@MISC{Delfour_two-personzero-sum, author = {Michel C. Delfour}, title = {Two-person zero-sum differential games}, year = {} }

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Abstract

In differential zero-sum two-player games the first player tries to minimize and the second player to maximize a utility function that depends on a state variable whose dynamics is governed by a system of differential equations. Two classical approaches are via open loop and closed loop strategies for the players. In this paper, we shall mainly restrict ourselves to the open loop case and a perfect knowledge of the state. (M. C. Delfour and S. K. Mitter 1969) stud-ied the dynamical Min-Sup problem for the fol-lowing perturbed control process in Rn dx dt (t) = A(t)x(t)+f(t, u(t))+ g(t, v(t)), (1) where A(t) is a n × n measurable and bounded matrix on [0, T], f is in C1 in R1+m and g is in C1 in R1+k (n, m, and k are integers ≥ 1), and furthermore, (i) the initial state x0 at time 0 is given, (ii) the admissible controllers F consist of all Lebesgue measurable functions t 7 → u(t) on the compact interval [0, T] such that u(t) ∈ U, (almost everywhere on [0, T]), where U is a com-pact set in Rm, (iii) the admissible disturbances G consist of all Lebesgue measurable functions t 7 → v(t) on the compact interval [0, T] such that v(t) ∈ V, almost everywhere on [0, T], where V is a compact set in Rk, (iv) the cost func-tion for each admissible u and v is given by C(u, v) = G(x(T)), where G is a continuous function in Rn. The fundamental theory of closed loop two-player zero-sum LQ games was given in (P. Bernhard 1979) followed by the seminal book in 1991 of (T. Başar and P. Bernhard 1995) that covered the H∞-theory. They considered two-player zero-sum game over the finite time inter-val [0, T] characterized by the quadratic utility function Cx0(u, v) def = Fx(T) · x(T)