Numeric algorithm for optimal impulsive control based on feedback maximum principle

• Published in: Optimization letters Vol. 13; no. 8; pp. 1953 - 1967
• Main Authors: Sorokin, Stepan P., Staritsyn, Maxim V.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2019
• Subjects: Computational Intelligence; Mathematics; Mathematics and Statistics; Numerical and Computational Physics; Operations Research/Decision Theory; Optimization; Original Paper; Simulation; Feedback necessary optimality conditions; Impulsive control; Discrete-time optimal control; Numeric algorithms for optimal control
• ISSN: 1862-4472; 1862-4480
• DOI: 10.1007/s11590-018-1344-9

In this short note, we present an approach for numeric analysis of a class of nonlinear optimal impulsive control problems with states of bounded variation. The approach is based on feedback necessary optimality condition laying in the formalism of the Pontryagin maximum principle (employing only the related standard constructions), but involving feedback control variations of an “extremal” structure. We make a “double reduction” of the impulsive control problem: first, we perform a well-known equivalent transform of the measure-driven system to an ordinary terminally-constrained control system and, second, pass to its discrete-time counterpart. For the resulted discrete control problem, we present a necessary condition of global optimality called the discrete feedback maximum principle. Based on this optimality condition, we elaborate a nonlocal numeric algorithm, which can, potentially, improve nonoptimal extrema of the discrete maximum principle. Due to a specific structure of the investigated model, the algorithm admits a deep specification. As an illustration of our approach, we present a numeric implementation of an academic example—a singular version of a generalized Sethi–Thompson investment problem from mathematical economics.