(u(t))dt < c1;

∞0

G′(u(t))h(t)dt < c2.

If the nonlinear system (1) is asymptotically stable, one can show that with G provided by Massera’slemma with suitable g and h, the integral (9) is well-defined and satisfies other conditions for V being aLyapunov function on some bounded region around the origin. Note that if the system is exponentiallystable, the function G in (9) can be taken as a quadratic one: G(z) = z2. The following statementsummarizes the converse Lyapunov theorem based on this construction.

Theorem 1 Suppose that nonlinear system (1) is asymptotically stable on Br, r < ∞. Suppose that the Jacobian matrix [∂f/∂x] is bounded on Br. Then there is a constant r0 ∈ (0, r), and a continuously differentiable function V : Br0 → [0, ∞) such that

α1(|ξ |) ≤ V (ξ ) ≤ α2(|ξ |)

Lf V (ξ ) ≤ −α3(|ξ |)∇V (ξ ) ≤ α4(|ξ |)

for all ξ ∈ Br0, where α1, α2, α3, α4 are class K functions on [0, r0).

The second construction, due to Kurzweil, comprises two primary steps. The first step creates a functiong : Rn → [0, ∞) as

g(ξ ) := inf t≤0

{|φ(t, ξ )|}

which is nonincreasing along solutions of the nonlinear system (1) for all initial states in forward time:

g(φ(t, ξ )) ≤ g(ξ ) ∀ t ≥ 0, ξ ∈ Rn. (10)

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The second step modifies the function g obtained in step 1 as

V (ξ ) := supt≥0

g(φ(t, ξ ))k(t)

(11)

so that the resulting function is strictly decreasing along solutions of the nonlinear system except at theorigin:

V (φ(t, ξ )) < V (ξ ) ∀ ξ ∈ Rn \ {0}, t ≥ 0 (12)

where k : [0, ∞) → R is a strictly increasing, smooth function satisfying the following properties:

K1. c1 ≤ k(t) ≤ c2 for all t ≥ 0 and some 0 < c1 < c2 < ∞.

K2. There is a decreasing continuous function τ : [0, ∞) → (0, ∞) such that

d

dtk(t) ≥ τ (t) ∀ t ≥ 0.

From this point onwards, when we write k, we refer to some fixed function k satisfying the above twoconditions (for example, k(t) = (1 + 2t)/(1 + t) is such a function). We state Kurzweil’s method in alemma. Because we frequently refer to functions with specific properties, for a function V : Rn → [0, ∞),we call the following two properties P1 and P2.

P1. Positive definite and radially unbounded. (This is equivalent to the existence of α1, α2 ∈ K∞ suchthat α1(|ξ |) ≤ V (ξ ) ≤ α2(|ξ |) ∀ξ ∈ Rn)

P2. Locally Lipschitz on Rn \ {0} and continuous on Rn.

Lemma 2 Given the GAS nonlinear system (1), suppose that a function g : Rn → [0, ∞) satisfies prop-erties P1, P2 and the inequality (10). Define a function V by (11). Then V satisfies properties P1, P2 and

Df V (ξ ) ≤ −α3(|ξ |) ∀ ξ ∈ Rn

\ {0}where α3 is some positive definite function on [0, ∞).

The function V is continuous and locally Lipschitz but may not be continuously differentiable. However,it can be smoothed to get a smooth Lyapunov function. In particular, it can be modified to be a con-tinuously differentiable function. Kurzweil’s construction leads to the following global converse Lyapunovtheorem.

Theorem 2 Suppose that the nonlinear system (1) is GAS. Then there is a continuously differentiable function V : Rn → [0, ∞) such that

α1(|ξ |) ≤ V (ξ ) ≤ α2(|ξ |)

Lf V (ξ ) ≤ −α3(|ξ |)

for all ξ ∈ Rn, where α1, α2 ∈ K∞ and α3 is some positive definite function on [0, ∞).

