ИСТИНА

In a Hilbert space, the problem of terminal control with linear dynamics and fixed ends of trajectory is considered. The integral objective functional has a quadratic form. In contrast to the traditional approach, the problem of terminal control is interpreted not as an optimization problem, but as a saddle-point problem. The solution to this problem is a saddle point of the Lagrange function with components in the form of controls, phase and conjugate trajectories. We need to find a control $u^*(\cdot)\in\mathrm {U}$ such that the corresponding trajectory $x^*(\cdot)$ connects some starting point $x_0$ with a given point $x_1\in X_1$ at the right end. Problem is a convex programming problem formulated in Hilbert space. We introduce the Lagrangian. By linearization of the Lagrangian and using the saddle-point inequalities system, we get the dual problem. By bringing together main elements of primal and dual problems, we obtain a final system. The solution to this system is a saddle point of the Lagrangian, and some of its components form the desired solution. In linear-convex case, this approach could be interpreted as strengthening the Pontryagin maximum principle. It provides the convergence of computing process to solution of the problem in all components: the convergence in controls is weak, the convergence in phase and conjugate trajectories is strong.