Existence of solutions to bilinear problems with a closed-loop control

Abstract

Here we prove the existence of solutions to nonlinear differential inclusion problems with closed-loop control z + A (z ) = B (u,z ) + f, u ∈ U (t,z ),z (0) = z 0 where the operator B is bilinear with respect to the control u and the state z in reflexive, separable Banach spaces denoted Y and V , respectively. The operator A is nonlinear in V , and given a positive real number T , the set-valued map U is defined in [ 0,T ] × V . Without making any assumptions about the convexity of U , its values are taken to be non-empty closed, decomposable subsets of Y .