Necessary Conditions in Optimal Control and in the Calculus of Variations

  • Francis Clarke
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 75)


The goal of this article is to find, for the two most standard paradigms in dynamic optimization, the simplest proofs that can be based on the techniques invented and refined over the last thirty years in connection with the nonsmooth analysis approach. Specifically, we present a proof of Theorem 2.1 below, which asserts all the first-order necessary conditions for the basic problem in the calculus of variations, and a proof of Theorem 3.1, which is the Pontryagin maximum principle in a classical context.


Dynamic Optimization Nonsmooth Analysis Pontryagin Maximum Principle Classical Context Temporary Hypothesis 
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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2007

Authors and Affiliations

  • Francis Clarke
    • 1
  1. 1.Institut universitaire de France et Institut Camille JordanVilleurbanneFrance

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