The equation of dynamic programming for a time-optimal problem with phase constraints

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.
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