To the problem of the pursuit in quasilinear differential lag games

In the field of the theory of differential games defined in a finite-dimensional space, fundamental works were carried out by L. S. Pontryagin, N. N. Krasovskiy, B. N. Pshenichny, L. S. Petrosyan, M. S. Nikol'skiy, N. Yu. Satimov and others. L. S. Pontryagin and his students consider differential games separately, from the point of view of the pursuer and from the point of view of the evader, which inevitably connects the differential game with two different problems. In this paper, in a Hilbert space, we consider the pursuit problem in the sense of L. S. Pontryagin for a quasilinear differential game, when the dynamics of the game is described by a differential equation of retarded type with a closed linear operator generating a strongly continuous semigroup. Two main theorems on the solvability of the pursuit problem are proved. In the first theorem, a set of initial positions is found from which it is possible to complete the pursuit with a guaranteed pursuit time. The second theorem defines sufficient conditions on the optimality of the pursuit time. The results obtained generalize the results of works by P. B. Gusyatnikov, M. S. Nikol'skiy, E. M. Mukhsinov, and M. N. Murodova, in which it is described by a differential equation of retarded type in a Hilbert space. Our results make it possible to study delayed-type conflict-controlled systems not only with lumped, but also with distributed parameters.