Differential Game with Integral Constraints in a Hilbert Space

We study a pursuit differential game problem involving one pursuer and one evader in a Hilbert space. The motion of the pursuer is governed by first-order differential equations, while that of the evader is described by a second-order differential equation. The control functions of the pursuer and the evader are subject to integral constraints. The pursuit is considered completed if the equality y(θ)=x(θ) is achieved. We formulate and prove two theorems that provide sufficient conditions for the completion of pursuit at a given time.