On differential games with long-time-average cost

The paper deals with the ergodicity of deterministic zero-sum differential games with longtime-average cost. Some new sufficient conditions are given, as well as a class of games that are not ergodic. In particular, we settle the issue of ergodicity for the simple games whose associated Isaacs equation is a convex-concave eikonal equation. Introduction We consider a nonlinear system in R controlled by two players ẏ(t) = f(y(t), a(t), b(t)), y(0) = x, a(t) ∈ A, b(t) ∈ B, (1) and we denote with yx(·) the trajectory starting at x. We are also given a bounded, uniformly continuous running cost l, and we are interested in the payoffs associated to the long time average cost (briefly, LTAC), namely, J∞(x, a(·), b(·)) := lim sup T→∞ 1 T ∫ T 0 l(yx(t), a(t), b(t)) dt, J∞(x, a(·), b(·)) := lim inf T→∞ 1 T ∫ T 0 l(yx(t), a(t), b(t)) dt. We denote with u− val J∞(x) (respectively, l − val J∞(x)) the upper value of the zero-sum game with payoff J∞ (respectively, the lower value of the game with payoff J∞) which the 1st player a(·) wants to minimize while the 2nd player b(·) wants to maximize, and the values are in the sense of Varaiya-Roxin-Elliott-Kalton. We look for conditions under which u− val J∞(x) = l − val J∞(x) = λ ∀x, for some constant λ, a property that was called ergodicity of the LTAC game in [3]. The terminology is motivated by the analogy with classical ergodic control theory, see, e.g., [30, 14, 28, 9, 25, 6, 7, 2]. Similar problems were studied for some games by Fleming and McEneaney [21] in the context of risk-sensitive control, by Carlson and Haurie [16] within the turnpike theory, by Kushner [29] for controlled nondegenerate diffusion processes. There is a large literature on related problems for discrete-time games, see the survey by Sorin [35]. More recently, several sufficient conditions for the ergodicity of the LTAC game were given by Ghosh and Rao [24] and by Alvarez and the author [3]. Among other things these papers clarified the connections with the solvability of the stationary Hamilton-Jacobi-Isaacs equation associated to the problem and with the long-time behavior of the value functions of the finite horizon games with the same running cost. In particular, under the classical Isaacs’ condition min b∈B max a∈A {−f(y, a, b) · p− l(y, a, b)} = max a∈A min b∈B {−f(y, a, b) · p− l(y, a, b)}, ∀y, p ∈ R, (2)