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Optimal Control and Pontryagin’s Maximum Principle

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Encyclopedia of Systems and Control
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Pontryagin’s Maximum Principle is a set of conditions providing information about solutions to optimal control problems; that is, optimization problems with differential equation constraints. It unifies and extends many classical necessary conditions from the calculus of variations. This article provides an overview of the Maximum Principle, including free end-time and nonsmooth versions. A time-optimal control problem is solved, to illustrate its application.

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  • Clarke FH (1976) The maximum principle under minimal hypotheses. SIAM J Control Optim 14:1078– 1091

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  • Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mischenko EF (1962) The mathematical theory of optimal processes. Tririgoff KN Transl., Neustadt LW Ed. Wiley, New York

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  • Warga J (1983) Optimization and controllability without differentiability assumptions. SIAM J Control Optim 21:239–260

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Recommended Reading

  • Berkovitz LD (1974) Optimal control theory. Applied mathematical sciences, vol 12. Springer, New York

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  • Bryson AE, Ho Y-C (1975),Applied optimal control (Revised edn), Halstead Press (a division of John Wiley and Sons), New York

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  • Fleming WH, Rishel RW (1975) Deterministic and stochastic optimal control. Springer, New York

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  • Ioffe AD, Tihomirov VM (1979) Theory of extremal problems. North-Holland, Amsterdam Reflections on the origins of the Maximum Principle appear in:

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  • Pesch HJ, Plail M (2009) The maximum principle of optimal control: a history of ingenious ideas and missed opportunities. Control Cybern 38:973–995 Expository texts that also cover advances in the theory of necessary conditions related to the Maximum Principle, based on techniques of Nonsmooth Analysis:

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  • Clarke FH (1983) Optimization and nonsmooth analysis. Wiley-Interscience, New York

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  • Clarke FH (2013) Functional analysis, calculus of variations and optimal control. Graduate texts in mathematics. Springer, London

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  • Vinter RB (2000) Optimal control. Birkhaüser, Boston Engineering text illustrating the application of the Maximum Principle to solve problems of optimal control and design, in flight mechanics and other areas

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  • Bryson AE (1999) Dynamic optimization. Addison Wesley Longman, Menlo Park

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Correspondence to Richard Vinter .

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Vinter, R. (2020). Optimal Control and Pontryagin’s Maximum Principle. In: Baillieul, J., Samad, T. (eds) Encyclopedia of Systems and Control. Springer, London.

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  • Publisher Name: Springer, London

  • Print ISBN: 978-1-4471-5102-9

  • Online ISBN: 978-1-4471-5102-9

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Chapter history

  1. Latest

    Optimal Control and Pontryagin’s Maximum Principle
    02 March 2020


  2. Original

    Optimal Control and Pontryagin’s Maximum Principle
    14 March 2014