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Strong invariance and one-sided Lipschitz multifunctions. (English) Zbl 1068.34010
The paper is devoted to the study of some viability issues concerning differential inclusions in the Euclidean space $\bbfR^n$. The authors study the Cauchy problem $$x'\in F(t, x), \quad x\in S, \quad t\in I= [t_0, t_1), \quad x(t_0)= x_0\in S,\tag $*$ $$ where $S\subset\bbfR^n$ 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 $H_F(t,x,x- y)- H_F(t,y,x- y)\le k(t)\Vert x- y\Vert$ 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 $N^P_S(x)$ at $x\in S$. The authors study also the situation of time dependent state constraints $S(t)$, $t\in I$, and show conditions equivalent to the weak and strong invariance. These conditions are stated in terms of Hamilton-Jacobi-type inequalities.

34A60Differential inclusions
49J53Set-valued and variational analysis
49L99Hamilton-Jacobi theories, including dynamic programming
Full Text: DOI
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