American Institute of Mathematical Sciences

Advanced
2012, 2(3): 619-630. doi: 10.3934/naco.2012.2.619

Noether's symmetry Theorem for variational and optimal control problems with time delay

Gastão S. F. Frederico 1, and  Delfim F. M. Torres 2,

1. 

Department of Science and Technology, University of Cape Verde, Praia, Santiago, Cape Verde, Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
2. 

CIDMA — Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

Received  March 2012 Revised  March 2012 Published  August 2012

We extend the DuBois--Reymond necessary optimality condition and Noether's symmetry theorem to the time delay variational setting. Both Lagrangian and Hamiltonian versions of Noether's theorem are proved, covering problems of the calculus of variations and optimal control with delays.
Keywords: DuBois--Reymond necessary optimality condition, Noether's theorem., Time delay, constants of motion, invariance, symmetries.
Mathematics Subject Classification: Primary: 49K05, 49S0.
Citation: Gastão S. F. Frederico, Delfim F. M. Torres. Noether's symmetry Theorem for variational and optimal control problems with time delay. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 619-630. doi: 10.3934/naco.2012.2.619
References:
[1]

O. P. Agrawal, J. Gregory and K. Pericak-Spector, A Bliss-type multiplier rule for constrained variational problems with time delay,, J. Math. Anal. Appl., 210 (1997), 702.  doi: 10.1006/jmaa.1997.5427.  Google Scholar

[2]

Z. Bartosiewicz and D. F. M. Torres, Noether's theorem on time scales,, J. Math. Anal. Appl., 342 (2008), 1220.  doi: 10.1016/j.jmaa.2008.01.018.  Google Scholar

[3]

M. Basin, "New Trends in Optimal Filtering and Control for Polynomial and Time-Delay Systems,", Lecture Notes in Control and Information Sciences, (2008).   Google Scholar

[4]

G. V. Bokov, Pontryagin's maximum principle in a problem with time delay,, J. Math. Sci. (N. Y.), 172 (2011), 623.  doi: 10.1007/s10958-011-0208-y.  Google Scholar

[5]

J. Cresson, G. S. F. Frederico and D. F. M. Torres, Constants of motion for non-differentiable quantum variational problems,, Topol. Methods Nonlinear Anal., 33 (2009), 217.   Google Scholar

[6]

G. S. F. Frederico, "Generalizations of Noether's Theorem in the Calculus of Variations and Optimal Control,", Ph.D. thesis, (2009).   Google Scholar

[7]

G. S. F. Frederico and D. F. M. Torres, Nonconservative Noether's theorem in optimal control,, Int. J. Tomogr. Stat., 5 (2007), 109.   Google Scholar

[8]

G. S. F. Frederico and D. F. M. Torres, A formulation of Noether's theorem for fractional problems of the calculus of variations,, J. Math. Anal. Appl., 334 (2007), 834.  doi: 10.1016/j.jmaa.2007.01.013.  Google Scholar

[9]

G. S. F. Frederico and D. F. M. Torres, Fractional Noether's theorem in the Riesz-Caputo sense,, Appl. Math. Comput., 217 (2010), 1023.  doi: 10.1016/j.amc.2010.01.100.  Google Scholar

[10]

L. Göllmann, D. Kern and H. Maurer, Optimal control problems with delays in state and control variables subject to mixed control-state constraints,, Optimal Control Appl. Methods, 30 (2009), 341.  doi: 10.1002/oca.843.  Google Scholar

[11]

P. D. F. Gouveia and D. F. M. Torres, Automatic computation of conservation laws in the calculus of variations and optimal control,, Comput. Methods Appl. Math., 5 (2005), 387.   Google Scholar

[12]

P. D. F. Gouveia and D. F. M. Torres, Computing ODE symmetries as abnormal variational symmetries,, Nonlinear Anal., 71 (2009).  doi: 10.1016/j.na.2008.10.009.  Google Scholar

[13]

