2015, 2015(special): 990-999. doi: 10.3934/proc.2015.990

Noether's theorem for higher-order variational problems of Herglotz type

1. 

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

2. 

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

3. 

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

Received  September 2014 Revised  July 2015 Published  November 2015

We approach higher-order variational problems of Herglotz type from an optimal control point of view. Using optimal control theory, we derive a generalized Euler--Lagrange equation, transversality conditions, DuBois--Reymond necessary optimality condition and Noether's theorem for Herglotz's type higher-order variational problems, valid for piecewise smooth functions.
Citation: 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
References:
[1]

G. S. F. Frederico and D. F. M. Torres, Fractional isoperimetric Noether's theorem in the Riemann-Liouville sense,, Rep. Math. Phys., 71 (2013), 291. Google Scholar

[2]

G. Herglotz, Berührungstransformationen,, Lectures at the University of Göttingen, (1930). Google Scholar

[3]

S. Lenhart and J. T. Workman, Optimal control applied to biological models,, Chapman & Hall/CRC, (2007). Google Scholar

[4]

E. Noether, Invariante Variationsprobleme,, Nachr. v. d. Ges. d. Wiss. zu Gttingen, (1918), 235. Google Scholar

[5]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The mathematical theory of optimal processes,, Interscience Publishers, (1962). Google Scholar

[6]

S. P. S. Santos, N. Martins and D. F. M. Torres, Higher-order variational problems of Herglotz type,, Vietnam J. Math., 42 (2014), 409. Google Scholar

[7]

S. P. S. Santos, N. Martins and D. F. M. Torres, An optimal control approach to Herglotz variational problems, Optimization in the Natural Sciences (eds. A. Plakhov, T. Tchemisova and A. Freitas),, Communications in Computer and Information Science, (2015), 107. Google Scholar

[8]

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

[9]

D. F. M. Torres, On the Noether theorem for optimal control,, European Journal of Control, 8 (2002), 56. Google Scholar

[10]

D. F. M. Torres, Quasi-invariant optimal control problems,, Port. Math. (N.S.), 61 (2004), 97. Google Scholar

[11]

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), 3 (2004), 491. Google Scholar

[12]

B. van Brunt, The calculus of variations,, Universitext, (2004). Google Scholar

show all references

References:
[1]

G. S. F. Frederico and D. F. M. Torres, Fractional isoperimetric Noether's theorem in the Riemann-Liouville sense,, Rep. Math. Phys., 71 (2013), 291. Google Scholar

[2]

G. Herglotz, Berührungstransformationen,, Lectures at the University of Göttingen, (1930). Google Scholar

[3]

S. Lenhart and J. T. Workman, Optimal control applied to biological models,, Chapman & Hall/CRC, (2007). Google Scholar

[4]

E. Noether, Invariante Variationsprobleme,, Nachr. v. d. Ges. d. Wiss. zu Gttingen, (1918), 235. Google Scholar

[5]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The mathematical theory of optimal processes,, Interscience Publishers, (1962). Google Scholar

[6]

S. P. S. Santos, N. Martins and D. F. M. Torres, Higher-order variational problems of Herglotz type,, Vietnam J. Math., 42 (2014), 409. Google Scholar

[7]

S. P. S. Santos, N. Martins and D. F. M. Torres, An optimal control approach to Herglotz variational problems, Optimization in the Natural Sciences (eds. A. Plakhov, T. Tchemisova and A. Freitas),, Communications in Computer and Information Science, (2015), 107. Google Scholar

[8]

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

[9]

D. F. M. Torres, On the Noether theorem for optimal control,, European Journal of Control, 8 (2002), 56. Google Scholar

[10]

D. F. M. Torres, Quasi-invariant optimal control problems,, Port. Math. (N.S.), 61 (2004), 97. Google Scholar

[11]

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), 3 (2004), 491. Google Scholar

[12]

B. van Brunt, The calculus of variations,, Universitext, (2004). 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]

Agnieszka B. Malinowska, Delfim F. M. Torres. Euler-Lagrange equations for composition functionals in calculus of variations on time scales. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 577-593. doi: 10.3934/dcds.2011.29.577

[3]

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

[4]

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

[5]

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

[6]

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

[7]

Hans Josef Pesch. Carathéodory's royal road of the calculus of variations: Missed exits to the maximum principle of optimal control theory. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 161-173. doi: 10.3934/naco.2013.3.161

[8]

Leonardo Colombo, David Martín de Diego. Higher-order variational problems on lie groups and optimal control applications. Journal of Geometric Mechanics, 2014, 6 (4) : 451-478. doi: 10.3934/jgm.2014.6.451

[9]

Anthony Bloch, Leonardo Colombo, Fernando Jiménez. The variational discretization of the constrained higher-order Lagrange-Poincaré equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 309-344. doi: 10.3934/dcds.2019013

[10]

Eduardo Martínez. Higher-order variational calculus on Lie algebroids. Journal of Geometric Mechanics, 2015, 7 (1) : 81-108. doi: 10.3934/jgm.2015.7.81

[11]

Min Zhu. On the higher-order b-family equation and Euler equations on the circle. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 3013-3024. doi: 10.3934/dcds.2014.34.3013

[12]

Giovanni Bonfanti, Arrigo Cellina. The validity of the Euler-Lagrange equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 511-517. doi: 10.3934/dcds.2010.28.511

[13]

Menita Carozza, Jan Kristensen, Antonia Passarelli di Napoli. On the validity of the Euler-Lagrange system. Communications on Pure & Applied Analysis, 2015, 14 (1) : 51-62. doi: 10.3934/cpaa.2015.14.51

[14]

Michał Jóźwikowski, Mikołaj Rotkiewicz. Bundle-theoretic methods for higher-order variational calculus. Journal of Geometric Mechanics, 2014, 6 (1) : 99-120. doi: 10.3934/jgm.2014.6.99

[15]

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

[16]

Stefano Bianchini. On the Euler-Lagrange equation for a variational problem. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 449-480. doi: 10.3934/dcds.2007.17.449

[17]

Bernard Dacorogna, Giovanni Pisante, Ana Margarida Ribeiro. On non quasiconvex problems of the calculus of variations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 961-983. doi: 10.3934/dcds.2005.13.961

[18]

Felix Sadyrbaev. Nonlinear boundary value problems of the calculus of variations. Conference Publications, 2003, 2003 (Special) : 760-770. doi: 10.3934/proc.2003.2003.760

[19]

David F. Parker. Higher-order shallow water equations and the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 629-641. doi: 10.3934/dcdsb.2007.7.629

[20]

Aliang Xia, Jianfu Yang. Normalized solutions of higher-order Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 447-462. doi: 10.3934/dcds.2019018

 Impact Factor: 

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (0)

[Back to Top]