ON A STABILIZATION PROBLEM FOR A CONTROL SYSTEM WITH AFTEREFFECT

摘要

In [5], N. N. Krasovskii formulated and studied an equilibrium problem for a control system with aftereffect that can be described by differential-difference retarded equations. This problem was reduced to a boundary-value problem for a system of differential-difference equations with divergent argument in the lowest terms. Boundary problems for differential-difference equations with shifts of argument in the highest terms were studied in [2, 3]. It was shown that if an equation contains shifts that map boundary points inside the region, then solutions that can be nonsmooth inside the region even for infinitely differentiable right-hand sides, and other essentially new properties appear. In [4], sufficient conditions for the smoothness of generalized solutions were obtained; they can be formulated as the orthogonality of the right-hand side to some finite-dimensional subspace. In [1], this result was generalized to the case of boundary-value problems for systems of differential-difference equations. A generalization of the Krasovskii problem to the case where the equation that describes the control system also contains leading terms with delay, i.e., belongs to the neutral type, was studied in [6]. However, in both cases, only a one-dimensional (model) problem was studied. To the real control system, a multidimensional problem corresponds.