Kurzweil’s construction is defined on the whole state space and provides a global result. In contrast,the integral method works on a bounded region only. However, with the additional assumption of bound-edness of the Jacobian matrix of the vector field, the latter construction gives a bound on the gradientof the converse Lyapunov function and hence, it is possible to infer about stability of the system underperturbations.

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2.3 Available results on common Lyapunov functions for pairwise commuting systems

A method for constructing a common Lyapunov function for a finite family of pairwise commuting linear systems was first proposed in [17]. For a family of pairwise commuting Hurwitz matrices {A p, p ∈ P}, acommon quadratic Lyapunov function V (ξ ) = ξ T Pξ, P > 0 such that AT

p P + P A p < 0 ∀ p ∈ P could beobtained as follows:

AT 1 P 1 + P 1A1 = −P 0

AT p P p + P pA p = −P p−1, 2 ≤ p ≤ m

P := P m

where P 0 is some positive definite matrix. For a family of pairwise commuting exponentially stable nonlinear systems, i.e., nonlinear systems with solutions satisfying

|φ p(t, ξ )| ≤ c|ξ |e−λt ∀ t ≥ 0, ξ ∈ Bcr, p ∈ P (13)

for some r, c, λ > 0, a common Lyapunov function is constructed as follows [18]:

V 1(ξ ) :=

T 0

|φ1(t, ξ )|2dt

V p(ξ ) :=

T 0

V p−1(φ p(t, ξ ))dt, 2 ≤ p ≤ m

V (ξ ) := V m(ξ )

where T > T ∗ and T ∗ is some appropriately chosen constant. With the additional assumption of uniformboundedness of the Jacobian matrices of the vector fields, i.e., if there exists L < ∞ such that

∂f p(x)

∂x

(ξ ) < L ∀ ξ ∈ Bcr, p ∈ P (14)

where · is the induced matrix 2-norm, one has the following theorem.

Theorem 3 [18] Consider the family (2) of pairwise commuting systems. Suppose that they are ex-ponentially stable as in (13) and have property (14). There is a continuously differentiable function V : Br/cm−1 → [0, ∞) satisfying the following inequalities:

a1|ξ |2 ≤ V (ξ ) ≤ a2|ξ |2

Lf pV (ξ ) ≤ −a3|ξ |2 ∀ p ∈ P ∇V (ξ )

≤ a4|ξ |

for all ξ ∈ Br/cm−1 , where a1, a2, a3, a4 are positive constants.

The theorem is stated locally. A finite T such that T > T ∗ ensures the existence of a4 > 0. If T = ∞,the first two inequalities are still valid but the third one on |∇V (ξ )| no longer holds. If r = ∞, the resultholds globally.

We see that the above construction is based on the special case of Massera’s construction with Gbeing quadratic. The next section provides general results on common Lyapunov functions for a family of pairwise commuting GAS systems, following both Massera’s construction and Kurzweil’s construction.

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3 Main results

3.1 Local common Lyapunov function

We need the following extension of Massera’s lemma for multivariable functions.

Lemma 3 Let g : [0, ∞) → [0, ∞) be a continuous and decreasing function with g(t) → 0 as t → ∞. Let h : [0, ∞) → (0, ∞) be a continuous and nondecreasing function. Then there exists a differentiable function G : [0, ∞) → [0, ∞) such that

• G and its derivative G′ are class K functions on [0, ∞).

• For every positive integer l, there exist positive real numbers c1 and c2 such that for all continuous function u : Rl → [0, ∞) satisfying

0 ≤ u(t1, . . . , tl) ≤ g(t1 + · · · + tl) ∀ ti ≥ 0, 1 ≤ i ≤ l

we have ∞0

. . .

∞0

G(u(s1, . . . , sl))ds1 . . . d sl < c1

and ∞0

. . .

∞0

G′(u(s1, . . . , sl))h(s1 + · · · + sl)ds1 . . . d sl < c2.

Proof The proof proceeds along the lines of the proof of Massera’s lemma and is included in the appendix.

