Discrete Approximations and Refined Euler–Lagrange Conditions for Nonconvex Differential Inclusions

SIAM J. CONTROL AND OPTIMIZATIONVol. 33, No. 3, pp. 882-915, May 1995

() 1995 Society for Industrial and Applied Mathematics011

DISCRETE APPROXIMATIONS AND REFINEDEULER-LAGRANGE CONDITIONS FOR NONCONVEX

DIFFERENTIAL INCLUSIONS*

BORIS S. MORDUKHOVICHt

This paper is dedicated to Terry Rockafeller on the occasion of his 60th birthday.

Abstract. This paper deals with the Bolza problem (P) for differential inclusions subject togeneral endpoint constraints. We pursue a twofold goal. First, we develop a finite difference methodfor studying (P) and construct a discrete approximation to (P) that ensures a strong convergence ofoptimal solutions. Second, we use this direct method to obtain necessary optimality conditions in arefined Euler-Lagrange form without standard convexity assumptions. In general, we prove neces-sary conditions for the so-called intermediate relaxed local minimum that takes an intermediate placebetween the classical concepts of strong and weak minima. In the case of a Mayer cost functionalor boundary solutions to differential inclusions, this Euler-Lagrange form holds without any relax-ation. The results obtained are expressed in terms of nonconvex-valued generalized differentiationconstructions for nonsmooth mappings and sets.

Key words, discrete approximations, differential inclusions, nonsmooth analysis, generalizeddifferentiation, Euler-Lagrange conditions

AMS subject classifications. 49K24, 49J52, 49M25

1. Introduction. This paper is mainly concerned with the following problem ofdynamic optimization: minimize the Bolza functional

(1.1)b

J[x] := (x(a),x(b)) + f(x(t),ic(t),t)dt

over absolutely continuous trajectories x’[a, b] --. Rn for the differential inclusion

it(t) e F(x(t), t) a.e. t e [a, b]

subject to general endpoint constraints

(1.3) (x(a), x(b)) E 2 C R2n.

Here T := [a, b] is a fixed time interval and F is a set-valued mapping (multifunction).We label this problem (P) and call it the Bolza problem for differential inclusions.

The formulated problem covers a broad range of other problems in dynamic opti-mization, in particular, both standard and nonstandard models in optimal control foropen-loop and closed-loop control systems (see, e.g., Clarke [8]). On the other hand,problem (P) can be imbedded in the so-called Generalized Problem of Bolza [39], [8]where the function f is allowed to take values in R R t {+cx}. In this paper we

Received by the editors March 16, 1993; accepted for publication (in revised form) January 3,1994. The results and techniques of this paper were presented at the Institute for Mathematics andIts Applications Workshop "Nonsmooth Analysis and Geometric Methods in Deterministic OptimalControl" on February 15, 1993. This research was partially supported by National Science Foundationgrant DMS-9206989 and the Career Development Chairs Award at Wayne State University and alsoby the Institute for Mathematics and Its Applications with funds provided by the National ScienceFoundation. A preliminary version of this paper has been published in Institute for Mathematicsand Its Applications Preprint Series 1115, March 1993.

Department of Mathematics, Wayne State University, Detroit, Michigan 48202(boris(C)math. wayne, edu).

882

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p

DISCRETE APPROXIMATIONS 883

prefer to consider the problem in form (1.1)-(1.3) and prove results depending on thespecific character of differential inclusions.

The mainstream in studying optimization problems for differential inclusions con-sists of obtaining necessary conditions for optimality (global or strong local minima).There are different approaches and various results in this area using one or anothertool in nonsmooth analysis; we refer the reader to [5]-[10], [16], [22]-[25], [27]-[29],[35], [37], [38], [46], [51] and the bibliography therein. Most of the results are obtainedfor the Mayer problem, which corresponds to (1.1)-(1.3) in the case where f 0.

In [24], Loewen and Rockafellar consider the Bolza problem (P) (with additionalstate constraints). Assuming that the function f(x,., t) and the sets F(x, t) of admis-sible velocities are convex, they obtain necessary optimality conditions under usualboundedness and Lipschitzness hypotheses but without imposing any constraint qual-ification such as calmness (cf. Clarke [5]-[10]). They prove that if 2(t) solves problem(1.1)-(1.3), then there exist a number A >_ 0 and an absolutely continuous functionp’[a, b] --. Rn, not both zero, such that

(1.4)([9(t),p(t)) e AOcf(2(t),:2(t),t) + Nc((2(t),;2(t));gphF(.,t)) a.e. t e [a,b],

(-ib(t),(t)) 6 OcH(2(t),p(t),t) a.e. t e [a,b],

(p(t), (t)} HA(2(t),p(t), t) a.e. t e [a, b],

(1.7) (p(a),-p(b)) e AOc(2(a),2(b)) + Nc((2(a),2(b));

where gph F(.,t) ((x,v) E R2nlv E F(x,t)},

H(x,p,t) := max{(p, v> Af(x,v,t)]v e F(x,t)},

and notation Nc and Oc, respectively, stand for Clarke’s normal cone to a closed setat a given point and the generalized gradient of a locally Lipschitz function [8].

Condition (1.4), which is called the Euler-Lagrange inclusion, was first obtainedby Clarke [5] for the Mayer problem under the calmness assumption, which ensuresnormality (A 1) in (1.4), (1.7). The Hamiltonian inclusion (1.5) was proved byClarke first under the calmness hypothesis and then without it; see [8], [9]. Observethat (1.6) is the Weierstrass-Pontryagin maximum condition, which is implied by eachof the conditions (1.4) and (1.5) under the convexity assumptions imposed. Note alsothat, in general, conditions (1.4) and (1.5) are independent; see examples in [22], [25].

Another version of necessary optimality conditions was obtained by Mordukhovichfor the Mayer problem (1.1)-(1.3) with f 0 under the convexity of F(x, t) and usualboundedness and Lipschitzness assumptions but without any calmness hypotheses or

something similar; see [27]-[29]. The conditions obtained are stated in the form:

(1.9) (iS(t), (t)) e co{(u, v)l(u,p(t)) e N((2(t), v);gphF(.,t)),

v e M(2(t),p(t),t)} a.e. t e [a,b],

(1.10) (p(a),-p(b)) e AO(2(a),2(b)) + N((x(a),x(b));t)

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p

884 BORIS S. MORDUKHOVICH

where "co" means the convex hull of a set,

(1.11) M(x,p,t) := {v e F(x,t)]{p,x) H(x,p,t)},

and H(x,p,t) coincides with the Hamiltonian (1.8) for f 0. Here N and 0 arenot Clarke’s normal cone and generalized gradient but their nonconvex counterpartswhose convex closures coincide with the corresponding constructions of Clarke; see 4for more details. These nonconvex constructions were first used in Mordukhovich [26]for obtaining transversality conditions like (1.10) in nonsmooth problems of optimalcontrol.

Observe that condition (1.9) implies both the maximum condition (1.6) and an

analogue of the Euler-Lagrange inclusion (1.4) in the form

(1.12) ig(t) e co{ul(u,p(t)) e N((2(t), v);gphF(.,t)),

v e M(2(t),p(t),t)} a.e. t e [a,b].

In comparison with (1.4) for f 0, condition (1.12) requires less convexification:only to the components involving derivatives of the adjoint function instead of toall components at once. This makes (1.12) essentially stronger than (1.4) in certainsituations. In particular, if the maximum set (1.11) is a singleton along (2(t),p(t))for a.e. t E [a, b] (it happens, for instance, if the sets F(x, t) are strictly convex along2(t)), then (1.12) is reduced to

(1.13) [9(t) co{ul(u,p(t)) N((2(t),(t));gphF(.,t))} a.e. t [a,b],

which is strictly better than (1.4).So (1.9) turns out to be an advanced version of the Euler-Lagrange inclusion

and the maximum condition for the Mayer problem involving convex (i.e., convex-

valued) differential inclusions (1.2). What relationships exist between (1.9) and theHamiltonian inclusion (1.5) under the usual convexity, boundedness, and Lipschitznessassumptions?

It follows from Rockafellar’s dualization result [43] that (1.5) implies (1.9). Onthe other hand, it has been recently proved by Ioffe (personal communication; see also[19, 3.5]) that (1.9) implies (1.5) under the mentioned assumptions. Therefore, ver-sion (1.9) of the Euler-Lagrange condition is equivalent to the Hamiltonian conditionin Clarke’s form (with the same adjoint function) for convex differential inclusionseThis prolongs the line of equivalency between the Hamiltonian and Euler-Lagrangeconditions, which is well known for smooth and fully convex problems (see, e.g., [39],[43]). Now one can conclude that any improvement of the necessary optimality condi-tions in form (1.9) provides a strengthening of the Hamiltonian conditions in Clarke’sform for convex differential inclusions.

In the recent paper [25], Loewen and Rockafellar establish that the Mayer problemfor convex differential inclusions can actually be reduced to the situation where the setsF(x, t) are strictly convex along the optimal trajectory. In the latter case, conditions(1.12) and (1.13) are equivalent while (1.9)is equivalent to the simultaneous fulfilmentof (1.6) and (1.13). In the general convex case, (1.13) always implies the maximumcondition in (1.8) with f 0; see Proposition 4.7 stated below. Therefore, (1.13) alsoimplies (1.12) as well as (1.9) where the improvement may be proper; see [25].

Using the mentioned strict convexification procedure and a Hamiltonian calculus,Loewen and Rockafellar prove in [25] that an optimal solution to the Mayer problem

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


for convex differential inclusions always satisfies conditions (1.10) and (1.13) as well as

(1.5) and (1.6)where f 0 with the same adjoint function p(t). Moreover, they alsoconsider the case of unbounded differential inclusions, truncating it to the boundedcase under suitable Lipschitzian assumptions and the convexity of F(x,t). Theirconsideration of the unbounded case leads to improvements of necessary conditionsfor the Bolza problem with convex velocities.

It follows from the discussion above that the refined Euler-Lagrange and transver-sality conditions (1.3) and (1.10) provide the best results for convex differential in-clusions. On the other hand, convexity assumptions appear to be restrictive in certainimportant situations, so it makes sense to release them as much as possible. Note thatif the admissible velocity sets F(x, t) are convex and the multifunction F(., t) is Lips-chitz continuous, then the differential inclusion (1.2) admits a control representationF(x, t) g(x, U, t) with a Lipschitzian function g and a control set U independent on

x; see, for example, [3]. This is no longer the case when F(x, t) are not convex. Sowhen considering nonconvex differential inclusions, one should definitely study themfor their own sakes.

The primary goal of this paper is to develop the theory of necessary optimalityconditions in the refined Euler-Lagrange form for Mayer and Bolza problems involvingnonconvex bounded differential inclusions. The results obtained below achieve thefollowing advancements in the state of art.

1. We study a new (to the best of our knowledge) concept of local minimum forthe considered variational problems involving differential inclusions. Previous resultsfor such problems were concerned with strong (or global) minima. In contrast to thestrong minimum, we compare a reference feasible trajectory 2(.) with other feasibleones close to it not only in the C-norm for arcs but also in the LP-norm (1 _< p < oc) forderivatives. This means that we consider a neighborhood of 2(.) in the Sobolev spaceWI’p equipped with a natural topology. Such a local minimum takes an intermediateplace between the classical weak and strong minima; we call it the intermediate localminimum. Note that the results obtained in this paper provide new information evenfor convex differential inclusions. In particular, they imply the maximum conditionfor an intermediate local minimum, which may not be strong.

2. We obtain refined necessary conditions for the Bolza problem (P) stated abovewith the Euler-Lagrange inclusion

(1.14) ib(t) E co{ul(u,p(t) Of(2(t),:2(t),t) + N((2(t),)(t));gphF(.,t))}

and the transversality inclusion (1.10) where an absolutely continuous function p(.)and a number >_ 0 are not equal to zero simultaneously.

