On the problem of the analytical design of regulators for distributed-parameter systems under boundary-function control: PMM Vol. 35, No. 4, 1971. pp. 598–608

Abstract

The method of extension of a differential operator is carried over to systems of second-order parabolic linear differential equations. The problem of the analytical design of regulators with controls by boundary functions is reduced by means of the method of extension of a differential operator to a problem with distributed controls and is solved by the dynamic programming method. In the theory of partial differential equations it is well known [1] that linear homogeneous equations with inhomogeneous boundary conditions are essentially equivalent to inhomogeneous equations with homogeneous boundary conditions. This can be shown by the method of extension of a differential operator [2–4]. Using delta-functions and their derivatives a linear homogeneous equation with inhomogeneous boundary conditions can be written as an inhomogeneous equation with homogeneous boundary conditions when certain continuity and differentiability conditions are fulfilled. In the case when the boundary conditions are the control functions, the inhomogeneous equation obtained can be treated as an optimal problem with distributed controls. An analytical solution was obtained in [3,4] by this method for the problem of bringing a rod's temperature upto a specified temperature distribution in a fixed interval of time with minimal energy, and an example with controls bounded in absolute value, analyzed earlier in [5], also was examined. The analytical design problem for regulators for partial differential systems with distributed controls was considered in [6, 7], Problems with boundary-function controls were studied in [7–9].