Consider the family (2) of pairwise commuting asymptotically stable systems on a ball Br. There exist

a class KL function β and a positive number r0 such that for every nonempty subset Q of P , we have

|φtQQ (ξ )| ≤ β (|ξ |, t1 + · · · + tq) ∀ ξ ∈ Br0 , tQ ≥ 0, q = card(Q). (15)

If the individual systems are GAS, the foregoing inequality holds for all ξ ∈ Rn. This is the uniform

asymptotic stability property of a switched system generated by a family of pairwise commuting asymp-totically stable systems [11]; details are in the proof below. Suppose that the vector fields are continuouslydifferentiable on Br. Since the ball Br is compact and P is a finite index set, there exists a positive numberL such that

∂f p(x)

∂x(ξ )

< L ∀ p ∈ P , ξ ∈ Br. (16)

For ξ ∈ Br0, construct a function V as follows:

V 1(ξ ) :=

∞0

G(|φ1(t, ξ )|)dt (17)

V p(ξ ) :=

∞0

V p−1(φ p(t, ξ ))dt, 2 ≤ p ≤ m (18)

V (ξ ) := V m(ξ ) (19)

for some function G satisfying Lemma 3 with g(t) = β (r, t), h(t) = exp(Lt). We have the following theorem.

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Theorem 4 Consider the family (2) of pairwise commuting asymptotically stable systems on Br. Suppose that for each p ∈ P , the vector field f p is continuously differentiable on Br. There is a constant r0 ∈(0, r), such that the continuously differentiable function V constructed in (17)– (19) satisfies the following inequalities:

α1(|ξ |) ≤ V (ξ ) ≤ α2(|ξ |) (20)

Lf pV (ξ ) ≤ −α3(|ξ |) ∀ p ∈ P (21)∇V (ξ ) ≤ α4(|ξ |) (22)

for all ξ ∈ Br0, where α1, α2, α3, α4 are class K functions on [0, r0).

Proof By the asymptotic stability assumption, there are class KL functions β p, p ∈ P such that for each p ∈ P , we have

|φ p(t, ξ )| ≤ β p(|ξ |, t) ∀ ξ ∈ Br, t ≥ 0. (23)

Let α p(s) := β p(s, 0), p ∈ P ; they are class K functions on [0, ∞). Define α(s) := max{α p(s), p ∈ P}.Let r∗0 = min{r, α−1(r), . . . , α−1 ◦ · · · ◦ α−1(r)} where α−1 ◦ · · · ◦ α−1 is the composition of α−1 with itself

l times, 1 ≤ l ≤ m. For every nonempty Q ⊆ P , since |φtQQ (ξ )| ≤ α1 ◦ · · · ◦ αq(|ξ |), q = card(Q), it is

guaranteed that |φtQQ (ξ )| < r, ∀tQ ≥ 0, ∀ξ ∈ Br0 if 0 < r0 < r∗0. It is easy to prove that for arbitrary

β 1, β 2 ∈ KL, there is a β ∈ KL such that β 1(β 2(r, s), t) ≤ β (r, s + t) ∀r,s,t ≥ 0 (see [11, Lemma 2.2]). Itfollows that for every nonempty Q ⊆ P , there exists a β Q ∈ KL such that

|φtQQ (ξ )| ≤ β Q(|ξ |, t1 + · · · + tq) ∀ξ ∈ Br0, ∀ tQ ≥ 0, q = card(Q).

Let β (r, t) := max{β Q(r, t), Q ⊆ P , Q = ∅}. We then have the inequality (15) for every nonempty Q ⊆ P .For a nonempty Q ⊆ P , let

V Q(ξ ) :=

∞0

. . .

∞0

G(|φtQQ (ξ )|)dt1 . . . d tq, q = card(Q). (24)

As can be seen from (17), (18) and (24), V Q(ξ ) is obtained by iteratively following (17), (18) on the indexset Q starting with φ j(t, ξ ) for some j ∈ Q. We have

V Q(ξ ) ≤

∞0

. . .

∞0

G(β (|ξ |, t1 + · · · + tq))dt1 . . . d tq =: αQ(|ξ |) ∀ ξ ∈ Br, q = card(Q) (25)

by virtue of (15). The function αQ is well-defined, in view of Lemma 3 applied with u = |φtQQ

|. It is clearthat αQ ∈ K.