In general, we prove the refined Euler-Lagrange inclusion (1.14) for any trajec-.tory 2(t) that is feasible for the original problem (P) and provides an intermediatelocal minimum for the so-called relazed problem obtained from (P) by some convex-ification procedure. Note that in this case, condition (1.14) is expressed in terms ofthe original data F, f and may be quite different from its counterpart in terms of theconvexifications. We discuss effective sufficient conditions when an optimal solutionto (P) solves the relaxed problem as well, so the conditions obtained characterizesolutions to the original problem of Bolza without any convexity.

3. In the case of a Mayer functional in (1.1)-(1.3), we prove that the refinedEuler-Lagrange inclusion (1.14), coinciding with (1.13) in this case, is fulfilled forevery strong minimum 2(t) without any relazation. This implies that if 2(t) solves theoriginal Mayer problem but may not solve the relaxed one, conditions (1.10) and (1.13)

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


still hold. Moreover, these conditions are proved to be necessary for the weak minimumto the Mayer problem under an additional Riemann integrability assumption thatmakes the technique used more transparent and in principle can be omitted.

We also establish the refined Euler-Lagrange inclusion (1.13) for any boundarytrajectory of (1.2) without either convexity or relaxation. The latter result essentiallystrengthens the recent one of Kaskosz and Lojasiewicz [22] who proved the Euler-Lagrange inclusion in Clarke’s form (1.4) with f 0 for boundary trajectories.

So we obtain that the refined Euler-Lagrange conditions, providing strongestresults for convex differential inclusions, also hold true for nonconvex problems. Thisis proved under the relaxability assumption (so far) for the general Bolza problemand without the latter assumption for the Mayer problem as well as for boundarytrajectories. Note, however, that the maximum condition no longer follows from(1.13) or (1.14)in the nonconvex case.

The Hamiltonian inclusion (1.5) always implies the maximum condition (1.6), butfor now (1.5) has been justified (as a necessary condition for the strong minimum orboundary trajectories) only in the convex case. This implies that the Hamiltoniancondition also holds under the relaxability assumption because (1.5), in contrast tothe Euler-Lagrange inclusions, is obviously invariant with respect to convexification.It follows from [25, Example 5.2] that the refined Euler-Lagrange inclusion may beessentially better than the Hamiltonian inclusion in the convex case. A series ofexamples in [22, 2] shows that the Hamiltonian inclusion does not imply even Clarke’sform (1.4) of the Euler-Lagrange inclusion in nonconvex and/or convexified problems.Therefore, the results obtained in this paper sharpen known conditions under therelaxability assumption and especially in the fully nonconvex setting.

Now let us explain the principal method that we use to obtain the mentionedresults. This is a direct method based on finite difference (discrete) approximations.Such an approach to variational problems goes back to Euler, who used it in 1744 toprove the classical Euler-Lagrange equation in the calculus of variations. (ActuallyLeibnitz was the first to employ a similar direct method to find the brachistochrone inthe very beginning of the calculus of variations; see, for example, [1]). The basic ideais as follows: (1) to replace (approximate) the original continuous-time variationalproblem by a "correct" sequence of finite-dimensional optimization problems thatcan be solved (studied) effectively, and then (2) passing to the limit with respect toapproximation parameters to obtain desirable characteristics of the original variationalproblem.

Finite difference methods turn out to be a powerful tool for numerical solutions ofinfinite-dimensional variational problems. We refer the reader to the book of Polak [36]and the survey paper of Dontchev and Lempio [13], which are devoted to numericalaspects of consistent discrete approximations in optimal control. Some results inthis paper are also concerned with numerical questions. In 3 we develop a discreteapproximation algorithm for nonconvex differential inclusions with strong convergenceproperties and error estimates. But our main interest here is to use finite differenceapproximations as a direct vehicle for obtaining necessary optimality conditions ininfinite-dimensional problem (P) via a variational analysis of nonsmooth problems infinite dimensions. Two issues are important in this approach:

(1) to construct a correct discrete approximation of problem (P) that ensuresa desirable convergence of optimal solutions for discrete problems to a given localminimum for (P);

(2) to choose "right" generalized derivative (normal) constructions for nonsmooth

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


mappings and sets that are suitable to the method. Such constructions need to beappropriate for obtaining optimality conditions in discrete problems and also possessrobustness and calculus properties for passing to the limit in the approximation pro-cedure. Let us observe that problem (P) and its discrete counterparts are definitelyobjects of nonsmooth analysis and optimization because of a special nature of dynamicconstraints like (1.2) even under smooth data in (1.1) and (1.3).

Note that not all differentiation constructions in nonsmooth analysis fit theserequirements. For example, Pshenichnyi in [38] employed some tangentially generatedconstructions related to the contingent cone. Such constructions possess the requiredproperties only in special situations. This allowed him to prove necessary conditionsfor global (actually strong) minima in autonomous differential inclusions under somerestrictive assumptions close to the graph convexity of F(., t). He used a discreteapproximation ensuring the uniform (C-) convergence of optimal trajectories.

In Mordukhovich [27]-[29], we used generalized normals and derivatives of anothernature and somewhat different algorithms to approximate Lipschitzian differential in-clusions. These generalized constructions appear in (1.9), (1.10) and possess requiredrobustness and calculus properties that are reviewed in 4. Note that if one employsthe convexification of the normal cone in (1.9), i.e., uses Clarke’s normal cone tothe graph of a Lipschitzian mapping, then such a construction does not ensure theconvergence of adjoint functions in discrete approximations (see Remark 4.6).

Although the approximation algorithm in [28] is used to prove the C-convergenceof optimal trajectories in discrete approximations, its slight modification provides thestrong L2-convergence of the velocities. It was first observed by Smirnov [48], who ob-tained the refined condition (1.13) for optimal solutions to a Mayer problem involvingconvex autonomous differential inclusions under some additional assumptions.

In this paper, we develop the method of discrete approximations to obtain theresults mentioned above in the general setting under consideration. The remainder ofthe paper is organized as follows.

Section 2 is devoted to the concept of intermediate local minimum for the originaland relaxed problems of Bolza. We consider sufficient conditions that ensure therelaxability of (P) when a given minimum for the original problem solves the relaxedproblem as well.

Section 3 deals with discrete approximations of problem (P). We provide a con-struction of discrete approximations and natural assumptions that ensure the strongconvergence of optimal solutions with respect to the value function, trajectories, andvelocities.

In 4 we describe the tools of the generalized differentiation for nonsmooth andset-valued mappings used in the paper to obtain necessary optimality conditions fordiscrete and differential inclusions. The reader can find there a brief review of the basicdifferentiation properties that are import:nt in the method of discrete approximations.

Section 5 is concerned with necessary optimality conditions for nonsmooth finite-dimensional problems. We obtain discrete analogues of the refined Euler-Lagrangeand tranversality conditions (1.14), (1.10) without the convexity operation in (1.14)and any Lipschitzian assumptions on F, f. These results turn out to be direct conse-quences of the Lagrange multiplier rule in nondifferentiable programming with manygeometric constraints.

Section 6 is devoted to the limiting procedure in discrete approximations thatallows us to prove conditions (1.10) and (1.14) for an intermediate relaxed local min-imum in (P). Under relaxation stability, the results obtained characterize optimal

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


solutions to the original problem without imposing convexity.In 7, we study a general Mayer problem for nonconvex differential inclusions

without relaxation. We prove the refined Euler-Lagrange and transversality condi-tions for strong (as well as for weak) local minima on the base of results in 6 andan approximation procedure involving Ekeland’s variational principle. The same ap-proach works to prove (1.13) for any boundary trajectory.

The notation in this paper is standard. The adjoint (transposed) matrix of A isdenoted by A*; the set B is always the unit closed ball of the space in question. Somespecial symbols are introduced and explained in 4.

2. Intermediate local minimum and relaxation. Recall that we considerproblem (P) stated above in the class of absolutely continuous functions x’[a, bR (arcs) satisfying constraints (1.2)-(1.3). Any solution to (1.2) is called an (original)trajectory for the differential inclusion, and any trajectory satisfying constraints (1.3)is called a feasible solution to problem (P). Let us introduce a notion of local minimumfor (P) studied in the paper.

DEFINITION 2.1. The arc 5c(.) is called an intermediate local minimum (i.l.rn.) ofrank p [1, oc) for (P) if 2(.) is a feasible solution to (P) and there exist numbers> 0 and c > 0 such that J[2] <_ J[x] for any other feasible solution x(.) to (P)

satisfying

(2.1) Ix(t) (t)l < v t [a, b],

b

(2.2) a 12(t)- :(t)lPdt < .If (2.2) is fulfilled, then instead of (2.1) one can obviously use Ix(a)- 2(a)l < .Relationships (2.1), (2.2) mean that we consider a neighborhood of 2(.) in the Sobolevspace W*’ of absolute continuous functions z [a, b] -- R equipped with a naturalnorm. If there is only requirement (2.1) in Definition 2.1 (i.e., a 0), then one getsa strong local minimum (with respect to the C-norm). This actually correspondsto the L1-weak topology for derivatives instead of the strong (LP-norm) topology in

(2.2). Obviously any optimal solution to problem (g) (global minimum) provides a

strong local minimum (and, therefore, an i.l.m.) for (P). As we know, most necessaryoptimality conditions for differential inclusions are obtained for strong local minima,but it is not clear a priori whether they hold for i.l.m.

If instead of (2.2) one sets the more restrictive requirement

I(t)- N(t)l < e a.e. t e [a, b],

then we have a weak local minimum in the framework of Definition 2.1. This corre-

sponds to considering a neighborhood of 2(.) in the space W1’ with the L-normfor derivatives (or the Cl-norm for continuously differentiable functions in classicalvariational problems). Therefore, the notion of i.l.m, that we introduced takes (forany rank p E [1, c)) an intermediate place between the familiar concepts of strong(a 0) and weak (p a) minima. Note that some aspects of this setting are relatedto the (local) Lavrentiev phenomenon in the calculus of variations.

The following example shows that the class of intermediate local minimizers differsfrom that of weak local minimizers in classical variational problems.

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


Example 2.2. Consider the simplest problem of the calculus of variations:

minimize J[x] 23(t)dt subject to x(0) 0, x(1) 1.

It is easy to conclude that the function 2(t) t provides a weak local minimum forthis problem; see [21, 2.2.2]. Now taking the functions xk(t) 2(t) + yk(t) withyk(O) yk(1) 0 and

-yr if 0 <_ t <_ 1/n,1) -1 if 1/n < t < 1,

one can check that

and

J[xk] V/ + O(1) -- --oc

]2k(t) ;2(t)lPdt --* 0 as k oc

for each p E [1, c). Therefore, the extremal 2(t) does not provide an intermediatelocal minimum of any rank p E [1, x) for the example considered.

On the other hand, intermediate local minimizers may not be strong local min-imizers even for convex and Lipschitzian differential inclusions. Such examples areconstructed by Vinter and Woodford [49] in both bounded (autonomous) and un-bounded cases. They also distinguish intermediate local minimizers of different rankfor some unbounded (but integrably bounded) differential inclusions.

Now we consider an extension of the original problem (P) in the line well knownin the calculus of variations and optimal control (cf., e.g., [4], [6], [15], [20], [50], [52]).Let

(2.3) fF(X, V, t) :-- f(x, V, t) + 5(V, F(x, t))

where 5(v, A) 0 if v e A and 5(v, A) o if v A (the indicator function). Denoteby IF(x, v, t) the convexification (the biconjugate function) for fF in the v variable,i.e., the largest convex function majorized by fF(x,., t) for each x and t. Along withthe original problem (P), we consider its relaxation (R) as follows:

b

(2.4) minimize [x] (x(a),x(b))+ ]F(x(t),ic(t),t)dt

over absolutely continuous functions on [a, b] under endpoint constraints (1.3). Notethat if J[x] < oe, then x(.) satisfies the convexified differential inclusion

2(t) e coF(x(t), t) a.e. t e [a, b].

