Stability of Solutions to Systems of Parabolic Equations with Delay

Abstract

This work is devoted to analysis of stability (in the Lyapunov sense) of solutions to systems of linear parabolic equations with coefficients depending on time and with delay depending on time. The cases of continuous and impulsive perturbations are considered. A method for studying the stability of solutions to systems of linear parabolic equations is as follows. Applying the Fourier transform to the original system of parabolic equations, we arrive at a system of nonstationary ordinary differential equations with delay depending on time, which is defined in the spectral region. First, the stability of the resulting system is studied by the method of frozen coefficients in the metric of space R_n of n-dimensional vectors. Then the resulting statements are extended to space L₂. The application of the Parseval equality allows us to return to the domain of the originals and to obtain sufficient conditions for the stability of solutions to systems of linear parabolic equations. An algorithm is proposed that allows one to obtain sufficient stability conditions for solutions of finite systems of linear parabolic equations with time-dependent coefficients and with time-dependent delays. Sufficient stability conditions are expressed in terms of the logarithmic norms of matrices composed of the coefficients of the system of parabolic equations. The algorithms are obtained in the metric of space L₂. Algorithms for constructing sufficient stability conditions are efficient both in the case continuous and in the case of impulsive perturbations. A method is proposed for constructing sufficient stability conditions for solutions to finite systems of linear parabolic equations with time-dependent coefficients and delays. The method can be used in the study of nonstationary dynamical systems described by systems of linear parabolic equations with delays depending from time.