Abstract
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.
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References
J.P. Aubin and A. Cellina. Differential Inclusions. Springer-Verlag, 1984.
F.H. Clarke. Necessary Conditions for Nonsmooth Problems in Optimal Control and the Calculus of Variations. Doctoral thesis, University of Washington, 1973. (Thesis director: R.T. Rockafellar).
F.H. Clarke. The Euler-Lagrange differential inclusion. J. Differential Equations, 19 (1975), 80–90.
F.H. Clarke. Maximum principles without differentiability. Bulletin Amer. Math. Soc., 81 (1975), 219–2221.
F.H. Clarke. Optimization and Nonsmooth Analysis. Wiley-Interscience, New York, 1983. Republished as Vol. 5 of Classics in Applied Mathematics, SIAM, 1990.
F.H. Clarke, Yu. S. Ledyaev, R. J. Stern, and P. R. Wolenski. Nonsmooth Analysis and Control Theory. Graduate Texts in Mathematics, Vol. 178. Springer-Verlag, New York, 1998.
F. Clarke. Necessary Conditions in Dynamic Optimization. Memoirs of the Amer. Math. Soc., No. 816, Vol. 173. 2005.
F. Clarke. The maximum principle in optimal control. J. Cybernetics and Control, 34 (2005), 709–722.
R.B. Vinter. Optimal Control. Birkhäuser, Boston, 2000.
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Dedicated to Arrigo Cellina and James Yorke
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© 2007 Birkhäuser Verlag Basel/Switzerland
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Clarke, F. (2007). Necessary Conditions in Optimal Control and in the Calculus of Variations. In: Staicu, V. (eds) Differential Equations, Chaos and Variational Problems. Progress in Nonlinear Differential Equations and Their Applications, vol 75. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8482-1_11
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DOI: https://doi.org/10.1007/978-3-7643-8482-1_11
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-8481-4
Online ISBN: 978-3-7643-8482-1
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