Optimal control of solutions to the showalter — Sidorov problem for the Sobolev type equation of higher order

Abstract

The theory of dynamical measurements is a wide and independent part of the metrology. Previously, dynamical measurements were studied mostly within the framework of inverse problems theory. Lastly there arose a new idea for the restoration of dynamically distorted signals, which evolved into a theory of optimal measurements. It was suggested to use methods of optimal control theory for the Sobolev type equations in solving this problem of dynamical measurements. In this article, a problem of optimal control of solutions of the Sobolev type equation of higher order with the Showalter — Sidorov condition is studied. All considerations are held under assumption of relative polynomial boundedness of an operator pencil. The theorem stating that there exists a unique strong solution to abstract equation with the Showalter — Sidorov condition is proved. The sufficient and necessary (in the case when infinity is a removable singularity of the A-resolvent of an operator pencil) conditions for existence of unique optimal control for strong solutions are found. We use the ideas and methods developed by G.A. Sviridyuk and his disciples.