A Hamilton-Jacobi type equation in control problems with hereditary information (Q2761278)

The author studies the problem of conflict control with uncertain information in the system with delay NEWLINE\[NEWLINE \begin{aligned} &\frac{dx[t]}{dt} = f(t,x[t_*[\cdot]t],u,v),\quad t_*\leq t_0\leq t\leq T,\\ &x\in R^n,\quad u\in U\subset R^r,\quad v\in V\subset R^m, \end{aligned}\tag{1} NEWLINE\]NEWLINE where \(x\) is the phase vector, \(u\) and \(v\) are the controls of the first and second players and \(U\) and \(V\) are the known compacts. The cost functionals NEWLINE\[NEWLINE \gamma = \sigma(x[t_0[\cdot]T]) NEWLINE\]NEWLINE are formed in terms of the notion of co-invariant derivatives. The author considers the Hamilton-Jacobi type equation and constructs the control strategies being extremal with respect to the generalized solution of the above mentioned equation.