Generalization of the Kalman equation to a linear-quadratic optimal control problem

In a Hilbert space, a linear-quadratic optimal control problem with a fixed left end and a movable right end is considered. At the right end of the time interval, a linear programming problem is formulated, the solution of which implicitly determines the terminal condition. A saddle point approach is proposed, which is reduced to calculating the saddle point of the Lagrange function. It is based on saddle point inequalities for both groups of variables: direct and dual. These inequalities are sufficient conditions for optimality. A control synthesis is constructed that implements feedback in the presence of constraints on controls as a convex closed set. This is a new result, since in the classical case in the theory of a linear controller (Kalman equation) a similar result is proven only in the absence of constraints on controls.