Any trajectory for (2.5) is called a relaxed trajectory for (1.2). It is well known thatunder natural assumptions involving Lipschitzness of F in x, the following approxi-ruction property holds: Every relaxed trajectory x(.) can be uniformly approximatedin [a, b] by original trajectories xk(.) starting with the same initial state (but may notsatisfy endpoint constraints) such that

(2.6) lim b f(xk(t), ic(t), t)dt <_ b ]F(x(t),2(t),t)dt as k--, c

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


DEFINITION 2.3. The arc 2(.) is called an intermediate relaxed local minimum(i.r.l.m.) of rank p E [1, c) for the original problem (P) if 2(.) is a feasible solution

of (P) and provides an intermediate local minimum of rank p for the relaxed problem(R) with J[2]

It is essential in Definition 2.3 that 2(.) is an original trajectory for (1.2). Obvi-ously, there is no difference between i.r.l.m, and i.l.m, if problem (P) is convex in thefollowing sense: for each t E [a, b] and x around 2(t), the function f is convex in v onthe convex set F(x, t). One can see that the refined Euler-Lagrange inclusion that isobtained for i.r.l.m, may be different from its counterpart for arbitrary i.l.m, in therelaxed problem. This happens because the normal cone to the graph of F and to thegraph of coF is not the same. On the other hand, we cannot guarantee, in principle,that necessary conditions for i.r.l.m, will work for arbitrary i.l.m. (or strong minima)for the original problem. Nevertheless, the latter holds in rather general settings with-out any convexity assumptions. Actually this is related to the property of "hiddenconvexity" inherent in continuous-time systems like (1.2). In this paper, we use onlyone result in this direction going back to the classical Bogoljubov theorem [4].

PROPOSITION 2.4. Let 2(.) Wl’[a, b] be a strong minimum for problem (1.1),(1.3) where the integrand f(x, v, t) is continuous in (x, v) around (2(t), ;2(t)) uniformlyin [a, b] and measurable in t. Then 2(.) is a strong minimum for the relaxed problem(1.3), (2.4) with iF ] and J[2]

Proof. According to the version of Bogoljubovs theorem in [21, 9.2.4], for anyx(.) e Wl’[a, b] one can find a sequence of xk(.) e W’[a, b] such that xk(a)x(a), xk(b) x(b), xk(.) converge to x(.) uniformly in [a,b], and (2.6) is fulfilledin the case where IF ]. If 2(’) is not a strong minimum for the relaxed problem(2.4), (1.3) or/and [2] < J[2], then there exists a function x(.) e W’[a,b] with[x] < J[2] such that x(.) belongs to a C-neighborhood of 2(.) and satisfies constraints(1.3). This contradicts the strong minimality of 2(.) in the original problem, thanksto the Bogoljubov approximation for x(.).

There are several generalizations and analogues of Bogoljubov’s theorem that havemany important applications to optimal control systems and differential inclusions;see, for example, [6], [15], [20], [28], [50] and references therein. They lead to theproperty of relaxation stability (or proper relaxation) when an optimal solution tothe original problem solves the relaxed problem as well with the same optimal value.For problems (P) involving differential inclusions with endpoint constraints at eithert a or t b, such a relaxation stability follows directly from the approximationproperty for relaxed trajectories stated above.

For problems with general endpoint constraints, the relaxation stability is ensuredby the calmness property in Clarke [6], [8]. The latter property is fulfilled for "almostall" endpoint constraints (at least of inequality type) and shows that the relaxationstability may fail only for ill-posed problems where small perturbations of boundaryconditions produce proportionally unbounded variations of the minimum. Accordingto Clarke [8], the calmness hypothesis implies that corresponding necessary optimal-ity conditions can be taken to be normal. A general result that "normality impliesrelaxation stability" for optional control systems has been obtained by Warga [51].

For special classes of problems (P) with arbitrary endpoint constraints, the relax-ation stability holds without any calmness or normality assumptions. In particular,let differential inclusion (1.2) be represented in the linear form

c(t) e F (t)x(t)+ F2(t) a.e. t e [a, b]

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


where the multifunctions F1 and F2 are integrable in [a, b], F1 is convex-valued while

F2 is not. If, Inoreover, the function f in (1.1) is convex in v, then any of such prob-lems (P) possesses the property of relaxation stability. This can be proved by usingAumann’s theorem about set-valued integrals; cf. arguments in [28, Thm. 19.7]. Thesame situation holds for problems (P) involving nonlinear one-dimensional differentialinclusions; see Remark 19.2 in [28].

3. Discrete approximations. In this section we construct a sequence of dis-crete approximations for the original problem of Bolza such that optimal solutions todiscrete approximations converge in W,p to a given i.r.l.m, for problem (P).

First let us consider a fixed original trajectory 2(.) for (1.2) and prove that itcan be approximated by trajectories for corresponding discrete inclusions. To do this,we assume that the multifunction F(x, t) is bounded and locally Lipschitzian in xaround 2(.) and is Hausdorff continuous in t a.e. on [a, b]. More precisely, we imposethe following hypotheses:

(H1) There are an open set U C Rn and positive numbers 1F, mE such that2(t) E U for any t E [a, b], the sets F(x, t) are closed for all (x, t) U x [a, b], and onehas

(3.1)

and

F(x, t) C mFB V(x, t) e U x [a, b]

F(x, t) C F(x2, t) + 1FIx x2lB Vx, x2 e U, t e [a, b].

(H2) The multifunction F(x, .) is Hausdorff continuous for a.e. t E [a, b] uniformlyinxU.

Following Dontchev and Farkhi [12], let us consider the so-called averaged modulus

of continuity for the multifunction F(x, t) in t [a, b] when x g. This modulusT(F; h) that depends on the parameter h > 0 is defined as follows:

b

(3.3) T(F; h) := a(F; t, h)dt

where a(F; t, h) sup{w(F; x, t, h)lx e U}, where

w(F; x, t, h) sup{haus(F(x, t’), F(x, t"))lt’, t" It h/2, t + h/2] C [a, b]},

and where hats(., .) is the Hausdorff distance between compact sets.It is proved in [12] that if F(x, .) is Hausdorff continuous for a.e. t [a,b]

uniformly in x G U, then T(F; h) --, 0 as h -- 0.Note that in the case of single-valued functions f(t) not depending on x, the

construction T(f; h) in (3.3) was developed in Sendov and Popov [47] under the nameof "averaged modulus of smoothness." It was proved in [47] that T(f; h) - 0 as h --. 0if and only if f is Riemann integrable on [a, b]. The latter is equivalent to f beingcontinuous for a.e. t [a, b]. In this paper, we use the name "averaged modulus ofcontinuity" for both single-valued and multivalued cases.

Now let us construct a finite difference (discrete) approximation for the givendifferential inclusion using the replacement of the derivative in (1.2) by the Euler

finite difference

it(t) [x(t + h) x(t)]/h.

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


For any natural number k 1,2,..., we consider a uniform grid Tk := {tj[j0, 1,...,k} with to a, tk b and stepsize

hk:=(b-a)/k=tj+l-tj (j O, ,k -1).

The associate discrete inclusion is as follows:

(3.4) F(x xj+ E xj + hk j O, k 1.

THEOREM 3.1. Let 2(.) be a trajectory for (1.2) under hypotheses (H1) and (H2).Then there is a sequence {z]lj 0,...,k}, k 1,2,..., of solutions to discrete

inclusions (3.4) such that Zo 2.(a) and the functions

(3.5) vk(t) := (Zjk+l z])/hk, t [tj,tj+l), j O,...,k- 1,

converge to :2(.) as k ec in the norm topology of Ll[a,b].Proof. Let {wk(.)}, k 1,2,..., be an arbitrary sequence of functions in [a,b]

such that w(t) are constant in [tj, tj+l) for every j 0, k- 1 and w(t) convergeto (t) as k oc in the norm of L[a, b]. Such a sequence always exists because ofthe density of step-functions in L[a, b]. Employing (3.1), one gets

(3.6) Iwk(t)l<_mE+l Vte[a,b] as k---,

In the arguments and estimates below, we use the number

(3.7)b

k := 1:2(t) -w(t)ldt --, 0 as k --,

klj 0 k} as follows:Let us define the discrete functions {yi

(3.8) Y)+I Yj + hkw j 0,..., k 1, Yo 2(a)

where wik := wk(ty), j 0,... ,k 1. Note that the functions

yk(t) := 2(a)+ wk(s)ds, a < t < b,

are piecewise linear extensions of (3.8) on the interval [a, b] and

lyk(t)- 2(t)l <_ k Vt e [a, b].

Therefore, yk(t) U for all t [a, b] if k is big enough.Denote by dist(w, F) the Euclidean distance between the point w and the closed

set F. It is well known that the Lipschitz condition (3.2) is equivalent to

dist(w, F(xl, t)) < dist(w, F(x2, t)) + 1Fix1 x2l Vw e Rn, Xl, X2 e U, t e [a, b].

For any w, x Rn and tl, t2 [a, b], one obviously has

dist(w,F(X, tl)) <_ dist(w,F(x, t2)) + haus(F(X, tl),F(x, t2)).

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


Now using (3.3), we get

k-1

[tj+lCa := E hadist(w, F(y, tj)) dist(w, F(y, tj)ldtj=o j=O J tj

k-1

ftlj+l dist(w], F(y, t))dt + T(F; ha).

It follows from (3.2), (3.6), and (3.9) that

dist(w],F(y,t)) <_ dist(wa(t),F(yk(t),t)) + 1F(mF + 1)(t- tj) Vt E [tj,tj+l)

and

dist(wa(t), F(ya(t), t)) <_ dist(wa(t), F(2(t), t)) + 1Flya(t)_< Iw(t) (t)l + IFg a.e. t [a, b].

Therefore, we have the estimate

(3.10) ca <_ % := (1 + 1F){a + lg(b a)(mF + 1)ha/2 + T(F;

Note that functions (3.8) are not trajectories for (3.4) because one does not geta a to define trajectories for (3.4) which are close to yWj F(y], tj). Now we use wj

and have the convergence property stated in this theorem.Let us construct the desirable trajectories {z]j 0,..., k} using the following

proximal algorithm:

(3.11) a (z) tj) with Iv) wjl dist ty))Zo- 2(a), vj F a (w] F(z

zj+l zj + hav j-O, k -1.

Note that in (3.11) we take projections of velocities as in [28], [48] instead of projectionsof states as in [381, [12]. This will allow us to prove a strong convergence of discreteapproximations with respect to velocities.

First we prove that algorithm (3.11) keeps {z]lj -0 k} inside the neighbor-hood U from (H1) if k is big enough. Indeed, we consider any number k such that2(t) + raB C U for all t e [a, b] where

ra := % exp[1F(b a)] + {a,

where {a and % are defined in (3.7) and (3.10), respectively. One can see that ra + 0as k o because T(F; ha) - 0 under assumption (H2).

kBy induction, let us show that if z, U for all m 0,..., j, then this also holdsfor m j + 1. Using (3.2), (3.10), and (3.11), one gets

() (z,t)) <_ Iz 1 + ( ,t))z+ +1 <_ Iz 1+ dist dit( (]k k W

k+ l[zj y[) <_... <_ ha E (1+ lFhk)Y-’dist(m,F(Ym,tm))m--O

J<_ exp[1F(b a)] E hkdist(wm’F(yk’tm)) <- % exp[/F(b- a)].

m--0

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


Due to (3.9), the latter implies

(3.12) IZjk+l 2(tj+l)l <-- "Yk exp[/(b a)] + k :=

kwhich proves that zj E U for all j E {0,... ,k}. Taking this into account, one canextract from the previous arguments the following estimate:

(3.13)k k-1

E ]z] y]] <_ (b- a)exp[1F(b- a)] E dist(w,F(y],tj)).j=o j=0

Now let us estimate the quantity Ok f: Ivk(t)- wk(t)ldt where the functionsv(t) are defined in (3.5). Employing (3.10), (3.11), and (3.13), we get

k-1 k-1 k-1k F(z )) _< dist(w,F(y,t) E hklv] wy I= E hdist(wy, ty E h ))

j=o j=0 j=o

k-1k_ k (l+IF(b a)exp[1F(b a)]).<

j=0

Thus we obtain the final estimate

b

(3.14) (k IvY(t) (t)ldt <_ := k + "yk(1 + 1F(b a) exp[1F(b a)]).

