Abstract
The aim of the present paper is to establish a few theorems for a class of optimization problems of a bilinear distributed parameter system. The word “bilinear system” means that it is a linear system and some coefficients of its system equation are control variables. A system described by a parabolic partial differential equation is considered here. The cost functional to be optimized takes a quadratic form of the state variable. First established is a theorem of Fréchet differentiability of the state variable with respect to the coefficients. Then the existence of gradient of the cost functional is easily proved. Introducing the adjoint system, a representation of the gradient is obtained. Further, on the assumption of the existence of an optimal solution, the maximum principle is proved by making use of the variational inequality. The representation of the gradient will be useful in application of certain algorithms such as the method of steepest descent.
