Stackelberg solutions of feedback type for differential games with random initial data (Q384053)

In the space \(\mathbb{R}^n\) a differential game is investigated described by the differential equation \[ \dot{x}(t)=f(t,x(t),u_1(t),u_2(t)),\qquad u_1(t)\in U_1,u_2(t)\in U_2,t\geq 0, \] with the initial condition \[ x(0)=\hat{x}, \] where \(U_1,U_2\subseteq \mathbb{R}^{m}\) are given sets. The goal of the first \(u_1(\cdot)\) and second \(u_2(\cdot)\) player is to minimize his own cost given by \[ \int_{0}^{T}L_1(t,x(t),u_1(t),u_2(t))dt \] and \[ \int_{0}^{T}L_2(t,x(t),u_1(t),u_2(t))dt, \] respectively. In the paper it is assumed that the first player announces his feedback strategy (Stackelberg equilibrium) in advance i.e., \(u_1=u_1(t,x).\) The second player chooses his strategy \(u_2=u_2(t;u_1,\hat{x})\) in order to minimize his own cost. An existence theorem of an optimal feedback strategy under natural assumptions on the cost functions \(L_1\), \(L_2\) and necessary optimality conditions of such a feedback strategy are proved.