Метод аппроксимации экстремальных решений в задачах оптимального управления с фазовыми ограничениями

A Method for Approximating Extremal Solutions in Optimal Control Problems with State Constraints

The paper considers an optimal control problem with state constraints via its approximation in the class of piecewise-constant controls. Using smooth cubic penalty functions that ensure the continuity of the second derivatives of the penalty func- tional, the original continuous state-constrained problem is equivalently reduced to a problem with a terminal equality constraint. Necessary optimality conditions for the approximating finite-dimensional model are formulated as a fixed point problem for a projection operator in the control parameter space. On this basis, a nonlocal iterative method is developed that is free from the labor-intensive procedure of a parametric one-dimensional search for an improving control approximation typical of traditional gradient algorithms. At each iteration of the method, an approximation of the extremal control satisfying the state constraints is determined. Narrowing the search space to the set of admissible controls makes it possible to efficiently identify control strategies sufficient for practical applications in state-constrained model problems during the iterative process. The performance and computational efficiency of the proposed ap- proach are demonstrated on a model problem of optimizing a fed-batch bioreactor for penicillin production via anaerobic glucose fermentation.