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The value function in optimal control: Sensitivity, controllability, and time-optimality. (English) Zbl 0601.49020

The following perturbed optimization problem is considered: find \[ \begin{split} (V(\alpha):=\min \{f(T,x(0),x(T),\alpha)\mid \dot x(t)\in F(x(t),\alpha)\text{ a.e. on }[0,T],\;x: [0,T]\to {\mathbb{R}}^ n \\ \text{is absolutely continuous, and }(T,x(0),x(T),\alpha)\in S\} \end{split} \] where both the function f and the multifunction F are locally Lipschitzian, the values of F are nonempty compact convex sets, and S is closed. Under some additional hypotheses, a formula for the generalized gradient \(\partial V(0)\) is proved. The result is similar to that obtained in Theorem 3.4.3 of F. H. Clarke’s textbook [Optimization and nonsmooth analysis (1983; Zbl 0582.49001)] for a fixed- time control problem with additive perturbations of the endpoint constraints.
Next, the authors derive a series of conditions under which, successively, V is locally finite, locally Lipschitzian, admits directional derivatives, or is actually differentiable. Finally, a special study is made of the time-optimal control problem, one consequence of which is a new criterion assuring local null- controllabilty of the system and continuity of the minimal time function at the origin.
Reviewer: M.Studniarski

MSC:

49K40 Sensitivity, stability, well-posedness
49J50 Fréchet and Gateaux differentiability in optimization
93B03 Attainable sets, reachability
49J45 Methods involving semicontinuity and convergence; relaxation
93B05 Controllability
93B35 Sensitivity (robustness)
93C15 Control/observation systems governed by ordinary differential equations

Citations:

Zbl 0582.49001
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