Abstract
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.
Highlights
Consider the optimal control problem ⎧ (P) ⎪⎪⎪⎪⎪⎪⎨Minimize g(T, x(0), x(T )) over T ≥ 0, absolutely continuous functions x(·)and measurable functions u(·) satisfying : [0, T ] → Rn
In the first we focus attention on a strong local minimizer which cannot 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 processes
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 to that of the relaxed strong local minimizer
Summary
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. 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. For details of definition and properties of these objects, we refer the reader to [2, 7, 8]
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