Stability Of Distributed Parameter Systems Research Articles

Stability of distributed-parameter systems with retarded argument

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.

The condition for sign-definiteness of integral quadratic forms and the stability of distributed-parameter systems

The stability of distributed- parameter systems described by linear partial differential equations is investigated by reducing the original equations by a change of variables to a system of first-order equations in time and in spatial coordinates. The Lyapunov functions are constructed in the form of single integral forms. New necessary and sufficient conditions for the sign-definiteness of these forms are obtained. These conditions, unlike the Sylvester criterion, do not require the calculation of determinants. The check for sign-definiteness is made using recurrence relationships and is a generalization of the results obtained in /1/. The proposed criteria are applied to derive sufficient conditions for the stability of distributed-parameter linear systems. The construction of functionals for the one-dimensional second-order linear hyperbolic equation is considered in more detail. As an example, we examine the stability of the torsional oscillations of an aircraft wing.