This ensures the Ll-convergence vk(.) --+ (.) due to (3.7) and 7(F; h)under (H2).

Remark 3.2. The result obtained provides a strong approximation with respect tovelocities of any absolutely continuous trajectory for the differential inclusion (1.2)by discrete trajectories for its Euler difference counterparts (3.4). Note that the errorestimate for velocities (3.14) immediately implies the following estimate

Izk(t) 2(t) <_ k Vt [a,b]

for the corresponding motions zk(t)"= 2(a)+ f: vk(s)ds, which are piecewise linearextensions of discrete trajectories (3.11). One can see that the numerical e]ficiency ofthe estimates obtained depends on the evaluation of T(F; h) and the approximationaccuracy in (3.7).

It has been proved in [12] that T(F; h) O(h) if F(x, .) is of bounded variationon [a, b] uniformly in x U. Using the technique for averaged moduli of continuity(smoothness) developed in [47], one can obtain effective estimates for % in (3.7).Indeed, if (.) is Riemann integrable on [a, b], then we always get {k 5 2(’; h),taking v(t) (tj) for t [tj, tj + hk) as j 0,..., k- 1.

Now we consider the given original trajectory 2(.), which is an i.r.l.m, of somerank p [1, oo) for problem (P). One can easily see that under boundedness assump-tion (3.1), the notion of i.r.l.m, for (P) does not depend on rank p. This means thatif 2(.) is an i.r.l.m, of some rank p E [1, oc), then it will be an i.r.l.m, of any otherrank from [1, oc). In what follows we always take p 2 and set a 1 in (2.2) forsimplicity.

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


Let us construct a sequence of optimization problems (Pk) for discrete inclusions(3.4) such that optimal solutions to (Pk) strongly (in Wl’2[a,b]) converge to 5:(.) ask --. oc. Take a number e in (2.1), (2.2) for the given i.r.l.m; and assume (H1),(H2) along 2(.). One can always suppose that 2(t)+ /2 E U for all t E [a, b].

lJ=0 k}andUsing Theorem 3.1, we approximate 2(.) by discrete trajectories {zjcompute the numbers k in (3.12). Now we define a sequence of discrete approximationproblems (Pk), k 1, 2,..., as follows:

(3.15) minimize Jk[x] := (Xko,X) + ]x 2(a)] 2

over discrete trajectories x (x0, xk,... ,x) for the difference inclusion (3.4) sub-ject to the constraints

(3.16) (Xo e a +

(3.17) k _< y o,..., k,

and

(xj+ x)/ha :2(t)ledt <_ el2.j=0 Jtj

Let xk(.) be the piecewise linear extension of the discrete trajectory {xlj0,..., k} on [a, b], and let 2k (.) denote the piecewise constant extension of the "veloc-

kity" (xj+ -x)/hk. One has

ck(t) k --x)/hk (tj) for any t ).(xj+ ck [tj, tj+l

We are going to consider the (strong) Wl’2-convergence of xk(’) to some absolutelycontinuous function x(.) in [a, hi. This means that xk(a) x --. x(a) and ka(.) - 2(.)in L2[a, b] as k --. oc. The latter obviously implies that xk(.) converge to x(.) uniformlyin [a, hi.

In addition to (H1) and (H2), now we impose the following hypotheses on f,and

(H3) f(x, v, .) is continuous for a.e. tU x (rnFB).

(H4) There exists u > 0 such that the function f(.,., t) is continuous on the set

(3.19) A(t):= {(x, v) e U x (rnF + u)BI v e F(x, t’) for some t’ e (t- u, t]}

uniformly in t [a, b].(H5) is continuous on U x U and ft is closed around (2(a), 2(b)).THEOREM 3.3. Let 2(.) be an i.r.l.m, for problem (P), and let hypotheses (H1)-

(H5) be fulfilled. Then any sequence {2k(.)}, k 1, 2,..., of optimal solutions to (Pk)converges to 2(.) in the space wl’2[a, b] as k

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


Proof. First let us prove that for each k big enough, the discrete trajectory

{z]lj 0 k} constructed in Theorem 3.1 is a feasible solution of (Pk). We needto check that this trajectory satisfies constraints (3.16)-(3.18). For the case of (3.16),it follows directly from (3.12). Taking k such that rk <_ e/2, we also get (3.17) from(3.12). By virtue of (3.1) and (3.14), constraint (3.18) for xk zk is reduced to

fa

b

IVk(t) (t)12dt <_ 2mFak <_ 2mFk <_ /2.

The latter is fulfilled for big numbers k due to the expression for 3 in (3.14). There-fore, xk is a feasible solution to (Pk) for all k big enough. According to the classicalWeierstrass theorem, we can conclude that there is an optimal solution 2k to (P) forsuch k under the assumptions made.

Let us prove that for any sequence of optimal solutions 2 to (Pk) one has

(3.20) lim Jk[2k]_

g[2] as k .To accomplish this, it suffices to show that

(3.21) Jk[zk]--* J[2] as k cx

for the sequence of discrete trajectories zk approximating 2(.) by virtue of Theorem3.1.

Let us consider expression (3.15) for J[zk]. Due to continuity of one has

(z0k, zk) (2(a), 2(b)) as k .rther, the second term in this expression vanishes; the fourth term tends to zero ask because of (3.5) and (3.14). To justify (3.21), it remains to prove that

k-1 rb

h f(z],(zj+ zf)/h, tj) / f(2(t), (t), t) dt kkj=0 a

under assumptions (H1)-(H4). Note that (H3) implies r(f; h) 0 as k formodulus (3.3). In what follows we use the sign "" for expressions that are equivalentas k . Due to (3.5), (3.12), and (3.14) one gets

a f(z, (t),t)dt f(z,v(t),t)dt + r(f;h)= j=0

f(2(t),vk(t),t)dt f(2(t),v(t),t)dt f(2(t),:2(t),t)dt.j--O d t

The last statement holds by virtue of the classical Lebesgue limiting theorem because{v(.)} contains a subsequence converging for a.e. t E [a,b]. Therefore, we obtain(3.21), which implies (3.20).

In the arguments above we have not actually used the property of 2(.) to be ani.r.l.m, for (P). Now let us prove that in the latter case, inequality (3.20) implies

(3.22) lim[c’=12(a)-2(a)l2+ (t)-(t) dt]=O,

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


i.e., 2k(.) converge to 2(.) in the norm of wl’2[a, b]. Suppose that it is not true, andconsider a limiting point c > 0 of the sequence {ck} in (3.22). Let, for simplicity,c lim ck for all k

Because of (3.17) and (3.18) we claim the existence of an absolutely continuous

function (t) in [a,b] such that 2k(.) --. (.) uniformly in [a,b] and k(.) --, (.)weakly in L2[a, b] as k -- cx (wc take all k without loss of generality). According to

the classical Mazur theorem, there is a sequence of convex combinations of k (.) thatconverges to (.) in the norm topology of L2[a, b]. Hence it contains a subsequenceconverging to (.) for a.e. t e [a, b].

Using these facts and taking into account that

k-1 b

j+l xj)/hk, tj)j=0

f(2k (t), :2k (t), t)dt as k 0c

and also the definition of fF for (2.3), we get

b k-1

(a.2a) ]F((t),(t),t)dt <limhkf(2,(2+l-Xj)/h,tj) as k--- cx,j=0

where 2(.) satisfies the convexified differential inclusion (2.5).Observe that the integral functional

Z[v] Iv(t) (t)12dt

is lower semicontinuous in the weak topology of L2[a, b] due to the convexity of theintegrand in v. Since

j=0

the latter implies that

(3.24 I(t) :(t)]2dt < lim j0’= xy-k)/h-:2(t)12dt as k-- oc.

Now passing to the limit in (3.16)-(3.18) as k ---, c and using (3.24) as well as

(Hh), we get that &(.) satisfies constraints (1.3) and

12(t) 2(t)l

_e/2 for t e [a, b], Ic(t) ;2(t)12dt <_ el2.

The latter means that (.) belongs to the given neighborhood of 2(.) in W1’2[a, b].Moreover, (3.23) implies

(3.25) ((a), 2(b)) + fF(c(t),)(t),t)dt / c

Due to (3.20), (3.25), and c > 0 we get [2] < g[2]. But this is impossible because2(.) is an i.r.l.m, for (P). Therefore, one has c 0, which establishes (3.22) and endsthe proof of the theorem.

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


Remark 3.4. In the convergence result of Theorem 3.3,. one can avoid the conti-nuity hypothesis (H3) on f in t by changing the approximation

k--1 k-1k kh E f(x, (x+ xj)/hk, tj) for E f(x, (x+ xj)/hk, t) dt.

j=0 j=0 t

Indeed, we can handle the latter approximations in the same way s the last term in

(3.15) under the measurability assumption on f in t.The convergence theorem that we proved allows us to make bridge between

variational problems for differential inclusions and dynamic optimization problems infinite dimensions. The latter can be reduced to finite-dimensional problems of mathe-maticM programming with many geometric constraints. The mathematical program-ming problems obtained in this way turn out to be objects of nonsmooth optimizationeven in the case of smooth initial data in the original problem (P).

For variational analysis of these problems and then for passing to the limit in

optimality conditions as k -. c, we need to use generalized differential constructionswith special properties. They are considered in the next section.

4. Tools of variational analysis. This section is concerned with tools of gen-eralized differentiation that are appropriate for the main objectives of the research.The results reviewed are mostly connected with the approach in Mordukhovich [26]-[28] and recent developments in [30]-[34] dealing with nonconvex-valued generalizeddifferential constructions. The reader can also consult Clarke [8], [10], Ioffe [7]-[19],Loewen [23], Rockafellar [40], [44], and Rockafellar and Wets [45] for related andadditional material.

Let gt be a nonempty set in Rn, and let

(4.1) II(x, gt) := {w E clot such that Ixbe the Euclidean projector of x on clot. In the following definition, "cone" stands forthe conic hull of a set and "Limsup" denotes the well-known Kuratowski-Painlevupper limit for multifunctions.

DEFINITION 4.1. Given 2 cl, the closed cone

(4.2) N(2; ) Limsup__.[cone(x H(x, ))]

is called the normal cone to the set at the pointIf t is convex, then (4.2) is reduced to the normal cone of convex analysis. In

general, the convex closure of (4.2) coincides with the Clarke normal cone:

(4.3) Nc(2; t) cleoN(2;

Note that, in contrast to (4.3), the normal cone (4.2) is always robust with respect toperturbations of 2, i.e., the multifunction N(.; ) has closed graph.

DEFINITION 4.2. Let f" Rn -- R be an extended-real-valued function, and letIf()l < . The set

(4.4) Of(2) := {x* e Rnl(x*, -1) e N((2, f(2)); epif)}

is called the subdifferentiM of f at 2. If If (2)lObserve that for continuous functions, the subdifferential (4.4) turns out to be the

upper limit (robust regularization) of the subdifferential mapping used in the theory

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


of viscosity solutions [11]. It is well known that if f is locally Lipschitzian aroundwith modulus i, then

(4.5)

Moreover, in this case Ocf(2) co0f(2) for the generalized gradient of Clarke. Notealso that

(4.6) 06(2, 9t)= N(2; 9t) if 2 e

for the indicator function 5(., 9t). We also use another representation of the normalcone (4.2) in terms of the subdifferential (4.4) subdifferential of the distance function(see [28, Prop. 2.7]):

(4.7) N(2; fl) cone[0dist(2, gt)] if 2 e clFt.

