On the solution of the optimal linear control problems under conflict of interest

Abstract
In the control of multi-input plants, the optimal choice of the feedback control inputs depends upon the choice of disturbance inputs by the competitor, enemy, or nature. Under that viewpoint, the formulation of the control problem under conflict of interest is translated to a differential game. This study considers a two-person conflicting situation, described by linear plant dynamics, while the performance indices are quadratic functionals. A theorem and an iterative numerical technique, based on Newton's method, are developed, for the actual computation of the closed-loop solution in the stationary case of the nonzero and zero-sum differential game. Explicit solutions are also presented for the finite terminal time problem arising in the zero-sum linear differential game, and a simple sufficient condition for the existence of such solutions is included. Two examples are solved to illustrate the procedures described.

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