SYNTHESIS OF A REGULATOR FOR A LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEM

In Hilbert space, we consider a linear-quadratic optimal control problem with a fixed left end and a movable right end, for a fixed period of time. The target functional is the sum of the integral and terminal components of the quadratic form. Each of the components searches for its minimum on its permissible set independently of each other. At the right end of the time interval, we have a linear programming problem. The solution to this problem implicitly defines a terminal condition for controlled dynamics. A saddle approach is proposed to solve the problem, which boils down to calculating the saddle point of the Lagrange function. The approach is based on saddle inequalities in both groups of variables: direct and dual. These inequalities represent sufficient conditions for optimality. A method for calculating the saddle point of the Lagrange function is formulated. Convergence in direct and dual variables is proved, namely: weak convergence in controls, strong convergence in phase and conjugate trajectories, as well as in terminal variables of the boundary value problem. On the basis of the saddle approach, control synthesis is built, i.e. feedback in the presence of control constraints 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 control constraints, which makes it possible to use the Riccati matrix equation. If there are restrictions on management, these arguments no longer pass. Therefore, the basis of the obtained result is the concept of a reference plane to a set of controls.

terminal control, boundary value problem, Lagrange function, saddle methods, control synthesis, convergence