Estimates of Reachable Set and Suﬃcient Optimality Condition for Discrete Control Problems
The paper follows the “canonical optimality theory” (in the terminology due to A. A. Milyutin) for discrete-time optimal control problems. In respect of optimality conditions, the feature of this approach is to employ sets of strongly monotone functions being solutions of the respective Hamilton-Jacobi inequality. This idea serves to improve the eﬃciency of the derived optimality conditions, extends the area of their application, and increases their “stability” with respect to certain peculiarities of a problem (e.g., the lack of the uniqueness of the normed collection of Lagrange multipliers etc.)In the paper, we consider an optimal control problem for a nonlinear discrete-time dynamic system with a nonlinear cost function under pointwise state and mixed-endpoint constraints. For this system, we obtain external estimates of the reachable set. Based on these estimates, we derive a suﬃcient optimality condition for the respective optimal control problems under no convexity assumptions on the input data. The results operate with a new class of feedback-parametric strongly monotone functions depending on initial, intermediate or terminal positions. The use of such functions brings an extra ﬂexibility to the formulated suﬃcient optimality condition compared to the standard approach. The derived conditions admit a natural modiﬁcation for problems of local (strong) minimum. One can expect that these results can be used for further strengthening of the discrete-time minimum principle up to a suﬃcient optimality condition, that would not require the convexity of the systems’ godograph.The work essentially relies on related results of Professor V.I. Gurman.