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An equation of dynamic programming for a time-optimal problem with phase constraints. (Russian) Zbl 0654.49013
The time-optimal control problem for the finite-dimensional, autonomous differential inclusion (1) $$\dot x\in F(x)$$ with target set M and with restricted phase coordinates, x(t)$$\in K\supset M$$ for all t, is studied. It is assumed that the sets F(x) are non-empty, compact and uniformly bounded, that F($$\cdot)$$ is upper semicontinuous and that M, K are closed sets. Let T(x) denote the optimal time in which the trajectory of system (1), starting from the initial state x, attains M. By using the dynamic programming method, some sufficient conditions for T(x) to be such an optimal time function are obtained. For continuous F($$\cdot)$$ and for T($$\cdot)$$ satisfying a one-sided Lipschitz condition these conditions are also necessary.
As an example, the time-optimal control problem for the system ẍ$$=u$$, $$| u| \leq 1$$ with target (0,0) and with K determined by the inequality $$| \dot x| \leq 1$$ is studied.
Reviewer: Z.Wyderka

##### MSC:
 49L20 Dynamic programming in optimal control and differential games 93B05 Controllability 90C39 Dynamic programming 49J45 Methods involving semicontinuity and convergence; relaxation 34A60 Ordinary differential inclusions 49K15 Optimality conditions for problems involving ordinary differential equations
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