Abstract
Lyapunov functions are used to investigate the stability of processes described by a system of linear partial differential equations with retarded argument (for example: magnetohydrodynamic processes, elastic vibrations in aircraft, etc.). Some equations of the system may not involve time derivatives (for example, the equation of continuity in incompressible fluid flow, and the equation for the magnetic induction vector in the theory of electromagnetic phenomena). Such equations also arise when the order of a partial differential equation is reduced by introducing new notation for the space derivatives. A method is developed for investigating the stability of processes described by a system of this kind, some of whose equations do not contain time derivatives. Two constructions of the Lyapunov functions, as different integral quadratic forms, are proposed. Sufficient conditions for stability, in the form of inequalities relating the coefficients of the system, are established. As an example, the stability of the vibrations of a stretched string in a viscoelastic medium due to a distributed control force is considered.
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