Among the most important advantages of constructions (4.2) and (4.4), one hasa rich calculus under general assumptions. We refer to [10], [17]-[19], [23], [28]-[33],[40], [45] for various results in this direction. For applications in this paper, we needthe two following basic rules:

(4.8) O(f + f2)(2) a Ofx(2) + 0f2(2)

if one of the functions fi is Lipschitz continuous around 2, and

(4.9) N(2;1 n9) C N(2;1) + N(2; Ft2) if N(2;1)n (-N(2;gt2)) {0}

for any closed sets 9tl and Vt2. Note also the useful chain rule equality

(4.10) o o(vv(9),

where (q)o g)(x) := g)(g(x)) with q): Rm -- R strictly differentiable at 3 := g(2) and9 :Rn Rm Lipschitz continuous around 2.

DEFINITION 4.3. Let F :Rn Rr be a multifunction of nonempty graph gphF,and let (2, fl) cl(gphF). The multifunction D’F(2, f) from Rm into Rn defined by

(4.11) D’F(2, )(y*) := {x* e anl(x*, -y*) N((2, ); gphF)}

is called the coderivative of F at (2, 9). The symbol D’F(2) is used when F is single-valued at 2 and t F(2).

Note that because it is nonconvex valued, the coderivative (4.11) is not dual to anytangentially generated derivatives of multifunctions (see, e.g., [3, Chap. 5]). Now wereview some properties of the coderivative (4.11) that are significant for applicationsin this paper. First we consider the multifunction F of a special form whose graph is

(4.12) gphF := {(x,y) e Rn x Rm[x e ft, g(x)-y e A}

where 9:Rn RTM. The following result is proved in [28, Thm. 3.3].PROPOSITION 4.4. (i) Let the set (4.12) be closed around the point (2, 9) gphF.

Then

[D*F(2, 9)(y*) 0] ==* y* e N(9(2) 9; A).

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


(ii) If g is Lipschitz continuous around 2, then

OxS(2, y*) for y* E N(g(2) ; A),(4.13) D’F(2, )(y*) 0 otherwise

:= + 5(x, a).Setting ft Rn and A {0}, we get from (4.13) the scalarization formula

obtained in [17, Prop. 8]. Moreover, one has the following relation with Clarke’sgeneralized Jacobian:

(4.14) coD*g(2,)(y*) coO(y*,g)(2,) (Jcg(2,))*y* Vy* e RTM.

It turns out that the coderivative (4.11) enjoys a rich calculus under natural (e.g.,Lipschitzian) assumptions in general multivalued settings; see [33] for various resultsin this direction. Furthermore, the coderivative construction allows us to get completedual characterizations of Lipschitzian properties of multifunctions that are of crucialimportance for applications below (see Corollary 5.3 and the proofs of Theorems 6.1and 7.1). Now we present results for the classical locally Lipschitz property that wereproved in [31, Thm. 5.11]. We refer the reader to [2], [31], [32], [34], [41], [45] forstudies and applications of more general Lipschitzian properties and related topics.

PROPOSITION 4.5. Let F be of closed graph and bounded around 2. with F(2) O.Then each of the following conditions is necessary and sufficient for F to be locallyLipschitzian around this point:

(i) there exist a neighborhood U of 2, and a constant >_ 0 such that

(4.15) sup{Ix* x* e D*F(x,y)(y*)} <_ lly* Vx e U, y e F(x), y* Rm;

(ii) D’F(2,, 9)(0) {0} V9 e F(2,).Remark 4.6. The estimate (4.15) is crucial to ensure the convergence of adjoint

functions in the approximation procedures of 6 and 7. If one replaces the normalcone (4.2) in the coderivative construction (4.11) by the Clarke normal cone (4.3),then such a counterpart DF of the coderivative (actually appearing in Clarke’sversion of the Euler-Lagrange inclusion) does not provide estimate (4.15) and the"null-condition" (ii) for Lipschitzian multifunctions in many important situations.This is related to the fact that Clarke’s normal cone to any Lipschitzian manifold(which is a set locally representable as the graph of a Lipschitz continuous vector

function) is a linear subspace; see Rockafellar [42]. It turns out that Lipschitzianmanifolds include not only Lipschitz continuous functions but also graphs of maximalmonotone operators, in particular, subdifferential mappings for convex and saddlefunctions. For such objects, estimate (4.15) in terms of D&F and the corresponding"null-condition" are fulfilled in fact only for "strictly smooth" multifunctions. Werefer to [42] and [34] for more information about these and related properties.

In conclusion of this section, we present a useful result for convex-valued multi-functions [28, Thm. 3.1], hence showing that the considered Euler-Lagrange con-

ditions for differential and discrete inclusions automatically imply the maximum

(minimum) conditions in problems with convex velocities.PROPOSITION 4.7. Let F be convex-valued around 2, and lower semicontinuous

at 2,. Then for any f/ F(2) one has

[D*F(2, f)(y*) : 0] [(y*,9) min{(y*,y)l y e F(2)}].

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


5. Necessary conditions for discrete approximations. In this section weobtain necessary optimality conditions in discrete approximation problems (Pk) foreach k 1, 2, These conditions will be derived from a generalized Lagrangemultiplier rule for finite dimensional problems in mathematical programming withmany geometric constraints.

Let Oj :Rd--forj =0,...,sandgj :Rd--,Rn forj =0,...,m. Considerthe following problem (MP):

(5.1) minimize 0(z) for z E Rd subject to

Cj(z) _< 0 for j 1, s,

(5.3) gj(z) 0 for j 0,...,m,

(5.4) zEAj for j=O,...,1.

PROPOSITION 5.1. Let 2 be an optimal solution to problem (MP). Assume thatthe functions dpy are Lipschitz continuous, the functions gy are smooth, and the setsAj are closed around 2. Then there exist real numbers {#jlJ 0,..., s} as well asvectors {. Rnlj- 0,..., rn} and {z Rdlj 0,... ,/}, not all zero, such that

z N(2;Aj) for j 0,...,/,

(5.6) tj

__0 for j 0,...,8,

(.7) for j 1,...,s,

(.s) -z () + _’.(())*.j=0

The proof of this result, which is based on the metric approximation method,can be found in Mordukhovich [27, Thm. 1] and [28, Cor. 7.5.1]. Note that thismethod facilitates obtaining necessary conditions for (MP) in more general forms inthe presence of nonsmooth equality and inequality constraints without Lipschitzianassumptions; see [28, 7] and 7 below.

Now we employ Proposition 5.1 and calculus rules for the generalized differentialconstructions in 4 to prove necessary optimality conditions for finite difference prob-lems (Pk) in the following Euler-Lagrange form. Considering problem (Pa) in (3.4),(3.15)-(3.18) for any fixed k 1, 2,..., we denote

Fy(.) := F(.,tj) and fy(.,.):= f(.,.,t) as j 0 ,k- 1.

THEOREM 5.2. Let 2 (5:ko,...,2) be an optimal solution to problem (P).Assume that the sets and gphFj are closed and the functions Z and fj are Lipschitzco.ti..o. ao.. t oi.t (o.) a. ( - -(X+l x)/h) tiv, o a

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


j --O, k- 1. Then there exist a number Aa >_ 0 and a vector pa (po,... ,p) ER(a+l)n, not both zero, such that

(po + 2a(2(a)- 20k), --p) E a0(20,2) +

(5.10)k k k k k((Pj+I pj)/hk, Pj+I A Oj /ha) e Ofj(2, (xy++ N((2, 2ky+l 2)/hk);gphFj) for j 0,...,k- 1

where

(5.11)tj+l

k -kOj := --2 ((t) (Xj_t_l 2)/hk)dt.

Proof. Let us introduce a new variable z (x0,... ,xk, Y0,..., Yk-1) R(2k+l)n

and consider the following problem of mathematical programming:

(5.12)

k-1

minimize o(z)"= (xo, xa) + Ixo 2(a)l 2 + hk E fj(xj,y)j=0

+ [yj )(t)I2dt subject toj=o

(5.13) Cj(Z) :--IXj_I- 2(tj-1)l- /2

_0 for j 1,...,k + 1,

(5.14) Ck+2(z) [yj (t)12dt /2 <_ O,j--O Jtj

(5.15) gj(z) Xj+l xj hayj 0 for j 0,..., k 1,

(5.16) z e Ay {(x0,... ,ya-) R(2k+l)n[yj Fj(xj)} for j 0,... ,k- 1,

z e {(x0 e e

It is easy to see that problem (5.12)-(5.17) as defined is equivalent to the discreteapproximation problem (Pk)in (3.4), (3.15)-(3.18). On the other hand, (5.12)-(5.17)is a problem (MP) in (5.1)-(5.4) with d (2k + 1)n, s k + 2, m k- 1, l= k,and the specified functions j, gj and sets Aj. Now we employ Proposition 5.1for the optimal solution 2 2a (2o,.. -k (2 2o)/h, (2 2 1)/h) ofXk(5.12)-(5.17) where 2 (2o,... ,2) is a given optimal solution of (Pk).

According to this result, one gets real numbers (#o,..., #+2) as well as vectorsCj a (j- 0, k 1) and zj --(Xj, "’’,Xkj, Yj,’" Ya-* j) R(2k+l)n (J0,..., k), not all zero, such that conditions (5.5)-(5.8) are fulfilled for the initial datain (5.12)-(5.17). Note that these #j, Cj, and z depend on k but we omit the index"k" for simplicity, considering k big enough.

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


First let us observe that thanks to Theorem 3.3, j(2k) < 0 as j 1,... ,k + 2for all big k. This implies #j 0 for j 1,..., k + 2 by virtue of the correspondingcomplementary slackness conditions in (5.7). Now denoting/kk := #0 >_ 0, we ensurethat ,k, (0,..., -1), and (z,... ,z) are not equal to zero simultaneously for kbig enough.

Purther, it follows from the structure of the sets Aj in (5.16) and (5.17) thatconditions (5.5) are equivalent to

(5.18) (xjj,yjj)* E N((2), (xj+- -2)/hk);gphFj) and

Xij Yij 0 if - j Vj 0,..., k 1;

(5.19) (X;k,Xkk) X); k) and xi y 0 otherwise.

Taking this into account and using calculus rule (4.8), we get from (5.7), (5.8),(5.12), and (5.15) the following relationships:

(5.20) -Xoo Xo uo + 2

--xjj* )khk)j + Cj-1 --j for j 1,..., k- 1,

* kuk --)k-1,--Xkk

(5.23) --yj ikkhwj +0 hj for j 0 k 1

where 0) is defined in (5.11),

(5.24) (u0, ua) e 0(2o, 2),and

(Xj+ Xj)/hk) for j 0,..., k 1.

Now denoting

p0 := x)k + )kUo + 2k(20 2(a)) and pj j-1 for j 1,..., k,

one can conclude that relationships (5.18)-(5.25) imply conditions (5.9) and (5.10)where/k and (p0,...,p) are not equal to zero simultaneously. This ends the proofof the theorem.

COROLLARY 5.3. In addition to the assumptions of Theorem 5.2, let us supposethat for each j 0,..., k- 1, the multifunction Fj is bounded and Lipschitz continuousaround xj.- The conditions (5.9) and (5.10) are fulfilled with (Ak,p) O, i.e., onecan set

(5.26) )a + IPI- 1 Vk 1,2,....

