The concept of value in differential games of survival and viscosity solutions of Hamilton-Jacobi equations (Q1198721)

The paper is concerned with a differential game with two players governed by the state equation \(y'=f\bigl(y,a(t),b(t)\bigr)\), \(y(0)=x\in{\mathbf R}^ N\), with target \({\mathcal T}\subset{\mathbf R}^ N\) and payoff \(\int_ 0^{\tau_ x}h\bigl(y(t),a(t),b(t)\bigr) dt+g\bigl(y(\tau_ x)\bigr)\), where \(\tau_ x\) is the first hitting time. The problem is studied without any restriction on \(\tau_ x\), set to \(+\infty\) if \(y\) never reachs the target. It is proved that the upper and lower value functions obtained by the Fleming and the Friedman approximation schemes converge, under suitable assumptions on \(f, h, g\) and \({\mathcal T}\) to a function satisfying the Hamilton-Jacobi-Isaacs equation.