Quasi-gradient method for solving optimal control problems (Q1390400)

Traditionally, the gradient methods are standard means for numerically solving optimal control problems. The machinery of their construction and analysis on variational level is developed rather well and completely corresponds to finite-dimensional analogs. However again and again, the specificity of optimal control problems enables us to insert certain change into conventional structures of gradient procedures. As a rule, these modifications have nontrivial character and reveal additional reserves for increase of the efficiency of the corresponding method. In the present article, we carry out the construction and justification of a quasi-gradient method, which uses a certain correction of customary gradient as a direction of functional descent. The basis of modification is a nonstandard formula for the functional increment. It has improved approximation characteristics. This improvement augments the quality of a corresponding method, which gains the second order with respect to phase variables. As a consequence, we obtain the property of non-local improvement in linear-quadratic problems, which removes the problem of parametric search on each iteration. In addition, this method is suitable for phase systems which are discontinuous with respect to the state. Consequently, it enables us to realize sliding modes (singular control). Thus, this method opens a way for guaranteed solving of degenerate problems of optimal control.