Proof. If ,kk -0, then (5.10) is represented as

-k k(5.27) (P+I-p)/ha D*Fj((2,(x+ -xj)/ha)(-pj+l) for j=0, k-

in terms of the coderivative (4.11). By virtue of (5.27), Pk 0 implies that pfor all j 0,..., k- 1, according to Proposition 4.5. This proves the corollary.

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


6. Necessary conditions for the Bolza problem. Now we come back to theoriginal Bolza problem (P) and prove necessary optimality conditions for an i.r.l.m.in the refined Euler-Lagrange form. To accomplish it, it remains to pass to .the limitin the necessary conditions for discrete approximation problems (Pk) as k --- oc (5),taking into account the W1,2-convergence of discrete optimal solutions (3) and someproperties of the generalized differential constructions in 4.

Here we keep assumptions (H1)-(H3), but instead of the continuity hypothesesin (H4) and (H5) we assume the corresponding Lipschitz continuity. Namely:

(H4’) There exist numbers u > 0 and ly >_ 0 such that f(.,.,t) is locally Lips-chitzian with modulus ly around any point of the set A(t) in (3.19).

(Hh’) is Lipschitz continuous on U x U and ft is closed around (2(a), 2(b)).In what follows, we denote by Of Of(.,.,t) the subdifferential (4.4) of the

function f(x, v,t) with respect to (x, v) under fixed t. Similarly, N((., .); gphF(., t))means the normal cone (4.2) to the set gphF(., t) at a given point (., .) when t is fixed.Note that the normal cone to the graph of F is related to the generalized derivative(coderivative) of F according to (4.11).

One of the fundamental properties of the generalized differential constructions un-der consideration is their robustness (upper semicontinuity) with respect to variablesof differentiation; see 4. This is of principal importance for the method of discreteapproximations. In the limiting procedure below, we also need such a robustness ofOf(.,., t) and N((., .); gphF(.,t)) with respect to the parameter t. More precisely, we

impose the following technical assumptions:(H6) For a.e. t E [a, b] one has

limsup Of(x,’ v’,t’) Of(2(t),:2(t),t).(re’ ,’ )--. (e(t), (t))

t’--,t, t’ <t

(H7) For a.e. t E [a, b] one has

limsup N((x’, v’); gphr(., t’)) N((2(t), :(t)); gphF(., t)).(x ,v)--.((t),(t))

U--,t, t<

Properties (H6) and (H7) are obviously fulfilled if f and F do not depend on t andalso if f fl (x, v) + f2(t), F F (x) + F2(t) with F2 satisfying (H2). Actually, (H6)and (H7) mean that the continuity in t holds under the generalized differentiation of fand F with respect to the other variables. In particular, this takes place when f andF are represented as compositions of mappings separated in t and (x, v). Note alsothat for functions f smooth in (x, v), (H6) means the classical continuity of Of/Oxand Of/Ov at (ff:(t), (t), t).

Now we prove the refined Euler-Lagrange conditions for the original Bolza prob-lem (P).

THEOREM 6.1. Let 2(.) be an i.r.l.rn, for problem (P) under assumptions (H1)-(H3), (H4’), (H5’), (H6), and (H7). Then there exist a number iX > 0 and an absolutelycontinuous function p" [a, b] --, Rn, not both zero, such that

(6.1) [9(t) e co{u[(u,p(t)) e AOf(2(t),)(t),t) + N((2(t),)(t));gphF(.,t))}

for a.e. t [a, b] and

(6.2) (p(a),-p(b)) AO(2(a),2(b)) + N((2(a),2(b));

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


Proof. Let us construct a sequence of discrete approximations (Pk) of problem(P), which approximates 2(.) in the sense of Theorem 3.3. Now employing Theorem5.2 for optimal solutions xk (x0,... ,x) to (P) as k c, we find sequences ofnumbers >_ 0 and vectors pk (Po,... ,Pkk) satisfying conditions (5.9), (5.10), and(5.26). One can always suppose that Ak __, >_ 0 as k c.

In what follows we use the notation 2k(t) and pk(t) for piecewise linear exten-sions of the corresponding discrete functions, on [a, b] with their piecewise constantderivatives )k (t) and iba(t). Let us consider a sequence of the functions

Ok(t) := O/h for t e [tj, tj+l), j 0,..., k 1,

generated by (5.11). Theorem 3.3 implies that

b k-1

ft+lIOk(t)ldt E I0 <- 2 1:2(t) -k(Xj+ xj)/hkldtj=0

(a.a)_-o

Without loss of generality we can suppose that

k(6.4) z (t))(t) and Ok(t)O a.e. t[a,b as .Let us estimate the adjoint functions p(.) for big k. According to (g.10) and

kDefinition 4.a of the coderivative DF, there exist vectors (), w Ofi(), (z+-z-)/h) such that

v(6.) (P+I pj)/hk j e j(zj, (zj+ )/h) + k

for all j 0,...,- 1. Now using (6.g), (a.2), and Proposition 4.g, one has

(6.6) I(P+ pj)/hk jl < 1Fl wj + Pj+I for j 0, 1.

It follows from (H4’) and estimate (4.g) that

k(6.) I1 II and I1 z for j 0 ,- 1.

Using (.26}, (6.a), (6.6), and (6.7), we get

k(6.8) IP[ (1 + hIF)lP)+ll + hklf(1 + l) + llOl

<_exp[1F(b--a)]+li(b--a)(l+lF)+lFk Vj=0,...,k-1 as k--,

This means that the adjoint functions pk (t) are uniformly bounded in [a, b]. Employing(6.6) and (6.7) to estimate the derivatives ibm(t), one has

(P+I Pj)/hkl

_If + 1F(lf + Ok(t) + p+) for tj t < tj+.

By virtue of (6.4) and (6.8) this implies that the sequence {pc(.)} is weakly compactin L[a,b]. Therefore, we can find an absolutely continuous function p(.) such thatpk (.) p(.) uniformly in [a, b] and k (.) (.) weakly in L[a, b] for k (as usualwe take all k 1, 2 ).

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


Let us rewrite (5.10) as follows:

(6.9)[9e(t) {u[(u,p(ty+)- O(t)) AOf(2(tj),(t),tj)

+ N((k(tj),:2(t));gphF(.,tj))} for t [tj,tj+l), j O,...,k-1.

According to the classical results, there is a sequence of convex combinations .of ibk (t)that converges to ib(t) for a.e. t E [a, b]. Now passing to the limit in (6.9) as k --, cand using (6.4) as well as hypotheses (H6) and (H7), we obtain the Euler-Lagrangeinclusion (6.1).

Taking the limit in (5.26), one has the normalization condition

+ Ip(b)l- 1,

which implies that and p(.) are not equal to zero simultaneously. It follows from(6.8) that if p(to) 0 at some point to E [a, b], then p(t) =_ 0 in [a, b].

It remains to establish the transversality inclusion (6.2). First note that

AkO(2O, 2k) -- AO(2(a), 2(b)) as k --, c

due to robustness of the subdifferential (4.4). Then observe that the set tk in (3.16)is represented as

(6.10) t {(xo,x) Rail dist((xo, xa), gt) _< /k}.

Now passing to the limit in (5.9) as k c and using (4.7), we obtain (6.2). Thiscompletes the proof of the theorem. [3

Remark 6.2. The Euler-Lagrange inclusion (6.1) can be expressed in terms of thecoderivative DF of the multifunction F(., t) under fixed t. Indeed, one has

(6.11) ib(t) e co { U(O,w)Of((t),(t),t)

[0 + DF(2(t), (t), t)(Aw p(t))] }for a.e. t e [a,b], which is equavalent to (6.1) by virtue of (4.11). Inclusion (6.11)implies the following:

[9(t) co[AOxf(2(t), :2(t), t) + DF(2(t), :2(t), t)(AOvf(2(t), /2(t), t) p(t))]

in the case when Of(2(t), :2(t), t) C Oxf(2(t), :2(t), t) Ovf(2(t), :2(t), t) for a.e. t[a, b]. In particular, it happens when either f(x, v, t) fl(x, t)+f2(v, t)or f possessessome regularity (e.g., f is smooth or convex in (x, v)).

Remark 6.3. Theorem 6.1 provides necessary optimality conditions for the Bolzaproblem (P) in the basic case where endpoint constraints are given in the. abstract(geometric) form (1.3). Using calculus rules available for the nonconvex subdiffer-ential constructions in 4, we can obtain refined transversality conditions for specialrepresentations of the set in (1.3). In particular, for the case of inequality andequality type constraints

(6.12) i(x(a), x(b)) <_ 0 if 1, q,

(6.13) i(x(a),x(b)) 0 if q + 1,...,q + r

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


defined by locally Lipschitz functions Ti, one has

(6.14) (p(a),-p(b)) e O (/ii)\i=o (2(a),2(b)),

(6.15) Ai>_0 for i=0,1,...,q,

(6.16) Aiai((a), (b)) 0 for 1,..., q

where a0 := and (A0,..., Aq+,p(.)) - 0; cf. [27]-[29]. Due to the subdifferentialsum rule, the transversality inclusion (6.14) implies the following "separated" one:

q q+r

(p(a),-p(b)) E E )iOi(Sc(a)’2(b)) + Ei--0 i--q+1

where Oa(2) := 0a(2) U [-0(-a)(2)] is a symmetric subdifferential construction.In the next section we prove unified transversality conditions in the presence of bothfunctional and geometric constraints for the case of Lipschitz as well as non-Lipschitzfunctions 99.

Remark 6.4. Developing the discrete approximation approach, we can extendnecessary optimality conditions in Theorem 6.1 to the case of "Bolza constraints"

b

(x(a), x(b)) + f(x(t), it(t), t)dt <_ 0 for 1,..., q,

b

a(x(a),x(b)) + f(x(t),ic(t),t)dt 0 for q + 1 ,q + r,

which unify endpoint constraints (6.12) and (6.13) with isoperimetric type constraintsin variational problems. In this case, the set of necessary conditions consists of (6.14),(6.15) and

Ti(Sc(a), 2(b)) f(2(t), (t),t)dt 0 for 1,..., q,

for a.e. t e [a, b] where f0 := f and (A0 Aq+, p(.)) 0.Remark 6.5. The approach and results obtained can be extended to the Bolza

problem for nonconvex differential inclusions with free time where a varying timeinterval is involved in constraints and optimization. We refer the reader to [35] formore details and discussions.

The necessary optimality conditions in Theorem 6.1 are proved for any i.r.l.m.in the Bolza problem (P). In particular, they hold for any feasible solution to (P),which turns out to be a strong local minimum for the relaxed problem. As we pointed

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


out in 2, an optimal solution of (P) automatically solves the relaxed problem as wellin some common settings. Let us present such a corollary of Theorem 6.1, which isused in the next section to obtain the refined Euler-Lagrange conditions without anyrelaxation.

COROLLARY 6.6. Let the arc 2(.) provide a strong local minimum for the Bolzaproblem (1.1) and (1.3) obtained by ignoring the differential inclusion (1.2). Supposethat for some number # > 0 and open set U C Rn one has:

(6.17) 2(t) E U Vt E [a,b] and I)(t)l <# a.e. t [a,b];

f(x,v,.) is bounded and continuous for a.e. t e [a,b] uniformly in (x,v) e U(#B); f(.,., t) is Lipschitz continuous on U (#B) uniformly in t e [a, b]; and (H5’),(n6) hold. Then there exists an absolutely continuous function p: [a, b] R suchthat one has (6.2) with )-- 1 and

(6.18) [9(t) e co{ul(u,p(t)) e Of(2(t),:2(t),t)} a.e. t e [a,b].

Proof. The boundedness of)(t) in (6.17) means that 2(.) e Wl’[a, b]. Accordingto Proposition 2.4, 2(.) is a strong minimum for the relaxed problem (1.3), (2.4). Letus consider the (trivial) differential inclusion

(6.19) 2 e F(x,t):= #B a.e. t e [a,b].

