Semigeodesics and the minimal time function. (English) Zbl 1114.49028

Nour, Chadi

ESAIM, Control Optim. Calc. Var. 12, 120-138 (2006).

A time-optimal control problem is considered on solutions of a differential inclusion \(\dot{x}\in F(x)\) with a terminal condition \(x(T)\in S\). Here \(F(x)\) is a convex-compact-valued locally Lipschitz-continuous multifunction satisfying a linear growth condition, and \(S\) is a nonempty compact set in \({\mathbb R}^n\). A solution to the differential inclusion, \(x(t)\), \(t\geq0\), is called semigeodesic if every its part is time-optimal. A trajectory is called geodesic, if the same holds true for \(t\in(-\infty,\infty)\).
The main focus is finding local-controllability conditions under which the optimal time as a function of initial point is a proximal solution to a Hamilton-Jacobi equation with the Hamiltonian expressed through the support function of the right-hand side \(F(x)\). The results generalize the conclusions of previous work of the author and F. H. Clarke [J. Convex Anal. 11, No. 2, 413–436 (2004; Zbl 1072.49018)]. In that earlier work a similar problem has been considered for a single-point terminal condition. In fact, majority of proofs are given in form of references to the respective proofs of the earlier work. Somewhat artificial examples illustrate the role of the assumptions in the theorems.

Reviewer: Dmitry Silin (Berkeley)

MSC:

49L20	Dynamic programming in optimal control and differential games
49J24	Optimal control problems with differential inclusions (existence) (MSC2000)
34A60	Ordinary differential inclusions

Keywords:

Hamilton–Jacobi equation; proximal solution; time-optimal control; differential inclusion

Citations:

Zbl 1072.49018

References:

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