On the stability of dynamical systems with distributed parameters at the time of integrally small perturbations. (Q2901698)

This paper addresses the issue of stability for a system of second-order nonlinear partial differential equations. The author assumes that any solution of the Cauchy problem may be extended to the whole positive semi-axis for such a system with disturbances in the right-hand side and boundary conditions. The notions of stability and asymptotic stability with respect to a measure are introduced for this system with integrally small disturbances. A particular class of systems of linear partial differential equations is considered, and sufficient stability conditions are proposed assuming that the disturbances act on a finite interval in the time domain. This approach is extended to a class of nonlinear partial differential equations as well. The stability proof is based on the direct Lyapunov method. More detail. Lyapunov's functional method is developed for studying the stability of processes with distributed parameters, i. e. processes, the parameters of which, except the time, will depend on the spatial coordinates and the systems differential equations in partial derivatives are described. The problem of stability of dynamical systems with distributed parameters is being studied in the work, in case of the system being a finite time interval, influenced by integrally small perturbing forces. The definition of stability for the integral small perturbations is introduced according earlier. The problems of stability in this sense are put and solved. The author obtains sufficient conditions under which the system with distributed parameters is stable at the time of integrally small perturbations.