Control systems of subdifferential type depending on a parameter

A. A. Tolstonogov

Izvestiya. Mathematics , Tome 72 (2008) no. 5, pp. 985-1022

Résumé

In a separable Hilbert space, we consider a control system with a subdifferential operator and a non-linear perturbation of monotonic type. The control is subject to a restriction that is a multi-valued map depending on the phase variables with closed non-convex values in a reflexive separable Banach space. The subdifferential operator, the perturbation, the restriction on the control and the initial condition depend on a parameter. Along with this system we consider a control system with convexified restrictions on the control. By a solution of such a system we mean a pair ‘trajectory–control’. We prove theorems on the existence of selectors that are continuous with respect to the parameter and whose values are solutions of the control system. We establish relations between the sets of selectors continuous with respect to the parameter whose values are solutions of the original system and solutions of the system with convexified restrictions on the control. We deduce from these relations various topological properties of the sets of solutions. We apply the results obtained to a control system described by a vector parabolic equation with a small diffusion coefficient in the elliptic term. We prove that solutions of the control system converge to solutions of the limit singular system as the diffusion coefficient tends to zero.