On Differential Games for Infinite-Dimensional Systems with Nonlinear, Unbounded Operators

Abstract

We consider a two-player, zero-sum differential game governed by an abstract nonlinear differential equation of accretive type in an infinite-dimensional space. We prove that the value function of the game is the unique viscosity solution of the corresponding Hamilton᎐Jacobi᎐Isaacs equation in the sense of M. G. Crandall Ž and P. L. Lions ''Evolution Equations, Control Theory and Biomathematics,'' .

Infinite horizon differential games for abstract evolution equations

“…This has been achieved by using the notion of viscosity solution proposed by Crandall-Lions in [5] and [6]. By using our dynamic programming inequalities and mimicking the arguments in [8], without using (A0), we can also characterize the value function in the class of bounded uniformly continuous functions by taking the definition of viscosity solution as in [7] which is a refinement of Tataru's notion (see [11] and [12]). In the Elliott-Kalton framework, this has been established by Kocan et.…”

A zero-sum electromagnetic evader–interrogator differential game with uncertainty

“…One approach is based on the theory developed by Elliott and Kalton [16] for differential games in Euclidean spaces. For example, an infinite-dimensional differential game on the infinite horizon was studied in [24] with strategies in the sense of Elliott and Kalton, and the value function of the differential game is characterized as the unique viscosity solution of the Hamilton-Jacobin-Isaacs equation. The other approach is based on the theory developed by Berkovitz [7] for differential games in Euclidean spaces, wherein the definition of strategy is a combination of "K strategies" discussed by Isaacs [22] and Friedman's lower strategy (e.g., see [19,20]) and the definition of payoff and saddle point follows that of Krasovskii and Subbotin [25].…”

“…From the above equation we see that From the definition for J defined in (2.9), to show J is separately continuous in each of its variables, it suffices to show that for given Φ ε ∈ R(E) and a sequence 24) and for given Φ ω ∈ R(Ω) and a sequence …”

Sub- and Super-optimality Principles and Construction of Almost Optimal Strategies for Differential Games in Hilbert Spaces

“…However the proof of this basically follows the arguments of the finite dimensional proof of [9] with necessary modifications using continuous dependence estimates for (1) and other techniques that can be found in [5,6,16]. The reader can also consult [15] for a complete proof in the infinite horizon case even though it uses a different definition of viscosity solution. Here we just state the result.…”

“…When B is compact existence of solutions of general equations like (1) was shown in [5] by finite dimensional approximations. Proofs using the value functions of the games are in [15,23] and in [4] when A = 0. Unfortunately none of these papers provides an exact reference to the fact that lower and upper value functions considered in this paper are viscosity solutions of (5) − and (5) + in the sense of Definition 1.…”