On Solvability of a Pursuit Game with Nonlinear Dynamics in Hilbert Space

We consider a pursuit differential game in a Hilbert space. The game dynamics is described by two semilinear evolutionary equations with a not necessarily bounded operator in the Hilbert space; each of these equations is controlled by its own player. The controls appear linearly on the right-hand sides of the equations, and their norms are bounded by given constants. Sufficient conditions for the solvability of the pursuit game are established in both linear and nonlinear cases. For this purpose, we use the Minty–Browder theorem and a chain technology of successive continuation of the solution to a controlled system to intermediate states. As examples of reduction to the abstract operator equation under study, we consider Oskolkov’s system of equations and a semilinear wave equation.