The Effect of the Cost Functional on Asymptotic Solution to One Class of Zero-Sum Linear-Quadratic Cheap Control Differential Games

A finite-horizon zero-sum linear-quadratic differential game is considered. The feature of this game is that the cost of the control of the minimizing player (the minimizer) in the game’s cost functional is much smaller than the cost of the control of the maximizing player (the maximizer) and the cost of the state variable. This smallness is due to a positive small multiplier (a small parameter) for the quadratic form of the minimizer’s control in the integrand of the cost functional. Two cases of the game's cost functional are studied: (i) the current state cost in the integrand of the cost functional is a positive definite quadratic form; (ii) the current state cost in the integrand of the cost functional is a positive semidefinite (but non-zero) quadratic form. For each of these cases, an asymptotic solution with respect to the small parameter of the considered game is formally constructed and justified. These solutions are compared with each other. Illustrative example is presented.