Necessary and sufficient conditions for local invariance for semilinear differential inclusions (Q763336)

The authors study local invariance of a set \(K\subseteq X\), where \(X\) is a separable Banach space, for the autonomous semilinear differential inclusion \[ {u^{\prime }}(t) \in Au(t)+F(u(t)), \quad {u(0) = \xi }.\tag{1} \] The assumptions generally include that \(A:D(A)\subseteq X\rightarrow X\) is the infinitesimal generator of a \(C_{0}\)-semigroup \(\left\{ S(t):X\rightarrow X\mid t\geq 0\right\} \), \(F:X\rightarrow 2^{X}\) is closed and bounded-valued, and satisfies a Lipschitz condition, and \(K\) is locally closed. The authors make use of a tangency concept in which a function \(f\in L_{\text{loc}}^{1}\left({R}_{+},X\right) \) is said to be \({A}\)-tangent to the set \({K}\) at the point \({\xi \in K}\) if \[ \lim \inf_{h\downarrow 0}\frac{1}{h}\operatorname{dist}\left( S(h)\xi +\int_{0}^{h}S\left( h-s\right) f(s)ds,K\right) =0. \] Both necessary and sufficient conditions are specified. The final section gives an application to an optimal control problem.