Journal of Applied Mathematics and Stochastic Analysis
Volume 12 (1999), Issue 3, Pages 233-252
doi:10.1155/S1048953399000222
Nonconvex evolution inclusions generated by time-dependent subdifferential operators
National Technical University, Department of Mathematics, Zografou Campus, Athens 15780, Greece
Received 1 June 1998; Revised 1 September 1998
Copyright © 1999 Kate Arseni-Benou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We consider nonlinear nonconvex evolution inclusions driven by time-varying subdifferentials ∂ϕ(t,x) without assuming that ϕ(t,.) is of compact
type. We show the existence of extremal solutions and then we prove a
strong relaxation theorem. Moreover, we show that under a Lipschitz condition on the orientor field, the solution set of the nonconvex problem is
path-connected in C(T,H). These results are applied to nonlinear feedback control systems to derive nonlinear infinite dimensional versions of
the “bang-bang principle.” The abstract results are illustrated by two
examples of nonlinear parabolic problems and an example of a differential
variational inequality.