Abstract
Most researches on the optimal control of time delay system in the past were concerned with the necessary condition for optimal control, that is, the maximum principle. The maximum principle, however, is not generally so effective for finding the optimal control, and the direct computing method is preferred. Almost all of the direct methods resort to the gradient function, and it is not too much to say that the problem for finding optimal control is half solved once the means for computing the gradient function have come to hand.
First, by making the concept of the state of time delay system clear, it is recognized that the time delay system is essentially a distributed parameter system. Then, the dynamics of the time delay system are described in the form of partial differential equation together with its boundary condition. After that, the gradient function is derived. The procedure of derivation is outlined below.
The transition matrix is defined first, and then the gradient function is obtained in terms of transition matrix. The calculation of the transition matrix, however, is extremely complicated, though not impossible in principle, so the expression is not suited for practical use. By introducing a suitable costate, the gradient function is rewritten in terms of the costate. Taking advantage of the particular property of the transition matrix, the partial differential equation and the boundary condition to be satisfied by the costate are derived. The costate is easily computed by integrating numerically the costate system with respect to backward time. Thus, the latter expression of gradient function is very convenient for practical use.
