Synthesis of a regulator for a linear-quadratic optimal control problem

In a Hilbert space, a linear-quadratic optimal control problem with a fixed left end and a moving right end is considered on a fixed time interval. The objective functional is the sum of the integral and terminal components of quadratic type. Each component seeks its minimum on its admissible set independently. At the right end of the time interval, we have a linear programming problem. The solution to this problem implicitly determines the terminal condition for controlled dynamics. A saddle point approach is proposed for solving the problem, which reduces to calculating the saddle point of the Lagrange function. The approach is based on saddle point inequalities in both groups of variables: primal and dual. These inequalities are sufficient optimality conditions. A method for calculating the saddle point of the Lagrange function is formulated. Convergence in primal and dual variables is proved, namely, weak convergence in controls and strong convergence in phase and adjoint trajectories, as well as in terminal variables of the boundary value problem. Based on the saddle point approach, the control synthesis is constructed, i.e., feedback in the presence of constraints on controls in the form of a convex closed set. This is a new result, since, in the classical case in the theory of a linear regulator, a similar statement is proved in the absence of constraints on controls, which makes it possible to use the matrix Riccati equation. In the presence of constraints on control, these arguments no longer work. Therefore, the result obtained is based on the concept of a support plane to the control set.