From the construction, the explicit formula for V p(ξ ) is

V p(ξ ) =

∞0

. . .

∞0

G(|φtI p I p

(ξ )|)dt1 . . . d t p = V I p(ξ ) ∀ ξ ∈ Br, p ∈ P (26)

where I p := {1, . . . , p}. We then have

V p(ξ ) ≤ α2,p(|ξ |) ∀ ξ ∈ Br, p ∈ P (27)

for some α2,p ∈ K by virtue of (25). For all p ∈ P , V p is positive definite since G ∈ K. The composition

of flows φtQQ (ξ ) for every nonempty Q ⊆ P is continuous since the flow φ p(t, ξ ) is continuous for all p ∈ P .

By commutativity, for each p ∈ P , we have

V (ξ ) =

∞0

V p(φ p(t, ξ ))dt ∀ ξ ∈ Br (28)

8



where

V p(ξ ) := V P p(ξ ), P p := P \ { p}. (29)

The equation (28) is justified by Fubini’s theorem (see, e.g., [13]); we can change the order of the integralssince for every nonempty Q ⊆ P , V Q(ξ ) is well-defined and G(|φ

tQQ (ξ )|) is continuous. By boundedness of

the Jacobian matrices of the vector fields as in (16), it is easy to show (see, e.g., [5, Excercise 3.17]) thatfor each p ∈ P , we have

|φ p(t, ξ )| ≥ |ξ | exp(−Lt) ∀t ≥ 0 (30)∂φ p(t, ξ )

∂ξ

≤ exp(Lt) ∀t ≥ 0 (31)

for all ξ ∈ Br0 . From (30), for every nonempty subset Q ⊆ P , we have

|φtQQ (ξ )| ≥ |ξ | exp(−L(t1 + · · · + tq)) ∀tQ ≥ 0, q = card(Q). (32)

The inequality (32) together with (24) yields

V Q(ξ ) ≥

T 0

. . .

T 0

G(12 |ξ |)dt1 . . . d tq = T qG(12 |ξ |) =: α1,q(|ξ |) (33)

where T = ln 2/(qL), q = card(Q). Since G ∈ K, the function α1,p is of class K. Combining (27) and (33)with Q = I p yields the inequality (20) for V , in which p = m; α1 := α1,m, α2 := α2,m.

For each p ∈ P , the derivative of V along φ p(t, ξ ) will be

Lf pV (ξ ) = V p(φ p(t, ξ ))∞0

= −V p(ξ ) ∀ξ ∈ Br0

since limt→∞ φ p(t, ξ ) = 0 and V (0) = 0. There is a class K function α p such that V p(ξ ) ≥ α p(|ξ |) ∀ξ ∈ Br

by using (32), (24) and (33) with Q = P p. If we define

α3(s) := min p∈P

α p(s), s ∈ [0, r)

then α3 ∈ K and the inequality (21) follows.

By the chain rule, the gradient of V is

∂V (ξ )

∂ξ =

∞0

. . .

∞0

G′(|φtP P (ξ )|)

φT 1

|φ1|[φ′1(t1, ξ )] . . . [φ′m−1(tm−1, ξ )]

∂φm

∂ξ (tm, ξ )dt1 . . . d tm (34)

where [φ′i(t, ξ )] denotes the partial derivative with respect to ξ of the solution φi(t, ξ ) evaluated at ξ =

φtQiQi

(ξ ), Qi = {i + 1, . . . , m} for 1 ≤ i ≤ m − 1. Taking the norm of both sides of (34) and using (31), wehave the inequality (22):

∂V (ξ )

∂ξ

≤

∞0

. . .

∞0

G′(|φtI I (ξ )|)eLt1 . . . eLtmdt1 . . . dtm

=

∞0

. . .

