Averaged Hamiltonians in Differential Games

Abstract

Consider a differential game G whose dynamics are given by the equations (x∈R ^m)

$$ \frac{{dx}}{{dt}} = f\left( {t,x,y,z} \right),\,x(0) = 0 \in {R^m} $$

Here t ∈ [0,1] and f is continuous; to simplify the argument we suppose f satisfies a uniform Lipschitz condition in t and x. Further, we suppose the payoff has the form

$$ P = g\left( {x(1)} \right) $$

where g is real valued, twice differentiable and $\frac{{\partial g}}{{\partial t}}$, $\frac{{\partial g}}{{\partial {x_i}}}$, $\frac{{{\partial^2}g}}{{\partial {x_i}\partial {x_j}}}$ satisfy uniform Lipschitz conditions in t and x. Write K for the Lipschitz constant in all cases. The situation where f satisfies weaker Lipschitz and continuity conditions and the payoff has a more general form, can be treated by approximation arguments as in [3].