Abstract
This chapter describes the optimal control of linear systems with time-varying lags. The optimal controls consist of the current and the previous values of the state and the control. The representation is a generalization of the hereditary system, with time-variable arguments in which X(t) is an n-dimensional vector, and U(t, ·) is an m-dimensional vector. A(t, ·) and B(t, ·) are matrices of compatible dimensions whose elements are continuous. Hiratsuka, suggested that a functional-differential equation with time-varying arguments can be represented as a partial-differential equation and another functional differential equation. Using this representation, he derived necessary conditions in the form of a maximum principle; however, he did not indicate how to solve the quadratic optimization problem. The results are of importance in the construction of point wise finite-dimensional controls for hyperbolic distributed-parameter systems.
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