Free End-Time Optimal Control Problems: Conditions for the Absence of an Infimum Gap

This paper concerns free end-time optimal control problems, in which the dynamic constraint takes the form of a controlled differential inclusion. Such problems may fail to have a minimizer. Relaxation is a procedure for enlarging the domain of an optimization problem to guarantee existence of a minimizer. In the context of problems studied here, the standard relaxation procedure involves replacing the velocity sets in the original problem by their convex hulls. It is desirable that the original and relaxed versions of the problem have the same infimum cost. For then we can obtain a sub-optimal state trajectory, by obtaining a solution to the relaxed problem and approximating it. It is important, therefore, to investigate when the infimum costs of the two problems are the same; for otherwise the above strategy for generating sub-optimal state trajectories breaks down. We explore the relation between the existence of an infimum gap and abnormality of necessary conditions for the free-time problem. Such relations can translate into verifiable hypotheses excluding the existence of an infimum gap. Links between existence of an infimum gap and normality have previously been explored for fixed end-time problems. This paper establishes, for the first time, such links for free end-time problems.

We say that a process (T ,x(·),ū(·)) is a minimizer if it achieves the minimum of g(T , x(0), x(T )) over all admissible processes (T , x(·), u(·)). It is called a strong local minimizer if, for some ε > 0,
We write inf(P ), 'the infimum cost', for the infimum value of the cost function g(T , x(0), x(T )) over the set of admissible processes (T , x(·), u(·)).
Hypotheses ensuring the existence of a unique state trajectory for every control function and initial state and the continuous dependence of this state trajectory on these quantities, the continuity of the cost function and closedness of C and the values of U(·), and the nonemptiness and boundedness of the set of admissible processes, are not, alone, sufficient to guarantee existence of solutions to (P). Existence of solutions is assured only if an additional 'geometric' hypothesis is satisfied; this typically places a convexity requirement on the velocity sets f (t, x, U(t)) := {f (t, x, u) | u ∈ U(t)}, namely: Relaxation is a procedure for enlarging the domain of an optimization problem, but not too much, to guarantee existence of minimizers. Here 'not too much' means, that elements in the enlarged domain can be closely approximated by elements in the original domain. Typically, the enlarged domain is the closure, in some sense, of the original domain. Relaxation opens a way, if not to solving the original problem (which might not be possible), but to generating a sub-optimal minimizer of the original problem, i.e an element whose cost is arbitrarily close to the infimum cost of the original problem. The idea is to solve the relaxed problem and approximate the solution by elements in the original domain.
The concept of relaxation makes sense for any optimization problem. In the context of optimal control, however, relaxation is usually taken to mean modifying the optimal control problem in such a manner that the original velocity sets are replaced by their convex hulls. It will be clear from the preceding discussion that the relaxed problem satisfies the extra geometric condition, and therefore has solutions. On the other hand, the Relaxation Theorem, which tells us that relaxed state trajectories can be approximately arbitrarily closely by original state trajectories, w.r.t. the supremum norm, means that relaxation via convexification meets the other 'not too much' requirement of the procedure.
Notice that the velocity sets for the relaxed problem are So the convexity hypothesis (C) is satisfied and, under unrestrictive additional hypotheses, the relaxed problem has a minimizer. Denote the minimum cost by min (R). While relaxed state trajectories can be approximated arbitrarily closely in the supremum norm by non-relaxed state trajectories, it is nonetheless possible that there is an 'infimum gap', i.e., min(R) < inf(P ).
The pathology here is that, while according to the Relaxation Theorem, feasible relaxed state trajectories can be approximated arbitrary closely by non-relaxed state trajectories, doing so may be incompatible with satisfying all the constraints in the original problem. It is important to identify situations where an infimum gap arises because, then, the earlierdiscussed justification for studying the relaxed problem is no longer valid. The aim of this paper is to derive sufficient conditions for absence of an infimum gap, for free end-time optimal control problems. These conditions require necessary conditions of optimality, expressed in terms of the free end-time Pontryagin Maximum Principle, to be satisfied in normal form, i.e. for any valid multiplier set, the cost multiplier component must be non-zero.
Two kinds of relationship between occurrence of an infimum gap and abnormality of associated necessary conditions of optimality are explored. In the first we focus attention on a strong local minimizer which cannot also be interpreted as a strong relaxed minimizer; in the second, on a relaxed minimizer, whose cost is strictly less than the infimum cost over admissible (non-relaxed) processes. The following terminology, relating these properties was introduced in [5]: Type A Relations: A strong local minimizer satisfies the Pontryagin Maximum Principle in abnormal form (i.e., with cost multiplier zero) if, when regarded as a relaxed admissible process, it is not also a relaxed strong local minimizer. Type B Relations: A relaxed strong local minimizer satisfies the relaxed Pontryagin Maximum Principle in abnormal form if its cost is strictly less than the infimum cost over all admissible processes, whose state trajectories are close (in the L ∞ sense) to that of the relaxed strong local minimizer.
The main results of this paper can be summarized as ) is a strong local minimizer (for the free end-time problem) but not a relaxed strong local minimizer' implies '(T ,x(·),ū(·)) satisfies an 'averaged' version of the Pontryagin Maximum Priniciple in abnormal form', and (B): '(T ,x(·),ū(·)) is a relaxed strong local minimizer with cost strictly less than that of any admissible (non-relaxed) process, whose state trajectory is close (in the L ∞ sense) to that of the relaxed strong local minimizer' implies '(T ,x(·),ū(·)) satisfies the relaxed Pontryagin Maximum Principle in abnormal form'.
(A) and (B) will be recognized as forms of Type A and Type B relations, respectively. The relations are proved here, for the first time, for general, non-smooth free end-time optimal control problems. When we limit attention to optimal control problems with fixed initial state x 0 , there is a connection with the conditions excluding an infimum gap, considered in this paper, and sensitivity relations in optimal control. Here, by 'sensitivity relations' we mean information about generalized derivatives of the value function V (·, ·) for the relaxed problem at x 0 . It can be deduced from results in [1] that, under a normality hypothesis on the data for the relaxed problem, the convexified limiting subdifferential of V (0, ·) at x 0 is bounded. This, in turn, implies Lipschitz continuity of V (0, ·) near x 0 . On the other hand, continuity of V (0, ·) near x 0 implies 'no infimum gap', in consequence of the Relaxation Theorem (see, e.g., [8,Theorem 2.7.2]). Combining these facts, we arrive at an independent proof of 'normality of the relaxed problem' implies 'no infimum gap'; this is a Type B property. By contrast, our analysis covers problems for which the initial state is not fixed and also supplements conditions for Type B properties by conditions for Type A properties.
Previous work has addressed, exclusively, fixed end-time optimal control problems. Warga was the first to investigate the relation between the existence of an infimum gap and validity of the Pontryagin Maximum Principle in abnormal form. Warga announced a Type B relation for optimal control problems with smooth data in his early paper [9]. In his monograph [10] he proved a Type B relation for optimal control problems with state constraints. In a subsequent paper [11], Warga generalized his earlier Type B results to allow for nonsmooth data, making use of local approximations based on 'derivative containers', developed in [12]. Type A relations were proved, for the first time, by Palladino and Vinter in [5] and [6], for problems in which the dynamic constraint takes the form of a differential inclusion and a controlled differential inclusion, respectively. Related work by Ioffe is reported in [4].
We point out that, when the time dependence of the dynamic constraint is Lipschitz continuous and the control constraint set is independent of time, free end-time problems can be reduced to fixed end-time problems, of standard form, by a change of independent variable. Thus free time optimal control problems, in this special case, are covered by earlier work. However this reduction is not possible in the setting of this paper, because we allow a measurably time dependent dynamic constraint and a measurably time dependent control constraint set.
The following notation will be employed: for vectors x ∈ R n , |x| denotes the Euclidean length. B denotes the closed unit ball in R n . Given a set A ⊂ R n and a point x ∈ R n , we denote by d A (x) the Euclidean distance of a point x ∈ R n from A: with norm |x(0)|+ ẋ(·) L 1 . Given a, b ∈ R, a∧b and a∨b denote min{a, b} and max{a, b}, respectively.
We write W 1,1 or W 1, when the meaning is clear.
We shall use several constructs of nonsmooth analysis. Given a closed set D ⊂ R k and a pointx ∈ D, the limiting normal cone N D (x) of D atx is defined to be Here For details of definition and properties of these objects, we refer the reader to [2,7,8].

