On the Value Function to Differential Games with Simple Motions and Piecewise Linear Data

Positional differential games with simple dynamics are considered under assumption that at least one of two input functions (the Hamiltonian and the cost terminal function) is piecewise linear and positively homogeneous. The structure of the value function of the differential game is investigated in the framework of the theory of minimax (or/and viscosity) solutions for Hamilton–Jacobi equations. Inequalities are provided to estimate the value function. Cases of explicit formulas for the value function are pointed out.