It is obvious that 2(.) is an i.r.l.m, for problem (P) in (1.1), (1.3), and (6.19) whereall the assumptions of Theorem 6.1 are fulfilled. Moreover, (2(t),)(t)) (int gph F)for a.e. t [a, b]. In this case, (6.1) is equivalent to (6.18).

Remark 6.7. Using the modified discrete approximation of the Bolza functionalin Remark 3.4 and developing the procedure in the proof of Theorem 6.1, one canextend the results obtained for the case of integrands f measurable in t.

7. The Euler-Lagrange inclusion without relaxation. In the concludingsection of the paper we consider the following Mayer problem (PM) for differentialinclusions"

minimize J[x] := o(x(a),x(b))

over absolutely continuous trajectories of (1.2) under geometric, inequality, and equal-ity type endpoint constraints (1.3), (6.12), and (6.13). Obviously, problem (PM) isa special case of the Bolza problem (P) with f 0. On the other hand, the Bolzaproblem can be reduced to the Mayer form involving unbounded differential inclusions;see, for example, [8], [24]. Here we consider problem (PM) under the boundednessassumption which is used in our technique.

The main objective of this section is to prove the refined Euler-Lagrange condi-tions for the nonconvex Mayer problem (PM) without any relaxability assumption. Toaccomplish it, we employ the results in the previous section (namely, Corollary 6.6)and an additional approximation procedure combining ideas in [7], [22], [28]. Thelatter procedure allows us to approximate the Mayer problem under considerationby a sequence of nonsmooth Bolza problems without differential inclusions and end-point constraints. In this way, we prove the refined Euler-Lagrange conditions for anystrong local minimum in (PM) and also for any boundary trajectory of a nonconvex

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


differential inclusion. Moreover, these conditions are justified for a weak local min-imum in (PM) under an additional Riemann integrability assumption that simplifiesthe technique employed.

Keeping here assumptions (H1), (H2), and (H6) on F around the given trajectoryfor (1.2), we relax the Lipschitz continuity of in (gh) and Remark 6.3. Namely, weassume the following.

(H5) The functions i are lower semicontinuous for 0,..., q and continuousfor q + 1,..., q + r; the set is closed around (2(a), 2(b)).

Consider the closed set

:=

i(Xo, Xl) <_ui for i=0,...,q and

i(xo, xl) Ui for i=q+l,...,q+r}

and the essential endpoint Lagrangian

(7.2)q+r

L(xo,xl,Ao,..., Aq+):= E Aii(x’xl) qt_ (((X0, Xl), a).i--0

THEOREM 7.1. Let 2(.) be a strong minimum for the Mayer problem (PM)under assumptions (nl), (H2), (Hh), and (HT). Then there exist a vector y*(Ao,..., Aq+r) E Rq+r+l and an absolutely continuous function p" [a, b] Rn, notboth zero, such that

(7.3) [9(t) e coDF(2(t), (t), t)(-p(t)) a.e. t e [a, b],

(7.4) (p(a), -p(b), -y*) e N((2(a),2(b), ); 3)

where := (o(2(a),2(b)), 0... ,0) e Rq++l. Condition (7.4) always implies (6.15)and (6.16). Moreover, (7.4) is equivalent to the simultaneous fulfilment of (6.15),(6.16), and

(7.5) (p(a), -p(b)) e OLa(., Ao,..., Aq+)(2(a),2(b))

if all are Lipschitz continuous around (2(a),2(b)).Proof. Let :-- o(2(a),2(b)). According to the metric approximation method

in Mordukhovich [26-28], we consider the parametric functional

(7.6) Iv[x dist((x(a),x(b), c),$a) with c (y, 0,..., 0) E Rq++ e R,

on trajectories for the differential inclusion (1.2). Let U C R be a bounded neigh-borhood of the strong minimum 2(.) where assumptions (HI), (H2), nd (Hh) arefulfilled. For every > 0, one has

if /is close to . On the other hand

L[x] > 0 for any ’ < "whenever the trajectory x(.) for (1.2) belongs to the neighborhood U of the strongminimum 2(.).

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


Following Clarke [7], let us consider the set X of all trajectories x(.) for (1.2)satisfying x(t) E clU in [a, b] and define a metric in X as follows:

b

(7.7) d(x, y) := Ix(a) y(a)l + lie(t) $(t)ldt.

One can easily see that X is a complete metric space with metric (7.7) and functional(7.6) is continuous in X for any 7. According to the constructions above, for every> 0 we find % < such that % as -- 0, I[2] < , and

Ye[x] >0 for Ye’-Ywhere x(.) is any trajectory for (1.2) with x(t) e U in [a, b]. Now one can apply theEkeland variational principle [14] and claim the existence of x(.) X such that

(7.9) d(x,2) <_ xfl and

IE[x] + x/d(x,x) >_ I[x] Vx(.) X.

Note that (7.9) implies x(t) U for small enough, so I[xE] > 0 by virtue of (7.8).Now for any positive numbers M, and the Lipschitz constant 1F in (3.2), we

define the functional

(7.10) JM[x] I[x] + v/d(x,x) + M(1 + 12) 1/ faD

dist((x(t), 2(t) ), gphF(., t) )dt

on the set of all arcs x(.) (not necessarily trajectories for (1.2)) satisfying x(t) Uin [a, b]. We omit the proof of the following lemma, which can be furnished by thearguments in Kaskosz and Lojasiewicz [22, Lemmas 1 and 2].

LEMMA 7.2. There exists a number M >_ 1 such that for every (0, l/M) thearc x(.) provides an unconditional strong local minimum for the functional (7.10).

Let us continue the proof of Theorem 7.1. Setting c (%, 0,..., 0), we considerany element (zo,Zl,e) II((x(a),x(b),cE),$a) from the Euclidean projector (4.1)of (x(a),x(b),c) on the set Sa. Using this projection and Leinma 7.2, one canconclude that x(.) provides a strong local minimun for the unconstrained Bolzaproblem

b

(7.11) minimize aE(x(a),x(b)) + f(x(t),ic(t),t)dt

with the endpoint function

i. /(7.12) W(xo, xl) [[xo zo + Ix1 Zl --I(E ee[ 2] -[- V/IX0 x(a)[

and the integrand

(7.13) f(x, v, t) M(1 + 12g)/2dist((x, v), gphF(., t)) + x/lv 2(t)[.

Now we employ Corollary 6.6 in the unconstrained Bolza problem (7.11)-(7.13)taking into account the modified discrete approximation in Remark 3.4 for the last(measurable) term in (7.13); el. the proof of Theorem 6.1. One can easily checkthat the assumptions in Theorem 7.1 ensure the fulfilment of the assumptions in

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


Corollary 6.6 around the solution xe(.) and the first term in (7.12) is smooth around(xe(a),xe(b)) by virtue of (7.8).

For each e > 0, using the result in Corollary 6.6 and also calculus formulas (4.7)and (4.8) for functions (7.12) and (7.13), we find an arc p(.) such that

(7.14) ge(t) e co{ul(u,p(t)) e N((xe(t),ce(t));gphF(.,t)) + /(0, B)}

for a.e. t E [a, b] and

(7.15) pc(a) (x(a) zo)/c + x/, -p(b) (xe(b) zle)/ce

where ce := [)xe(a) z0e] 2 + Ixe(b) Zlel 2 --lee et] 1/ > 0.Denoting y := (ee- ce)/ce, one has

(7.16) [pe(a)l 2 + Ipe(b)[ 2 + [yl 2 1

and

(7.17)

Now let us consider the limiting procedure in (7.14)-(7.17) as e --. 0. By virtue of(7.7), relationship (7.9) means that xe(’) --, (’) in wl’c[a, b] as e 0. This impliesthat xe(t) --, 2(t) uniformly in [a, b] and 2e(t) )(t) for a.e. t [a, b].

Further, using (7.14), (7.16), (4.11), and Proposition 4.5, one can conclude (el.the proof of Theorem 6.1) that there exist an arc p(.) and a vector y* Rq+r+l, notboth zero, such that y converges to y*, pc(t) converges to p(t) uniformly in [a, b],and a convex combination of/Se(t) converges to ib(t) for a.e. t [a, b] as e --, 0 alongsome subsequence.

Now passing to the limit in (7.14), (7.16), (7.17) and using Definitions 4.1 and 4.3as well as robustness of the normal cone (4.2), one gets the main conclusions (7.3) and(7.4) of the theorem. Representing y* (),0,...,,q+r), we obtain (6.15) and (6.16)directly from Proposition 4.4(i) where

gphF gn, g (o,...,q+), 2 (2(a),2(b)), (0 0),

A ((#o,...,#q+,)l#i <_ 0 for 0,...,q and #i 0 for q + 1 ,q + r}.

If all i are locally Lipschitzian around (2(a),2(b), then the equivalence of (7.4)to the simultaneous fulfilment of (6.15), (6.16), and (7.5) follows from Proposition4,4(ii) where the scalarization function s(2, y*) is reduced to the essential endpoint La-grangian (7.2) in the case under consideration. This ends the proof of thetheorem, cl

COROLLARY 7.3. Let all i be Lipschitz continuous around (2(a),2(b)) in theframework of Theorem 7.1. Then, in addition to (7.3), one has the transversalityinclusion

(p(a),-p(b)) O ( (Sc(a),2(b)) + N((Sc(a),2(b));a)

where ,ki satisfies (6.15), (6.16) and (/o,... ,Aq+,p(’)) 7 O.

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


Proof. One can derive (7.18) directly from the sum rule (4.8) and representa-tion (4.6). Note that (7.18) implies transversality conditions of the corresponding"separated" form in Remark 6.3.

COROLLARY 7.4. Let 2(.) be a weak local minimizer for problem (PM)- If, inaddition to the assumptions of Theorem "/.t, :2(.) is Riemann intregrable on [a,b],then all the conclusions of the theorem hold true.

Proof. By definition of the weak local minimum for (PM), there exists e > 0 suchthat 2(.) provides the strong (actually global) minimum for the auxiliary problem(P) of the same structure involving the differential inclusion

e t) := t) t)

where G(x, t) := {v R" such that Iv :)(t)l _< e}. Due to a.e. continuity of :)(.)on [a, b], the differential inclusion (7.19) satisfies all the assumptions in Theorem 7.1.

Now we can employ the necessary conditions of Theorem 7.1 in problem (P).Taking into account the structure of G in (7.19) and using the intersection rule (4.9)for representing DF, we arrive at the required Euler-Lagrange inclusion (7.3) forthe weak local minimizer 2(.) to the original Mayer problem (PM). [-I

Remark 7.5. One can see that the Riemann integrability (a.e. continuity) assump-tion on )(.) in Corollary 7.4 is required to satisfy the Hausdorff continuity assumption(H2) on F(x, .) for a.e. t [a, b]. This is essential in the proof of general results inTheorem 6.1 based on discrete approximations. However, for applications to theMayer problem in the proof of Theorem 7.1, we need to use only the result of Corol-lary 6.6 that was obtained for the classical (but nonsmooth) Bolza problem withoutdifferential inclusions. The latter result could be extended to the case of integrandsf measurable in t; see Remark 6.7. In this way, one can obtain the Euler-Lagrangeconditions in Theorem 7.1 for Mayer problems involving differential inclusions withthe measurable t-dependence corresponding to the measurability of the distance func-tion integrands (7.13) in the approximation problems. Now using the procedure inCorollary 7.4, we could justify the refined Euler-Lagrange inclusion (7.3) for any(Lipschitz continuous) weak minimizer to the nonconvex Mayer problem (PM) underconsideration.

