Mixed semicontinuous perturbations of nonconvex sweeping processes
The paper studies sweeping processes of the form \[ -u'(t)\in N(C(t),u(t))+G(t,u(t))\quad a.e.\; ([T_0,T]),\quad u(T_0)=u_0\in C(T_0),\tag{1} \] where for any \(t\in [T_0,T]\), \(C(t)\) are closed prox-regular sets of a finite dimensional Hilbert space \(H\), \(N(C(t),u(t))\) is the normal cone of \(C(t)\) at \(u(t)\in C(t)\) and \(G(.,.):[t_0,T]\times H\to {\mathcal P}(H)\) is a set-valued map. When the set-valued map \(G(.,.)\) is measurable, satisfies a linear growth inclusion condition and \(G(t,.)\) satisfies a certain mixed semicontinuity property the authors prove the existence of a global solution of problem (1). This result allows to obtain an existence and uniqueness theorem of a periodic solution of problem (1) whenever \(C(T_0)=C(T)\) and the set-valued map \(G(t,.)\) satisfies some monotonicity conditions.
scientific article; zbMATH DE number 5675094