Quasilinear Pursuit Differential Game with a Simple Dynamics under Phase Constraints

On a fixed closed time interval we consider a quasilinear pursuit differential game with a convex compact target set under a phase constraint in the form of a convex closed set. We construct a convex compact guaranteed capture set similar to an alternating Pontryagin sum and define the guaranteed piecewise-programmed strategy of the pursuer ensuring the hitting of the target set by the phase vector satisfying the phase constraint in finite time. Under certain conditions, we prove the convergence of the constructed alternating sum in the Hausdorff metric to a convex compact set, which is an analog of the alternating Pontryagin integral for the differential game.