Averaging and linear programming in some singularly perturbed problems of optimal control

Vladimir Gaitsgory*, Sergey Rossomakhine

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    11 Citations (Scopus)


    The paper aims at the development of an apparatus for analysis and construction of near optimal solutions of singularly perturbed (SP) optimal controls problems (that is, problems of optimal control of SP systems) considered on the infinite time horizon. We mostly focus on problems with time discounting criteria but a possibility of the extension of results to periodic optimization problems is discussed as well. Our consideration is based on earlier results on averaging of SP control systems and on linear programming formulations of optimal control problems. The idea that we exploit is to first asymptotically approximate a given problem of optimal control of the SP system by a certain averaged optimal control problem, then reformulate this averaged problem as an infinite-dimensional linear programming (LP) problem, and then approximate the latter by semi-infinite LP problems. We show that the optimal solution of these semi-infinite LP problems and their duals (that can be found with the help of a modification of an available LP software) allow one to construct near optimal controls of the SP system. We demonstrate the construction with two numerical examples.

    Original languageEnglish
    Pages (from-to)195-276
    Number of pages82
    JournalApplied Mathematics and Optimization
    Issue number2
    Publication statusPublished - Apr 2015

    Bibliographical note

    Erratum can be found in Applied Mathematics and Optimization volume 71(2), p 277-278, https://doi.org/10.1007/s00245-014-9265-1


    • singularly perturbed optimal control problems
    • averaging and linear programming
    • occupational measures
    • numerical solution


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