*(English)*Zbl 1068.34010

The paper is devoted to the study of some viability issues concerning differential inclusions in the Euclidean space ${\mathbb{R}}^{n}$. The authors study the Cauchy problem

where $S\subset {\mathbb{R}}^{n}$ is closed, $F$ has convex compact values, is almost upper semicontinuous and integrably bounded. In order to establish the strong invariance (strong viability), that is to show that any solution to $(*)$ with ${x}_{0}\in S$ remains in $S$, they introduce the max-Hamiltonian ${H}_{F}(t,x,y)={sup}_{z\in F(t,x)}\langle z,y\rangle $ and, instead of the Lipschitz continuity of $F$ which has been usually assumed to get the strong invariance, they assume that ${H}_{F}$ satisfies the so-called one sided Lipschitz estimates of the form ${H}_{F}(t,x,x-y)-{H}_{F}(t,y,x-y)\le k\left(t\right)\parallel x-y\parallel $ where $k$ is locally integrable. They further prove that, under these assumptions, $S$ is strongly invariant, provided that the for almost all $t$, asymptotically $H(t,x,y)\le 0$ for all vectors $y$ in the proximal normal cone ${N}_{S}^{P}\left(x\right)$ at $x\in S$.

The authors study also the situation of time dependent state constraints $S\left(t\right)$, $t\in I$, and show conditions equivalent to the weak and strong invariance. These conditions are stated in terms of Hamilton-Jacobi-type inequalities.

##### MSC:

34A60 | Differential inclusions |

49J53 | Set-valued and variational analysis |

49L99 | Hamilton-Jacobi theories, including dynamic programming |