This article is cited in
2 papers
The equation of dynamic programming for a time-optimal problem with phase constraints
V. A. Komarov
Abstract:
The time-optimal problem with a phase constraint given by a compact set
$K$ is considered for a differential inclusion
$\dot x\in F(x)$ with right-hand side that is upper semicontinuous, convex, and compact for all
$x\in F^n$. It is shown that a nonnegative lower semicontinuous function
$\tau(x)$ vanishing only on the terminal set
$M$ and continuous on the solutions of the differential inclusion
$\dot x\in-F(x)$ is the optimal time in this problem if it satisfies the relation
$$
\min_{f\in F_K(x)}D^+\tau(x;f)=-1.
$$
for all
$x$ with
$\tau(x)<\infty$. Here
$D^+\tau(x;f)$ is the upper contingent derivative of
$\tau$ in the direction of
$f$,
$F_K(x)=T_K(x)\cap F(x)$, and
$T_K(x)$ is the lower contingent tangent cone to
$K$ at the point
$x$. It is also shown that if
$F$ is continuous and
$\tau$ satisfies a one-sided Lipschitz condition, then the conditions given are necessary.
Figures: 1.
Bibliography: 14 titles.
UDC:
517.9
MSC: Primary
34A60,
49B10,
49C20; Secondary
49E10,
49E15 Received: 16.10.1986