Differential games of inf-sup type and Isaacs equations
In the paper the dynamics of the form \[ \frac{d\,x}{d\,s}=f(s,x,a,b)\tag{1} \] are studied, dependent on a pair of control functions \(a(s)\), \(b(s)\), along with a game payoff function \[ J(t,x,a,b)=\int_t^Tl(s,x(s),a(s),b(s))\,ds+\Phi(x(T)).\tag{2} \] The functions \(a(s)\) and \(b(s)\) are regarded as, respectively, the control of the minimizing and the control of the maximizing player. The maximizing player is interpreted as a disturbance in \(H^{\infty}\) control, therefore the inf-sup game value function \[ V(t,x)=\inf_{\alpha}\sup_b J(t,x,\alpha(b),b) \] is examined, where the supremum is taken over all feasible \(b\), while the infimum is computed with respect to a class of strategies of the minimizing player, i.e. the maps of the function \(b\) into \(a\). The main results of the paper are concerned with establishing a relationship between the Dynamic Programming Principle (DDP) and the infinitesimal generator for the game value function, and with obtaining this generator. It has been shown that, if DDP holds and the infinitesimal generator has been obtained, then the value function of the inf-sup type game is a viscosity solution of the corresponding Isaacs equation. More specifically, in the case of progressive strategies the infinitesimal generator is the lower Hamiltonian related with (1) and (2) (Theorem 3.3), and in the case of strictly progressive strategies this generator is equal to the upper Hamiltonian (Theorem 3.5). Additionally, for strategies situated between strictly progressive and progressive strategies the infinitesimal generator is defined, in an asymptotic sense, by a convex combination of the lower and the upper Hamiltonians (Theorem 4.2).
scientific article; zbMATH DE number 5004511