On solutions of evolution inclusions. II

Irkutsk. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 33, No. 4, pp. 163–174, July–August, 1992.     

Maximal Monotonicity of a Nemytskii Operator

Functional Analysis and Its Applications - A family of maximally monotone operators on a separable Hilbert space is considered. The domains of these operators depend on time ranging over an...     

A Multivalued History-Dependent Operator and Implicit Evolution Inclusions. I

Siberian Mathematical Journal - On the space of continuous functions from a line segment to a reflexive Banach space, we consider some operator whose values are closed convex...     

Continuity in the parameter of the minimum value of an integral functional over the solutions of an evolution control system

The problem of minimization of an integral functional with an integrand that is nonconvex with respect to the control is considered. We minimize our functional over the solution set of a nonlinear evolution control system with a time-dependent subdifferential operator in a Hilbert space. The control constraint is given by a nonconvex closed bounded set. The integrand, the control constraint, the initial conditions and the operators in the equation describing the control system all depend on a parameter. We consider, along with the original problem, the problem of minimizing an integral functional with an integrand convexified with respect to the control over the solution set of the same system, but now subject to the convexified control constraint. By a solution of the control system we mean a “trajectory–control” pair. We prove that for each value of the parameter the convexified problem has a solution, which is the limit of a minimizing sequence of the original problem, and the minimum value of the functional of the convexified problem is a continuous function of the parameter.