4. A. E. Rapaport, R. B. Vinter,
Invariance Properties of Time Measurable Differential Inclusions and
Dynamic Programming, JDCS vol. 2, No. 3, p. 423--448
Take a multifunction $F: [0,T]\times\R^{n}\leadsto \R^{n}$ and
$P: [0,T] \rightarrow \R^{n}$. $P$ is said to be weakly invariant (or
``viable'') for $F$ if for any $x_{0} \in P(0)$ there exists a
solution $x$ to $\dot{x} \in F(t,x)$ satisfying the constraint $x(t)
\in P(t)$ for all $t$. If all solutions with initial value in $P(0)$
satisfy the constraint then $P$ is strongly invariant for $F$. Weak
and strong invariance are important concepts connected with existence
of optimal controls and stabilizing controls, and dynamic
programming. Weak and strong invariance have previously been proved
for multifunctions $F$, which are assumed merely to be measurable with
respect to the time variable, under a regularity hypothesis on $P$
(``local absolute continuity from the left''). We show how the
regularity hypothesis can be reduced when {\sl a priori\/} knowledge
is available of a closed subset $I \subset [0,T]$ such that $F$ is
upper semicontinuous at all points in $I\times\R^{n}$. The invariance
theorems are applied to characterize the value function for optimal
control problems with endpoint constraints as the unique generalized
solution of the Hamilton--Jacobi equation, under hypotheses which are
in certain cases less restrictive than those imposed hitherto.
See the title page of this paper