An extremum problem for a linear integro-differential system describing creeping flows of a viscoelastic fluid

Abstract

We consider an optimal control problem for an integro-differential system (with a quadratic cost functional) modeling a three-dimensional creeping flow of an incompressible viscoelastic fluid in a bounded domain with impermeable solid walls. The fluid flow is controlled by the time-dependent external force. The concept of the control operator is proposed. We prove a theorem on the existence of a unique optimal control under the assumption that the set of admissible controls is convex and that it is closed in a suitable function space. Moreover, we obtain a variational ine-quality for the optimal control. The proof of this theorem is based on the application of the Faedo–Galerkin approximation scheme taking into account energy estimates of approximate solutions and using the lemma on the existence and uniqueness of the metric projection of a point onto a closed convex set in a real Hilbert space.