Conditions and methods for improving control in quadratic systems with constraints

A new approach to the nonlocal improvement of admissible controls in the class of quadratic in state and linear in control optimal control problems with terminal constraints is considered. The approach under consideration allows avoiding the time-consuming operation of parametric changes to improve control, which ultimately leads to an increase in the efficiency of the developed optimization procedures. The nonlocality of control improvement is achieved by solving a special system of functional equations equivalent to the boundary value improvement problem, for the solution of which an iterative algorithm is proposed with the fulfillment of all terminal constraints at each iteration. At each iteration of the proposed iterative algorithm, the usual Cauchy problems are solved, in contrast to the methods requiring the solution of special Cauchy problems (with the right-hand side discontinuous in state variables), which significantly simplifies the implementation of the proposed procedure. In addition, the initial guess of the iterative process may not be an admissible control. Within the framework of the proposed approach, new necessary conditions for optimality are constructed that strengthen the maximum principle in the class of problems under consideration. The work contains the corresponding illustrative examples.