Conditions for Non-Existence of an Infimum Gap
In this section we state two theorems relating the existence of a gap between the infimum costs for the optimal control problem (P) and its relaxed counterpart (R), and the validity of a Pontryagin Maximum Principle in abnormal form. The following hypotheses, in which (T ,x(·),ū(·)) is a given process andε > 0, will be invoked. See [8,Chapter 8], where the main properties of the essential value are stated and proved.

Comments
(1) Part (a) is a known version of the free end-time Pontryagin Maximum Principle for problems with measurably time dependent data (see, e.g. [8,Chapter 8]). (2) Observe that the contrapositive statement of part (b) is a sufficient condition for the absence of an infimum gap (in a local sense): if (T ,x(·),ū(·)) is a strong local minimizer such that, given any multipliers p(·) ∈ W 1,1 , and λ ≥ 0 satisfying conditions (i), (ii) , (iii)-(iv), we have λ = 0, then (T ,x(·),ū(·)) is also a strong local relaxed minimizer. This is a Type A relation. (3) Part (b) can be regarded as a pathway to providing sufficient conditions, of a verifiable nature, for the absence of an infimum gap. The idea is to give verifiable sufficient conditions for normality, which translate, via part (b), into verifiable sufficient conditions for absence of an infimum gap. (4) It is not known whether, in part (b), the 'averaged' costate inclusion (ii) , in part (b), can be replaced by the stronger pointwise condition (ii).

Proofs of Theorems 1 and 2
We first prove Theorem 2, as the proof of Theorem 1 uses that of Theorem 2.
Now apply the free end-time Pontryagin Maximum Principle [8, Chapter 8] to a reformulation of (Q i ) in which the integral cost term has been eliminated by state augmentation. Since right endpoints of state trajectories in (Q i ) are unconstrained, we are justified in setting the cost multiplier to 1. We conclude: for each i there exist an arc p i (·) ∈ W 1,1 (0, T i ), h i ∈ R, and λ i ≥ 0 such that (Notice that the costate arc associated with the state component η i (·) is the zero function, and so does not appear in the above conditions.) Since 'ξ ∈ ∂d C (z) and z / ∈ C implies |ξ | = 1', we deduce from (6) that Now the (extended) functions p e i (·) ∈ W 1,1 (0,T +ε) are uniformly bounded and have uniformly integrably bounded derivatives. It follows that, along some subsequence (we do not relabel), p e i (·) → p e (·) strongly in L ∞ (0,T +ε) andṗ e i (·) →ṗ e (·) weakly in L 1 (0,T +ε) for some p e (·) ∈ W 1,1 . We can also arrange that h i → h, for some h ∈ R.
For any selector u(t) ∈ U(t) we have, from (iii),