P. D. F. Gouveia, D. F. M. Torres and E. A. M. Rocha, Symbolic computation of variational symmetries in optimal control,, Control Cybernet., 35 (2006), 831.   Google Scholar

[14]

D. K. Hughes, Variational and optimal control problems with delayed argument,, J. Optimization Theory Appl., 2 (1968), 1.  doi: 10.1007/BF00927159.  Google Scholar

[15]

G. L. Kharatishvili, A maximum principle in extremal problems with delays,, in, (1967), 26.   Google Scholar

[16]

G. L. Kharatishvili and T. A. Tadumadze, Formulas for the variation of a solution and optimal control problems for differential equations with retarded arguments,, J. Math. Sci. (N. Y.), 140 (2007), 1.  doi: 10.1007/s10958-007-0412-y.  Google Scholar

[17]

N. Martins and D. F. M. Torres, Noether's symmetry theorem for nabla problems of the calculus of variations,, Appl. Math. Lett., 23 (2010), 1432.  doi: 10.1016/j.aml.2010.07.013.  Google Scholar

[18]

M. N. Oğuztöreli, "Time-Lag Control Systems,", Mathematics in Science and Engineering, (1966).   Google Scholar

[19]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes,", Translated from the Russian by K. N. Trirogoff; edited by L. W. Neustadt Interscience Publishers John Wiley & Sons, (1962).   Google Scholar

[20]

E. A. M. Rocha and D. F. M. Torres, Quadratures for Pontryagin extremals for optimal control problems,, Control Cybernet., 35 (2006), 947.   Google Scholar

[21]

D. F. M. Torres, Conservation laws in optimal control,, in, (2001), 287.   Google Scholar

[22]

D. F. M. Torres, The role of symmetry in the regularity properties of optimal controls,, in, (2004), 1488.   Google Scholar

[23]

D. F. M. Torres, Carathéodory equivalence, Noether theorems, and Tonelli full-regularity in the calculus of variations and optimal control,, J. Math. Sci. (N. Y.), 120 (2004), 1032.  doi: 10.1023/B:JOTH.0000013565.78376.fb.  Google Scholar

[24]

D. F. M. Torres, Proper extensions of Noether's symmetry theorem for nonsmooth extremals of the calculus of variations,, Commun. Pure Appl. Anal., 3 (2004), 491.  doi: 10.3934/cpaa.2004.3.491.  Google Scholar

show all references

References:
[1]

O. P. Agrawal, J. Gregory and K. Pericak-Spector, A Bliss-type multiplier rule for constrained variational problems with time delay,, J. Math. Anal. Appl., 210 (1997), 702.  doi: 10.1006/jmaa.1997.5427.  Google Scholar

[2]

Z. Bartosiewicz and D. F. M. Torres, Noether's theorem on time scales,, J. Math. Anal. Appl., 342 (2008), 1220.  doi: 10.1016/j.jmaa.2008.01.018.  Google Scholar

[3]

M. Basin, "New Trends in Optimal Filtering and Control for Polynomial and Time-Delay Systems,", Lecture Notes in Control and Information Sciences, (2008).   Google Scholar

[4]

G. V. Bokov, Pontryagin's maximum principle in a problem with time delay,, J. Math. Sci. (N. Y.), 172 (2011), 623.  doi: 10.1007/s10958-011-0208-y.  Google Scholar

[5]

J. Cresson, G. S. F. Frederico and D. F. M. Torres, Constants of motion for non-differentiable quantum variational problems,, Topol. Methods Nonlinear Anal., 33 (2009), 217.   Google Scholar

[6]

G. S. F. Frederico, "Generalizations of Noether's Theorem in the Calculus of Variations and Optimal Control,", Ph.D. thesis, (2009).   Google Scholar

[7]

G. S. F. Frederico and D. F. M. Torres, Nonconservative Noether's theorem in optimal control,, Int. J. Tomogr. Stat., 5 (2007), 109.   Google Scholar

[8]

