Stability of Distributed-parameter Processes with Time-delays†

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Abstract
In this paper it is shown that time-delayed variables can enter into distributed-parameter processes due to the presence of both internal and external delayed-action energy sources. The dynamic behaviour of such processes is deseribable by a system of partial differential-difference equations. Particular attention is focused on the class of equations which admit product solutions so that their time-dependent equations are reducible to danumerably infinite systems of ordinary differential-difference equations. An extended version of Lyapunov's stability theory for such equations is given and its application is illustrated by the study of a one-dimensional diffusion process with non-linear delayed-action sources.

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