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

Université Paris-Sorbonne (Paris IV), 10 rue Molitor, 75016 Paris, France.
Jean-Marc.Clerin@paris-sorbonne.fr

Received: 13 June 2013
Revised: 17 September 2014

Abstract

Here we prove the existence of solutions to nonlinear differential inclusion problems with closed-loop control ż + A(z) = B(u,z) + f,u ∈ U(t,z),z(0) = z⁰ 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.