Functional evolution equations governed by nonconvex sweeping process (Q2764705)

The authors study the functional-differential inclusion \((P_\tau)\) governed by a nonconvex sweeping process for \(u : [-r, T]\to\mathbb R^n\) NEWLINE\[NEWLINE\dot u(t)\in -N_{C(t)}(u(t))+F(t,\tau(t)u)\text{ a.e. on }[0, T],NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(s)=\phi(s)\text{ for }s\in [-r,0],\quad u(t)\in C(t)\text{ for }t\in [0,T],NEWLINE\]NEWLINE with \(r,T>0\); \(C:[0,T]\to 2^{\mathbb R^n}\) ``varies in an absolutely continuous way'' as defined in section 1 and is compact, \(p\)-proximally regular-valued for some \(p\in [0,\infty]\); \(N_{C(t)}(u(t))\), represents the Clarke normal cone of \(C(t)\) at \(u(t)\); \(F(t,x) : [0,T]\times C_{\mathbb R^n}([-r,0])\to 2^{\mathbb R^n}\) is measurable in \(t\), upper semicontinuous in \(x\), satisfies a growth condition and is convex, compact-valued; for each \(t \in [0,T]\), \(\tau(t) : C_{\mathbb R^n}([-r,t])\to C_{\mathbb R^n}([-r,0])\) is defined by \((\tau(t)u)(s) =u(t+s)\) and \(\phi\in C_{\mathbb R^n}([-r,0])\) with \(\phi(0)\in C(0)\). In section 1, questions of existence, uniqueness and continuous dependence of solutions along with compactness of the solution set are considered for differential inclusions without delay which are related to \((P_\tau)\). In section 2, these results are used to prove that the solution set to \((P_\tau)\) is nonempty and compact. Finally, applications to optimal control are given in section 3.