부산대학교 도서관

The authors consider the differential game described by the state equation $\dot x=f(t, x, y, z)$ where $t\in[t_0, T]$ is the time variable, $x\in{\bf R}^m$ is the state variable, $y\in P_y\subset{\bf R}^p$ and $z\in P\subset{\bf R}^p$ are the control variables of the players, $P_y$ and $P_z$ are compact sets. The payoff functional is defined by $$P(y, z)=g(x(T))+\int_{t_0}^Th(t, x(t),y(t), z(t))\,dt.$$ It is assumed that the functions $f, g$ and $h$ satisfy the usual\break smoothness and growth requirements and Isaacs' condition.\break It is assumed also that the Hamiltonian function $H(t, x, l)=\break \min_y\max_z[\langle l, f(t, x, y, z)\rangle+ h(t, x, y, z)]$ is continuously differentiable on $x$ and the function $(t, x, l)\to(\partial H/\partial x)(t, x, l)$ is Lipschitz-continuous on $x$ and $l$. The authors study properties of the value function $(t_0, x_0)\to V(t_0, x_0)$. They state the following conjecture: the functions $(t, x)\to(\partial V/\partial x_i) (t, x)\;(i=1, 2,\cdots,m)$ are of bounded variation. In this paper the conjecture is proved in case $m=1$.