∞0

G′(|φtI I (ξ )|)h(t1 + · · · + tm)dt1 . . . dtm =: α4(|ξ |)

for all ξ ∈ Br0 , where h(t) = exp(Lt). The function α4 exists by the choice of G as in Lemma 3 withg(t) = β (r, t), h(t) = exp(Lt), and u(t1, . . . , tm) = |φtP

P (ξ )|.

9



We stated the theorem for a family of pairwise commuting asymptotically stable systems. If individualsystems are GAS, clearly they satisfy the conditions of the theorem. Further, if the Jacobian matrices of the vector fields are uniformly bounded on the whole state space Rn, then there is no restriction on howlarge r0 is, provided it is a finite number. Note also that the boundedness of the Jacobian matrices helps

establish the bound (22) on the gradient of V . If we do not have the inequality (16), we still have theconstructed V satisfying conditions (20) and (21) on a bounded region Br0, which imply local asymptoticstability of the corresponding switched system. The existence of a function α1 ∈ K in (20) can be concludedfrom the positive definiteness of V , without relying on (33).

Theorem 4 enables one to infer about stability of a switched system generated by a family of pair-wise commuting asymptotically stable systems under perturbations, similarly to well-known results fornon-switched nonlinear systems (see, e.g., [5, Lemma 9.3]). Consider a family of pairwise commutingasymptotically stable systems in the presence of perturbations,

x(t) = f p(x(t)) + f p(x(t)), p ∈ P (35)

where f p : Rn

→ R

n

is a locally Lipschitz function on Br for each p ∈ P . The corresponding switchedsystem is

x(t) = f σ(t)(x(t)) + f σ(t)(x(t)).

For non-vanishing perturbations, the origin is no longer a common equilibrium. However, if the perturba-tion is small in some sense and the initial state is close enough to the origin, the trajectory of the perturbedswitched system is ultimately bounded for arbitrary switching.

Corollary 1 Consider the perturbed switched system (35) on Br. Assume that the family of non-perturbed systems (2) satisfies the hypotheses of Theorem 4. There are constants r > 0, δ > 0 such that if the perturbation terms f p(x) satisfy

|f p(ξ )| ≤ δ ∀ξ ∈ Br, p ∈ P

for some δ ∈ (0, δ ), then there exist M > 0 and β ∈ KL such that for every initial state ξ ∈ Br, and every switching signal σ ∈ S , the solution ψσ(t, ξ ) of the perturbed switched system satisfies

|ψσ(t, ξ )| ≤ β (|ξ |, t) 0 ≤ t ≤ T

and

|ψσ(t, ξ )| ≤ M ∀t > T.

for some finite T > 0.

Proof By Theorem 4, there exist a positive constant r0 < r and a function V satisfying (20), (21)and (22). Let δ := θα3(α−12 (α1(r0)))/α4(r0) for some positive constant θ < 1, r := α−12 (α1(r0)), andM := α−11 (α2(α−13 (δα4(r0)/θ))). The proof is similar to the proof for non-switched systems but thecommon Lyapunov function V is used in place of a single Lyapunov function. Due to space limitation,details are omitted (cf. [5, Lemma 9.3]).

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3.2 Global common Lyapunov function

In this section, we construct a global common Lyapunov function for the family of pairwise commutingGAS systems (2), following Kurzweil’s construction. In the previous section, we only required individualsystems to be forward complete and utilized GUAS property of the corresponding switched system. Here,we do not invoke GUAS property but assume that the individual systems are complete1. Hence, ourconstruction yields a Lyapunov-based proof of the fact that a switched system generated by a finite familyof pairwise commuting GAS complete subsystems is GUAS.

Define a function g : Rn → [0, ∞) as

g(ξ ) := inf

|φtP P (ξ )|, tP ≤ 0

. (36)

Thus, the function g is the infimum of the solutions ψσ(t, ξ ) of the switched system (3) running backwardin time for all possible switching signals σ ∈ S . Construct a function V as follows:

V 1(ξ ) := supt≥0

g(φ1(t, ξ ))k(t)

(37)

V p(ξ ) := supt≥0{V p−1(φ p(t, ξ ))k(t)}, 2 ≤ p ≤ m (38)

V (ξ ) := V m(ξ ) (39)

where k is some function satisfying conditions K1 and K2 as in Section 2.2. We have the following theorem.

