Stabilization of solutions to a linear Sobolev-type equation with a relatively sectorial operator

The article considers a linear equation of the Sobolev type with a relatively sectoral operator. This type of equation arises when modeling various processes: the evolution of the free surface of a liquid, the flow of a viscous incompressible liquid, plane-parallel thermoconvection of a viscoelastic incompressible liquid, etc. The paper considers the following stabilization problem: it is necessary to find a control action on the equation so that it becomes uniformly asymptotically stable. The solution to this problem is based on the theory of semigroups and groups of operators with kernels. In the case when the relative spectrum consists of two parts, one of which lies in the left half-plane of the complex plane, and the second in the right half-plane of the complex plane, it is possible to construct a resolving semigroup and a group of operators, and to carry out their exponential estimates. In this case, the solution of the equation can be represented as the sum of a stable and unstable solutions. The stabilization of an unstable solution is based on the feedback principle. An equation describing the evolution of the free surface of a liquid is considered as an application.