About one differential game of neutral type with integral restrictions in Hilbert space

In the theory of differential games, when the game is defined in a finite-dimensional space, the fundamental works belong to academicians L.S. Pontryagin and N.N. Krasovskii. The works by N.N. Krasovskii and his students are mostly devoted to position games. In works by L.S. Pontryagin and his students the differential game is considered separately from the point of views of the pursuer and the evader and this unavoidably relates the differential game with two different problems. It is topical to study the games in finite-dimensional spaces since many important problems on optimal control under the conditions of a conflict or uncertainty governed by distributed systems, the motion of which is described by integro-differential equations and partial differential equations, can be studied as differential games in appropriate Banach spaces.
In the present work, in a Hilbert space, we consider a pursuit problem in the Pontryagin sense for a quasilinear differential game, when the dynamics of the game is described by a functional-differential equation of neutral type in the form of J. Hale with a linear closed operator and on the control of the players integral restrictions are imposed. We prove an auxiliary lemma and four theorems on sufficient conditions ensuring the solvavility of the pursuit problem. In the lemma we show that the corresponding inhomogeneous Cauchy problem for the considered game has a solution in the sense of J. Hale. In the theorems we employ a construction similar to the Pontryagin first direct method and the idea by M.S. Nikolskii and D. Zonnevend on dilatation of time $J(t)$ and describe the sets of initial positions, from which the termination of the pursuit is possible.