Theorem 5 Consider the family (2) of pairwise commuting GAS complete systems. The function V constructed in (37)– (39) is continuous on Rn and locally Lipschitz on Rn \ {0} and satisfies the following inequalities:

α1(|ξ |) ≤ V (ξ ) ≤ α2(|ξ |) ∀ξ ∈ Rn (40)

Df pV (ξ ) ≤ −α3(|ξ |) ∀ξ ∈ Rn \ {0}, p ∈ P (41)

where α1, α2 ∈ K∞ and α3 is some positive definite function on [0, ∞).

Proof Firstly, we prove that the function g constructed as in (36) has properties P1, P2 and satisfies thefollowing inequality:

g(φ p(t, ξ )) ≤ g(ξ ) ∀ t ≥ 0, ξ ∈ Rn, p ∈ P . (42)

From the definition of g in (36), we have g(ξ ) ≤ |ξ |, as can be seen by taking tP = 0. By commutativity,

we have φtP P (ξ ) = φ

tP pP p

(φ p(t, ξ )) = φt p ◦ φ

tP pP p

(ξ ) with P p defined as in (29), and the inequality (42) is verifiedas

g(φ p(t, ξ )) = inf {|φtp+t p ◦ φtP pP p (ξ )|, t p ≤ 0, tP p ≤ 0}

≤ inf {|φtp p ◦ φ

tP pP p

(ξ )|, t p ≤ 0, tP p ≤ 0} ≡ g(ξ ) ∀t ≥ 0, ξ ∈ Rn, p ∈ P .

Since |φ p(t, ξ )| ≤ β p(|ξ |, t) ≤ β p(|ξ |, 0) =: α p(|ξ |) ∀ tP ≥ 0, ξ ∈ Rn, we have

|φtP P (ξ )| ≤ α1 ◦ · · · ◦ αm(|ξ |) =: α(|ξ |) ∀ tP ≥ 0, ξ ∈ Rn.

1Note that completeness was not assumed in Theorem 2 because it can always b e achieved by time rescaling [5, p. 669];however, we cannot apply this technique to the systems (2) because it does not preserve commutativity.

11



Replacing ξ by φτ P P (ξ ), by commutativity property, the above inequality becomes

|φ(t+τ )P P (ξ )| ≤ α(|φτ P

P (ξ )|) ∀ tP ≥ 0, ξ ∈ Rn.

Letting (t + τ )P = 0, we obtain

|ξ | ≤ α(|φτ P

P (ξ )|) ≤ α(g(ξ )) ∀ τ

P ≤ 0, ξ ∈ Rn.

We then have property P1 for g,

α−1(|ξ |) ≤ g(ξ ) ≤ |ξ | ∀ξ ∈ Rn.

We now prove that g has property P2. Consider a compact set H = {ξ ∈ Rn : a1 ≤ |ξ | ≤ a2} where

0 < a1 < a2. From the GAS property of the individual systems, we have |ξ | ≤ β p(|φ p(t, ξ )|, −t), ∀ t <0, p ∈ P , ξ ∈ Rn. Then,

a1 ≤ |ξ | ≤ β −t1m ◦ · · · ◦ β −tm

1 (|φtP P (ξ )|), ∀ξ ∈ H, tP < 0, (43)

where β t p(x) := β p(x, t). Since β 0 p(r) ≥ r ∀r ≥ 0, there exists τ P ≥ 0 such that

β

τ 1

m ◦ · · · ◦ β

τ m

1 (2a2) = a1. (44)For example, we can take some τ P such that β i(a1 + ∆i, τ i) = a1 + ∆(i − 1) where ∆ = (2a2 − a1)/m,1 ≤ i ≤ m; note that there may be more than one τ P ≤ 0 such that (44) holds. For all tP ≤ −τ P ,

a1 ≤ β τ 11 ◦ · · · ◦ β τ mm (|φtP P (ξ )|) ∀ξ ∈ H

by virtue of (43). The foregoing inequality together with (44) yield

|φtP P (ξ )| ≥ 2a2 ≥ 2|ξ | ≥ 2g(ξ ) ∀ξ ∈ H, tP ≤ −τ P . (45)