G. S. F. Frederico and D. F. M. Torres, A formulation of Noether's theorem for fractional problems of the calculus of variations,, J. Math. Anal. Appl., 334 (2007), 834.  doi: 10.1016/j.jmaa.2007.01.013.  Google Scholar

[9]

G. S. F. Frederico and D. F. M. Torres, Fractional Noether's theorem in the Riesz-Caputo sense,, Appl. Math. Comput., 217 (2010), 1023.  doi: 10.1016/j.amc.2010.01.100.  Google Scholar

[10]

L. Göllmann, D. Kern and H. Maurer, Optimal control problems with delays in state and control variables subject to mixed control-state constraints,, Optimal Control Appl. Methods, 30 (2009), 341.  doi: 10.1002/oca.843.  Google Scholar

[11]

P. D. F. Gouveia and D. F. M. Torres, Automatic computation of conservation laws in the calculus of variations and optimal control,, Comput. Methods Appl. Math., 5 (2005), 387.   Google Scholar

[12]

P. D. F. Gouveia and D. F. M. Torres, Computing ODE symmetries as abnormal variational symmetries,, Nonlinear Anal., 71 (2009).  doi: 10.1016/j.na.2008.10.009.  Google Scholar

[13]

P. D. F. Gouveia, D. F. M. Torres and E. A. M. Rocha, Symbolic computation of variational symmetries in optimal control,, Control Cybernet., 35 (2006), 831.   Google Scholar

[14]

D. K. Hughes, Variational and optimal control problems with delayed argument,, J. Optimization Theory Appl., 2 (1968), 1.  doi: 10.1007/BF00927159.  Google Scholar

[15]

G. L. Kharatishvili, A maximum principle in extremal problems with delays,, in, (1967), 26.   Google Scholar

[16]

G. L. Kharatishvili and T. A. Tadumadze, Formulas for the variation of a solution and optimal control problems for differential equations with retarded arguments,, J. Math. Sci. (N. Y.), 140 (2007), 1.  doi: 10.1007/s10958-007-0412-y.  Google Scholar

[17]

N. Martins and D. F. M. Torres, Noether's symmetry theorem for nabla problems of the calculus of variations,, Appl. Math. Lett., 23 (2010), 1432.  doi: 10.1016/j.aml.2010.07.013.  Google Scholar

[18]

M. N. Oğuztöreli, "Time-Lag Control Systems,", Mathematics in Science and Engineering, (1966).   Google Scholar

[19]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes,", Translated from the Russian by K. N. Trirogoff; edited by L. W. Neustadt Interscience Publishers John Wiley & Sons, (1962).   Google Scholar

[20]

E. A. M. Rocha and D. F. M. Torres, Quadratures for Pontryagin extremals for optimal control problems,, Control Cybernet., 35 (2006), 947.   Google Scholar

[21]

D. F. M. Torres, Conservation laws in optimal control,, in, (2001), 287.   Google Scholar

[22]

D. F. M. Torres, The role of symmetry in the regularity properties of optimal controls,, in, (2004), 1488.   Google Scholar

[23]

D. F. M. Torres, Carathéodory equivalence, Noether theorems, and Tonelli full-regularity in the calculus of variations and optimal control,, J. Math. Sci. (N. Y.), 120 (2004), 1032.  doi: 10.1023/B:JOTH.0000013565.78376.fb.  Google Scholar

[24]

D. F. M. Torres, Proper extensions of Noether's symmetry theorem for nonsmooth extremals of the calculus of variations,, Commun. Pure Appl. Anal., 3 (2004), 491.  doi: 10.3934/cpaa.2004.3.491.  Google Scholar

[1]

Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Variational problems of Herglotz type with time delay: DuBois--Reymond condition and Noether's first theorem. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4593-4610. doi: 10.3934/dcds.2015.35.4593
[2]

Colin J. Cotter, Michael John Priestley Cullen. Particle relabelling symmetries and Noether's theorem for vertical slice models. Journal of Geometric Mechanics, 2019, 11 (2) : 139-151. doi: 10.3934/jgm.2019007
[3]

