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Numeric algorithm for optimal impulsive control based on feedback maximum principle

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Abstract

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.

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Acknowledgements

The work is partially supported by the Russian Foundation for Basic Research, Grants Nos. 16-31-60030, 16-08-00272 and 17-01-00733. We thank professor A.Yu. Gornov for cooperation, which lets us provide validation and comparison of our algorithm.

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  1. Matrosov Institute for System Dynamics and Control Theory, Siberian Branch of the Russian Academy of Sciences, Irkutsk, Russia

    Stepan P. Sorokin & Maxim V. Staritsyn

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  1. Stepan P. Sorokin
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  2. Maxim V. Staritsyn
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Correspondence to Stepan P. Sorokin.

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Sorokin, S.P., Staritsyn, M.V. Numeric algorithm for optimal impulsive control based on feedback maximum principle. Optim Lett 13, 1953–1967 (2019). https://doi.org/10.1007/s11590-018-1344-9

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  • DOI: https://doi.org/10.1007/s11590-018-1344-9

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