This implies that g is well-defined (infimum is achieved for some ti ∈ [−τ i, 0] and is unique for each fixedξ ∈ H ). Since f p is locally Lipschitz for all p ∈ P , φtP

P (ξ ) is Lipschitz for ξ ∈ H for any compact timeinterval. Since the norm function | · | is locally Lipschitz on H , it follows that g is Lipschitz on H and

hence, g is locally Lipschitz on Rn \ {0}. It is clear that g(ξ ) → 0 as ξ → 0 following (45). Since g(0) = 0,it follows that g(ξ ) is continuous everywhere. Thus, g has property P2.

The function V 1 defined by (37) is well-defined and has properties P1 and P2 by Lemma 2 with thefunction g as in (36). We have

V 1(φ2(t, ξ )) = supt1≥0

{g(φt11 ◦ φt

2(ξ ))k(t1)}

= supt1≥0

{g(φt2 ◦ φt1

1 (ξ ))k(t1)}

≤ supt1≥0

{g(φt11 (ξ ))k(t1)} ≡ V 1(ξ ) ∀t ≥ 0, ∀ξ ∈ Rn

by virtue of (42). It follows that V 2 is well-defined and has properties P1 and P2 by Lemma 2. Continuingthis procedure, we see that V p is well-defined and has properties P1 and P2 for every p ∈ P . It remains toprove (41). For each p ∈ P , we have

V (ξ ) = suptm≥0

{. . . sup

t1≥0{g(φt1

1 ◦ · · · ◦ φtmm (ξ ))k(t1)} . . . }k(tm)

= suptP ≥0

g(φtP

P (ξ ))k1(t1) . . . k(tm)

= suptP ≥0

{V p(φ p(t p, ξ ))k(t p)} ∀ p ∈ P

12



where

V p(ξ ) := suptP p≥0

g(φ

tP pP p

(ξ ))k(t1) . . . k(tm−1)

, P p = P \ { p}.

The function V p(x) is obtained by following the procedure in (37) and (38) along the index set P p starting

with φ j(t, x) for some j ∈ P p. As proved previously, V p(x) is well-defined and has properties P1 and P2.In particular, it is continuous and locally Lipschitz on Rn \ {0} and there exist α1,p, α2,p ∈ K∞ such that

α1,p(|ξ |) ≤ V p(ξ ) ≤ α2,p(|ξ |) ∀ξ ∈ Rn.

For all p ∈ P , we have

V p(φ p(t, ξ )) = supt≥0,tP p≥0

g(φ

tP pP p

(ξ ) ◦ φt p(x))k(t1) . . . k(tm−1)

= supt≥0,tP p≥0

g(φt

p ◦ φtP pP p

(ξ ))k(t1) . . . k(tm−1)

≤ suptP p≥0

g(φtP pP p

(ξ ))k(t1) . . . k(tm−1) ≡ V p(ξ ) ∀ t ≥ 0, ξ ∈ Rn

by virtue of (42). Applying Lemma 2 with g = V p, we then have that V satisfies

Df pV (ξ ) ≤ −α3,p(|ξ |) ∀ ξ ∈ Rn \ {0}, p ∈ P

where α3,p ∈ K. Define

α3(r) := min p∈P

α3,p(r)

, r ∈ [0, ∞) (46)

so that α3 ∈ K∞ and the inequality (41) is verified.

Remark 1 The function V constructed in Theorem 5 is not necessarily continuously differentiable. For asingle nonlinear system, smoothing a locally Lipschitz Lyapunov function is a well-known result (see, e.g.,[20]). The function V constructed here can be smoothed by the smoothing procedure described in [10].In particular, we can obtain a global continuously differentiable Lyapunov function. It is noted that wecan also smooth the local Lyapunov function V constructed in Section 3.1 to get a local smooth commonLyapunov function.

