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Applicable Analysis
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Volume 93, 2014 - Issue 1
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Articles

Noether’s theorem for non-smooth extremals of variational problems with time delay

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Pages 153-170 | Received 07 Nov 2012, Accepted 19 Dec 2012, Published online: 29 Jan 2013

References

  • Kosmann-Schwarzbach Y. The Noether theorems, translated, revised and augmented from the 2006 French edition by Bertram E. Schwarzbach, Sources and Studies in the history of mathematics and physical sciences. New York(NY): Springer; 2011.
  • Bartosiewicz Z, Torres DFM. Noether’s theorem on time scales. Journal of Mathematical Analysis and Applications. 2008;342(2):1220–6.
  • Frederico GSF. Generalizations of Noether’s theorem in the calculus of variations and optimal control [Ph.D. thesis], University of Cape Verde; 2009.
  • Frederico GSF, Torres DFM. A formulation of Noether’s theorem for fractional problems of the calculus of variations. Journal of Mathematical Analysis and Applications. 2007;334(2):834–46.
  • Frederico GSF, Torres DFM. Fractional Noether’s theorem in the Riesz-Caputo sense. Applied Mathematics and Computation. 2010;217(3):1023–33.
  • Gouveia PDF, Torres DFM, Rocha EAM. Symbolic computation of variational symmetries in optimal control. Control Cybernetics. 2006;35(4):831–49.
  • Malinowska AB, Torres DFM. Introduction to the fractional calculus of variations. Singapore: Imperial College Press, London & World Scientific Publishing; 2012.
  • Martins N, Torres DFM. Noether’s symmetry theorem for nabla problems of the calculus of variations. Applied Mathematics Letters. 2010;23(12):1432–8.
  • Torres DFM. On the Noether theorem for optimal control. European Journal of Control. 2002;8(1):56–63.
  • Frederico GSF, Torres DFM. Noether’s symmetry theorem for variational and optimal control problems with time delay. Numerical Algebra, Control and Optimization. 2012;2(3):619–30.
  • Göllmann L, Kern D, Maurer H. Optimal control problems with delays in state and control variables subject to mixed control-state constraints. Optimal Control Applications and Methods. 2009;30(4):341–65.
  • Basin, Michael, New trends in optimal filtering and control for polynomial and time-delay systems, Lecture Notes in Control and Information Sciences, vol. 380, Berlin: Springer-Verlag; 2008.xxiv+206.
  • Bokov GV. Pontryagin’s maximum principle in a problem with time delay. Journal of Mathematical Sciences (NY). 2011;172(5):623–34.
  • Hughes DK. Variational and optimal control problems with delayed argument. Journal of Optimization Theory and Applications. 1968;2:1–14.
  • Kharatishvili GL. A maximum principle in extremal problems with delays. in: Mathematical Theory of Control (Proc. Conf., Los Angeles, Calif., 1967), New York (NY): Academic Press; 1967; p. 26–34.
  • Kharatishvili GL, Tadumadze TA. Formulas for the variation of a solution and optimal control problems for differential equations with retarded arguments. Journal of Mathematical Sciences (NY). 2007;140(1):1–175.
  • Rosenblueth JF. Systems with time delay in the calculus of variations: the method of steps. IMA Journal of Mathematical Control and Information. 1988;5(4):285–99.
  • Jost J, Li-Jost X. Calculus of variations. Cambridge studies in advanced mathematics. Vol. 64. Cambridge: Cambridge University Press; 1998.
  • Troutman JL. Variational calculus and optimal control. 2nd ed. Undergraduate texts in mathematics. New York(NY): Springer; 1996.
  • Torres DFM. Proper extensions of Noether’s symmetry theorem for non-smooth extremals of the calculus of variations. Communications on Pure and Applied Analysis. 2004;3(3):491–500.

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