In a Hilbert space, we consider the problem of terminal control with linear dynamics, fixed left end and moving right end of the trajectories. On the reachability set(under additional linear constraints) the objective functional as the sum of integral and terminal components of the quadratic form is minimized. To solve the problem, we do not use the classical approach based on the consideration of the optimal control problem as an optimization problem. Instead, the saddle-point method for solving the problem is proposed. We prove its convergence.

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