4 Conclusion

We have presented iterative constructions of common Lyapunov functions for a family of pairwise com-muting GAS nonlinear systems, both local and global. Based on the iterative procedure proposed in [17],our constructions relax the exponential stability assumption imposed in [18] by employing general con-verse Lyapunov theorems for nonlinear systems. The local construction leads to the result that for theperturbed switched system, the state is ultimately bounded for arbitrary switching if perturbations areuniformly bounded and the initial state is sufficiently small. This result is not possible with the comparisonfunction approach [11]. The global construction provides a common Lyapunov function for the switchedsystem generated by a family of pairwise commuting GAS complete subsystems, thereby directly impliesGUAS property for this class of switched systems (this fact is known in [11] using comparison functions).

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A Appendices

A.1 Proof of Lemma 3

Proof There exists a sequence tn such that g(tn) ≤ 1/(1 + n), n = 1, 2, . . . since g(t) is decreasing. We

construct η(t) as follows:

• η(tn) = 1n

• between tn and tn+1, η(t) is linear

• in the interval 0 < t ≤ t1, η(t) = (t1/t) p where p is a large enough p ositive integer such thatη′(t−1 ) < η′(t+1 ).

The function η is decreasing by construction and g(t) < η(t) for t ≥ t1. We also have η(t) → ∞ as t → 0+.The inverse function η−1(t) is a decreasing function and η−1(s) → ∞ as s → 0+. Then for all si ≥ 0 suchthat s1 + · · · + sl ≥ t1, we have

η−1

(u(s1, . . . , sl)) ≥ η−1

(g(s1 + · · · + sl)) > η−1

(η(s1 + · · · + sl)) = s1 + · · · + sl. (47)Define

H (s) :=exp[−η−1(s)]

h(η−1(s)), s > 0, and H (0) := 0

then H is of class K. Define

G(r) :=

r0

H (s)ds

then G is well-defined and is also of class K. Its derivative G′(r) = H (r) is of class K. By virtue of (47),we have

G′(u(s1, . . . , sl)) = exp[−η−1(u(s1, . . . , sl))]h(η−1(u(s1, . . . , sl)))

≤ e−(s1+···+sl)

h(s1 + · · · + sl)∀s1 + · · · + sl ≥ t1. (48)

The foregoing inequality leads to ∞t1

. . .

∞t1

G′(u(s1, . . . , sl))h(s1 + · · · + sl)ds1 . . . d sl ≤

∞t1

. . .

∞t1

e−s1 . . . e−slds1 . . . d sl ≤ 1.

For a given index i, 1 ≤ i ≤ l, if si ≥ t1 we then have s1 + · · · + sl ≥ t1 since s j ≥ 0, 1 ≤ j ≤ l. Thus theinequality (48) holds and hence, the integral ∞

0. . .

∞0

G′(u(s1, . . . , sl))h(s1 + · · · + sl)ds1...dsl

is bounded by some constant c2 (loosely speaking, the integral ∞0 . . .

∞0 is the sum of multiple integrals,

each of which is a combination of t10 and

∞t1

; there are 2l of them; t10 is always bounded).

For all si ≥ 0 such that s1 + · · · + sl ≥ t1, we have

G(u(s1, . . . , sl)) =

u(s1,...,sl)0

exp[−η−1(s)]

h(η−1(s))ds <

e−(s1+···+sl)

h(0)u(s1, . . . , sl)

≤e−(s1+···+sl)

h(0)g(s1 + · · · + sl) ≤

e−(s1+···+sl)

h(0)

14



since h is non-increasing and by virtue of (47). Then,

∞t1

. . .

∞t1

G(u(s1, . . . , sl))ds1 . . . d sl ≤1

h(0)

∞t1

. . .

∞t1

e−(s1+···+sl)ds1 . . . d sl < ∞.

It follows that ∞0

. . .

∞0

G(u(s1, . . . , sl))ds1 . . . d sl

is bounded by some constant c1.

Acknowledgement We thank Michael Malisoff for useful comments on an earlier draft.

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Common Lyapunov Functions