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

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

Simão P. S. Santos 1, , Natália Martins 2, and  Delfim F. M. Torres 3,

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.
Keywords: invariance, Euler-Lagrange equations, Noether's theorem., optimal control, DuBois-Reymond condition, Herglotz's problems, higher-order calculus of variations.
Mathematics Subject Classification: Primary: 49K15, 49S05; Secondary: 49K05, 34H0.
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), no. 3, 291-304.  Google Scholar

[2]

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

[3]

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

[4]

E. Noether, Invariante Variationsprobleme, Nachr. v. d. Ges. d. Wiss. zu Gttingen, (1918), 235-257. Google Scholar

[5]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The mathematical theory of optimal processes, Interscience Publishers, John Wiley and Sons Inc, New York, London, 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), no. 4, 409-419.  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, Vol. 499, Springer, (2015), 107-117. Google Scholar

[8]

D. F. M. Torres, Conservation laws in optimal control, in Dynamics, bifurcations, and control (Kloster Irsee, 2001), 287-296,  Google Scholar

[9]

D. F. M. Torres, On the Noether theorem for optimal control, European Journal of Control, 8 (2002), no. 1 , 56-63. Google Scholar

[10]

D. F. M. Torres, Quasi-invariant optimal control problems, Port. Math. (N.S.), 61 (2004), no. 1, 97-114.  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), no. 3, 491-500.  Google Scholar

[12]

B. van Brunt, The calculus of variations, Universitext, Springer-Verlag, New York, 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), no. 3, 291-304.  Google Scholar

[2]

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

[3]

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

[4]

E. Noether, Invariante Variationsprobleme, Nachr. v. d. Ges. d. Wiss. zu Gttingen, (1918), 235-257. Google Scholar

[5]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The mathematical theory of optimal processes, Interscience Publishers, John Wiley and Sons Inc, New York, London, 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), no. 4, 409-419.  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, Vol. 499, Springer, (2015), 107-117. Google Scholar

[8]

D. F. M. Torres, Conservation laws in optimal control, in Dynamics, bifurcations, and control (Kloster Irsee, 2001), 287-296,  Google Scholar

[9]

D. F. M. Torres, On the Noether theorem for optimal control, European Journal of Control, 8 (2002), no. 1 , 56-63. Google Scholar

[10]

D. F. M. Torres, Quasi-invariant optimal control problems, Port. Math. (N.S.), 61 (2004), no. 1, 97-114.  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), no. 3, 491-500.  Google Scholar

[12]

B. van Brunt, The calculus of variations, Universitext, Springer-Verlag, New York, 2004.  Google Scholar

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