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 discretetime 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.
1. Gurman V.I. Printsip rasshireniya v zadachakh upravleniya [The Extension Principle in Control Problems]. Moscow, Nauka, 1997. (in Russian)
2. Dykhta V.A., Sorokin S.P. Hamilton-Jacobi inequalities and the optimality conditions in the problems of control with common end constraints. Automation and Remote Control, 2011, vol. 72, no 9, p. 1808-1821.
3. Krasovskii N.N.,Subbotin A.I. Game-theoretical control problems. New York, Springer, 1988.
4. Krotov V.F. Global Methods in Optimal Control Theory. NewYork,Marcel Dekker, 1996.
5. Clarke F.H., Ledyaev Yu.S., Stern R.J., Wolenski P.R. Nonsmooth Analysis and Control Theory. Grad. Texts in Math., New York, Springer-Verlag, vol. 178, 1998.