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Optimal Control and Pontryagin’s Maximum Principle

Encyclopedia of Systems and Control


Pontryagin’s Maximum Principle is a collection of conditions that must be satisfied by solutions of a class of optimization problems involving dynamic constraints called optimal control problems. It unifies many classical necessary conditions from the calculus of variations. This article provides an overview of the Maximum Principle, including free-time and nonsmooth versions. A time-optimal control problem is solved as an example to illustrate its application.

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Recommended Reading

  • Berkovitz LD (1974) Optimal control theory. Applied mathematical sciences, vol 12. Springer, New York For references that also cover advances in the theory of necessary conditions related to the Maximum Principle, based on techniques of Nonsmooth Analysis, we refer to the following:

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  • Bryson AE, Ho YC (1969) Applied optimal control. Blaisdell, New York. And (in revised edition) Halstead Press (a division of John Wiley and Sons), New York, 1975

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

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  • Clarke FH (1983) Optimization and nonsmooth analysis. Wiley-Interscience, New York. Reprinted as volume 5 of Classics in applied mathematics. SIAM, Philadelphia (1990)

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  • Clarke FH (2013) Functional analysis, calculus of variations and optimal control. Graduate texts in mathematics. Springer, London Engineering texts illustrating the application of the Maximum Principle to solve problems of optimal control and design, in flight mechanics and other areas, include the following:

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

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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 on the Maximum Principle and related control theory include:

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  • Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mischenko EF (1962) The mathematical theory of optimal processes (trans: Tririgoff KN; Neustadt LW (ed)). Wiley, New York It bears the name of L S Pontryagin, because of his role as leader of the research group at the Steklov Institute, Moscow, which achieved this advance. But the first proof is attributed to Boltyanskii.Reflections on the origins of the Maximum Principle appear in

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  • Ross IM (2009) A primer on Pontryagin’s principle in optimal control. Collegiate Publishers, San Francisco

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  • Vinter RB (2000) Optimal control. Birkhaüser, Boston

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Correspondence to Richard B. Vinter .

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© 2013 Springer-Verlag London

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Vinter, R. (2013). Optimal Control and Pontryagin’s Maximum Principle. In: Baillieul, J., Samad, T. (eds) Encyclopedia of Systems and Control. Springer, London.

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    Optimal Control and Pontryagin’s Maximum Principle
    02 March 2020


  2. Original

    Optimal Control and Pontryagin’s Maximum Principle
    14 March 2014