Remark 7.6. If F is convex-valued, then the Euler-Lagrange inclusion (7.3) auto-matically implies the Weierstrass-Pontryagin maximum condition

(7.20) (p(t),:2(t)} max{(p(t),v)l v E F(2(t),t)} a.e. t [a,b]

due to Proposition 4.7. It is no longer true in the nonconvex setting for arbitraryweak or intermediate local minimizers. But we believe that for the case of strongminimum, the refined Euler-Lagrange conditions in Theorem 7.1 are fulfilled simul-taneously with the maximum condition (7.20) for nonconvex differential inclusions.This statement can be obtained directly from the proof of Theorem 7.1 if one estab-lishes the classical Weierstrass necessary condition for the strong local minimum to thesimplest variational problem like (7.11)-(7.13) with no smoothness and/or convexityassumptions.

Now let us consider an analogue of the results obtained for the case of boundarytrajectories. Given a nonempty closed set A c Rn, we denote by R(A) the reachableset for (1.2) from A, i.e., the set of all points x(b) where x(.) is a trajectory for (1.2)with x(a) A.

THEOREM 7.7. Let 5c(.) be a trajectory for (1.2) with 5c(a) A, and let as-sumptions (H1), (H2), (H7) hold. If g: an --+ aTM i8 a locally Lipschitzian function

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


around 2(b) and if g(2(b)) is a boundary point of the set R(A), then there exist an arcp: [a, b] an and a ’unit vector e Rm such that p(.) satisfies the refined Euler-Lagrange inclusion (7.3) with the following boundary transversality) conditions:

(7.21) p(a) e N(2(a); A), -p(b) e

Remark 7.8. The result formulated generalizes the recent one in Kaskosz and Lo-jasiewicz [22] where (7.3) is replaced by Clarke’s form of the Euler-Lagrange inclusion(see (1.4) in the case where f 0) and conditions (7.21) are replaced by

p(a) E Nc(2(a); A), p(b) [Jcg(2(b))]*2

in terms of Clarke’s normal cone and generalized Jacobian; cf. (4.3) and (4.14).Proof of Theorem 7.7. Following the arguments in [22], for every > 0 we can

find a vector c R" and a trajectory x(.) of (1.2) with x(a) A such that> 0,

c--, g(2(b)), x(.) (.)in Wl’[a,b] as 0,

and x(.) is an unconditional strong local minimizer for problem (7.11) with integrand(7.13) and the endpoint functional

(7.22) e(x0, xl) := Ig(xl) c + v/lxo x(a) + Mdist(xo, A).

Now employing Corollary 6.6 in problem (7.11), (7.13), (7.22) and using calculus rules(4.7), (4.8), and (4.10) to compute the subdifferential of (7.22), one gets an arc p(.)and a unit vector Rm such that (7.14) holds and

(7.23) p(a) v/B + N(x(a); A), -p(b) 0(, g}(x(b)).

Following the proof of Theorem 7.1, we obtain conditions (7.3) and (7.21) by passingto the limit in (7.14) and (7.23) as

Remark 7.9. Using the method of metric approximations (as in the proof ofTheorem 7.1), one can extend Theorem 7.7 to a more general setting when g(2(b)) isa locally extremal point of the set R(A) relative to other given sets. We refer to [28,6] and [33, 3] for more details about this concept.

Remark 7.10. Following the proof of Theorem 7.7, we can obtain, in addition to(7.3) and (7.21), the maximum condition (7.20) for boundary trajectories of nonconvexdifferential inclusions if one has the Weirestrass necessary condition for the strongmimimum to the simplest variational problem (7.11) with nonsmoothness (7.13) and(7.22).

Acknowledgments. The author is indebted to Asen Dontchev, Alexander Ioffe,Richard Vinter, and two anonymous referees for their helpful remarks.

REFERENCES

[1] V. M. ALEKSEEV, V. M. TIKHOMIROV, AND S. V. FOMIN, Optimal Control, Nauka, Moscow,1979 (in Russian); English transl, in Consultants Bureau, New York, 1987.

[2] J.-P. AUBIN, Lipschitz behavior of solutions to convex minimization problems, Math. Oper.Res., 9 (1984), pp. 87-111.

[3] J.-P. AUBIN AND H. FRANKOWSKA, Set-Valued Analysis, Birkhuser, Boston, 1990.[4] N. N. BOGOLJUBOV, Sur quelques methodes nouvelles dans le calculus des variations, Ann.

Math. Pura Appl., 7 (1930), pp. 249-271.

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


[5] F. H. CLARKE, Optimal solutions to differential inclusions, J. Optimization Theory Appl., 19(1976), pp. 469-478.

[6] , The generalized problem o Bolza, SIAM J. Control Optim., 14 (1976), pp. 682-699.[7] , Necessary conditions for a general control problem, in Calculus of Variations and Con-

trol Theory, D. Russel, ed., Academic Press, New York, 1976, pp. 257-278.[8] , Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, 1983.[9] , Hamiltonian analysis of the generalized problem of Bolza, Trans. Amer. Math. Soc.,

301 (1987), pp. 385-400., Methods of Dynamic and Nonsmooth Optimization, Society for Industrial and Applied

Mathematics, Philadelphia, PA, 1989.M. G. CRANDALL AND P.-L. LIONS, Viscosity solutions of Hamilton-Jacobi equations, Trans.

Amer. Math. Soc., 277 (1983), pp. 1-42.A. L. DONTCHEV AND E. M. FARKHI, Error estimates for discretized differential inclusions,

Computing, 41 (1989), pp. 349-358.A. L. DONTCHEV AND F. LEMPIO, Difference methods for differential inclusions: a survey,

SIAM Rev., 34 (1992), pp. 263-294.!. EKELAND, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324-353.I. EKELAND AND R. TEMAN, Convex Analysis and Variational Problems, North-Holland, Am-

sterdam, 1976.H. FRANKOWSKA, The maximum principle for an optimal solution to a differential inclusion

with endpoint constraints, SIAM J. Control Optim., 25 (1987), pp. 145-157.A. D. IOFFE, Approximate subdifferentials and applications, I: the finite dimensional theory,

Trans. Amer. Math. Soc., 281 (1984), pp. 389-416., Approximate subdifferentials and applications, III: the metric theory, Mathematika, 36

(1989), pp. 1-38., Nonsmooth subdifferentials: their calculus and applications, in the Proceedings of the

1st World Congress of Nonlinear Analysts, W. de Gruyter, Berlin, 1995.A. D. IOFFE AND V. M. TIKHOMIROV, Extensions of variational problems, Trans. Moscow Math.

Soc., 18 (1968), pp. 207-273. (in Russian.), Theory of Extremal Problems, Nauka, Moscow, 1974 (in Russian); English transl.,

North-Holland, Amsterdam, 1979.B. KASKOSZ AND S. LOJASIEWICZ JR., Lagrange-type extremal trajectories in differential inclu-

sions, Systems Control Lett., 19 (1992), pp. 241-247oP. D. LOEWEN, Optimal Control via Nonsmooth Analysis, CRM Proceedings and Lecture

Notes, Vol. 2, American Mathematical Society, Providence, RI, 1993.P. D. LOEWEN AND R. T. ROCKAFELLAR, The adjoint arc in nonsmooth optimization, Trans.

Amer. Math. Soc., 325 (1991), pp. 39-72., Optimal control of unbounded differential inclusions, SIAM J. Control Optim., 34

(1994), pp. 442-470.B. S. MORDUKHOVICH, Maximum principle in problems of time optimal control with nonsmooth

constraints, J. Appl. Math. Mech., 40 (1976), pp. 960-969.Metric approximations and necessary optimality conditions for general classes of non-

smooth extremal problems, Soviet Math. Dokl., 22 (1980), pp. 526-530.Approximation Methods in Problems of Optimization and Control, Nauka, Moscow,

1988; revised English transl, to appear, Wiley-Interscience., On variational analysis of differential inclusions, in Optimization and Nonlinear Anal-

ysis, A. Ioffe et al., eds., Pitman Res. Notes Math. Ser. 244, Longman Sci. Tech., Harlow,1992, pp. 199-213.

[30] , Sensitivity analysis in nonsmooth optimization, in Theoretical Aspects of IndustrialDesign, D. A. Field and V. Komkov, eds., Society for Industrial and Applied Mathematics,Philadelphia, PA, 1992, pp. 32-46.

[31] , Complete characterization of openness, metric regularity, and Lipschitzian properties

of multifunctions, Trans. Amer. Math. Soc., 340 (1993), pp. 1-35.[32] , Lipschitzian stability of constraint systems and generalized equations, Nonlinear Anal.

Theory Methods Appl., 32 (1994), pp. 173-206.[33] , Generalized differential calculus for nonsmooth and set-valued mappings, J. Math. Anal.

Appl., 182 (1994), pp. 250-288.[34] , Stability theory for parametric generalized equations and variational inequalities via

nonsmooth analysis, Trans. Amer. Math. Soc., 343 (1994), pp. 609-658.[35] Optimization and finite difference approximations of differential inclusions with free

time, in the Proc. IMA Workshop on Nonsmooth Analysis and Geometric Nethods inDeterministic Optimal Control, B. S. Mordukhovich and H. J. Sussmann, eds., Springer-Verlag, New York, 1995.

[10]

[11]

[12]

[13]

[14]

[16]

[17]

[18]

[19]

[2o]

[21]

[22]

[23]

[24]

[25]

[26]

[27]

[28]

[29]

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


[36] E. POLAK, Computational Methods in Optimization: A Unified Approach, Academic Press,New York, 1971; 2nd revised edition, Academic Press, to appear.

[37] E. S. POLOVINKIN AND G. V. SMIRNOV, An approach to differentiation of multifunctionsand necessary optimality conditions for differential inclusions, Differential Equations, 22(1986), pp. 660-668.

[38] B. N. PSHENCHN, Convex Analysis and Extremal Problems, Nauka, Moscow, 1980. (Russian)[39] R. T. ROCKAFELLAR, Conjugate convex functions in optimal control and the calculus of vari-

ations, J. Math. Anal. Appl., 32 (1970), pp. 174-222.[40] , Extensions of subgradient calculus with applications to optimization, Nonlinear Anal.

Theory Methods Appl., 9 (1985), pp. 665-698.[41] , Lipschitzian properties of multifunctions, Nonlinear Anal. Theory Methods Appl., 9

(1985), pp. 867-885.[42] , Maximal monotone relations and the second derivatives of nonsmooth functions, Ann.

Inst. H. Poincar. Anal. Non Lindare, 2 (1985), pp. 167-184.[43] , Dualization of subgradient conditions for optimality, Nonlinear Anal. Theory Methods

Appl., 20 (1993), pp. 627-646.[44] , Lagrange multipliers and optimality, SIAM Rev., 35 (1993), pp. 183-238.[45] R. W. ROCKAFELLAR AND R. J.-B. WETS, Variational Analysis, Springer-Verlag, New York, to

appear.[46] J. D. L. ROWLAND AND R. B. VINTER, Dynamic optimization problems with free time and

active state constraints, SIAM J. Control Optim., 31 (1993), 677-691.[47] B. SENDOV AND V. A. POPOV, The Averaged Moduli of Smoothness, Wiley-Interscience, New

York, 1988.[48] G. V. SMIRNOV, Discrete approximations and optimal solutions to differential inclusions,

Kibernetika (1991), pp. 76-79. (Russian)[49] R. B. VINTER AND P. D. WOODFORD, On the occurence of intermediate local minimizers that

are not strong local minimizers, Preprint, April 1994.[50] J. WARGA, Optimal Control of Differential and Functional Equations, Academic Press, New

York, 1972.[51] , Controllability, extremality, and abnormality in nonsmooth optimal control, J. Optim.

Theory Appl., 41 (1983), pp. 239-260.[52] L. C. YOUNG, Lectures on the Calculus of Variations and Optimal Control Theory, Saunders,

Philadelphia, PA, 1969.

Dow

nloa

ded

09/1

7/13

to 1

30.1

5.24

1.16

7. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p

Documents

Discrete Approximations and Refined Euler–Lagrange Conditions for Nonconvex Differential Inclusions