Božzidar Jovanović. Symmetries of line bundles and Noether theorem for time-dependent nonholonomic systems. Journal of Geometric Mechanics, 2018, 10 (2) : 173-187. doi: 10.3934/jgm.2018006
[4]

Gaoxi Li, Zhongping Wan, Jia-wei Chen, Xiaoke Zhao. Necessary optimality condition for trilevel optimization problem. Journal of Industrial & Management Optimization, 2020, 16 (1) : 55-70. doi: 10.3934/jimo.2018140
[5]

Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Noether's theorem for higher-order variational problems of Herglotz type. Conference Publications, 2015, 2015 (special) : 990-999. doi: 10.3934/proc.2015.990
[6]

Delfim F. M. Torres. Proper extensions of Noether's symmetry theorem for nonsmooth extremals of the calculus of variations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 491-500. doi: 10.3934/cpaa.2004.3.491
[7]

Monika Dryl, Delfim F. M. Torres. Necessary optimality conditions for infinite horizon variational problems on time scales. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 145-160. doi: 10.3934/naco.2013.3.145
[8]

RazIye Mert, A. Zafer. A necessary and sufficient condition for oscillation of second order sublinear delay dynamic equations. Conference Publications, 2011, 2011 (Special) : 1061-1067. doi: 10.3934/proc.2011.2011.1061
[9]

Clara Carlota, António Ornelas. The DuBois-Reymond differential inclusion for autonomous optimal control problems with pointwise-constrained derivatives. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 467-484. doi: 10.3934/dcds.2011.29.467
[10]

Gianluca Gorni, Gaetano Zampieri. Lagrangian dynamics by nonlocal constants of motion. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020216
[11]

Dirk Pauly. On Maxwell's and Poincaré's constants. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 607-618. doi: 10.3934/dcdss.2015.8.607
[12]

Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Noether currents for higher-order variational problems of Herglotz type with time delay. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 91-102. doi: 10.3934/dcdss.2018006
[13]

B. Cantó, C. Coll, A. Herrero, E. Sánchez, N. Thome. Pole-assignment of discrete time-delay systems with symmetries. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 641-649. doi: 10.3934/dcdsb.2006.6.641
[14]

Jehad O. Alzabut. A necessary and sufficient condition for the existence of periodic solutions of linear impulsive differential equations with distributed delay. Conference Publications, 2007, 2007 (Special) : 35-43. doi: 10.3934/proc.2007.2007.35
[15]

Francesco Fassò, Andrea Giacobbe, Nicola Sansonetto. On the number of weakly Noetherian constants of motion of nonholonomic systems. Journal of Geometric Mechanics, 2009, 1 (4) : 389-416. doi: 10.3934/jgm.2009.1.389
[16]

Ferenc A. Bartha, Ábel Garab. Necessary and sufficient condition for the global stability of a delayed discrete-time single neuron model. Journal of Computational Dynamics, 2014, 1 (2) : 213-232. doi: 10.3934/jcd.2014.1.213
[17]

John Fogarty. On Noether's bound for polynomial invariants of a finite group. Electronic Research Announcements, 2001, 7: 5-7.

[18]

Shahlar F. Maharramov. Necessary optimality conditions for switching control problems. Journal of Industrial & Management Optimization, 2010, 6 (1) : 47-55. doi: 10.3934/jimo.2010.6.47
[19]

Piernicola Bettiol, Nathalie Khalil. Necessary optimality conditions for average cost minimization problems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2093-2124. doi: 10.3934/dcdsb.2019086
[20]

Francesco Fassò, Andrea Giacobbe, Nicola Sansonetto. Linear weakly Noetherian constants of motion are horizontal gauge momenta. Journal of Geometric Mechanics, 2012, 4 (2) : 129-136. doi: 10.3934/jgm.2012.4.129

 Impact Factor: 

Tools

Metrics

  • PDF downloads (27)
  • HTML views (0)
  • Cited by (12)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences