Open Access
Issue |
ESAIM: COCV
Volume 26, 2020
|
|
---|---|---|
Article Number | 35 | |
Number of page(s) | 38 | |
DOI | https://doi.org/10.1051/cocv/2019021 | |
Published online | 25 June 2020 |
- A.A. Agrachev and Y.L. Sachkov, Control theory from the geometric viewpoint, in Vol. 87 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin (2004). [Google Scholar]
- O.P. Agrawal, Formulation of Euler-Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272 (2002) 368–379. [Google Scholar]
- O.P. Agrawal, A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 38 (2004) 323–337. [Google Scholar]
- O.P. Agrawal, O. Defterli and D. Baleanu, Fractional optimal control problems with several state and control variables. J. Vib. Control 16 (2010) 1967–1976. [Google Scholar]
- H.M. Ali, F.L. Pereira and S.M.A. Gama, A new approach to the Pontryagin maximum principle for nonlinear fractional optimal control problems. Math. Methods Appl. Sci. 39 (2016) 3640–3649. [Google Scholar]
- R. Almeida and D.F.M. Torres, Calculus of variations with fractional derivatives and fractional integrals. Appl. Math. Lett. 22 (2009) 1816–1820. [Google Scholar]
- R. Almeida and D.F.M. Torres, Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives. Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 1490–1500. [Google Scholar]
- R. Almeida and D.F.M. Torres, A discrete method to solve fractional optimal control problems. Nonlinear Dyn. 80 (2015) 1811–1816. [Google Scholar]
- D. Baleanu and S.I. Muslih, Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives. Phys. Scripta 72 (2005) 119–121. [CrossRef] [MathSciNet] [Google Scholar]
- R.K. Biswas and S. Sen, Free final time fractional optimal control problems. J. Franklin Inst. 351 (2014) 941–951. [Google Scholar]
- J. Bonnans, The shooting approach to optimal control problems. IFAC Proc. Vol. 46 (2013) 281–292. [CrossRef] [Google Scholar]
- L. Bourdin, Existence of a weak solution for fractional Euler-Lagrange equations. J. Math. Anal. Appl. 399 (2013) 239–251. [Google Scholar]
- L. Bourdin, Cauchy-Lipschitz theory for fractional multi-order dynamics – state-transition matrices, Duhamel formulas and duality theorems. Differ. Int. Equ. 31 (2018) 559–594. [Google Scholar]
- L. Bourdin, Weighted Hölder continuity of Riemann-Liouville fractional integrals – application to regularity of solutions to fractional Cauchy problems with Carathéodory dynamics. To appear in Fract. Cal. Appl. Anal. (2019). Preprint hal-01726069. [Google Scholar]
- L. Bourdin and D. Idczak, A fractional fundamental lemma and a fractional integration by parts formula—applications to critical points of Bolza functionals and to linear boundary value problems. Adv. Differ. Equ. 20 (2015) 213–232. [Google Scholar]
- A. Bressan and B. Piccoli, Introduction to the mathematical theory of control, in Vol. 2 of AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences (AIMS), Springfield, MO (2007). [Google Scholar]
- H. Brezis, Functional analysis, Sobolev spaces and partial differential equations. Universitext, Springer, New York (2011). [Google Scholar]
- A.E. Bryson, Jr. and Y.C. Ho, Applied optimal control distributed by Halsted Press [John Wiley & Sons, New York-London-Sydney, 1975]. Optimization, estimation, and control, Revised printing. Hemisphere Publishing Corp. Washington, D.C. (1975). [Google Scholar]
- F. Bullo and A.D. Lewis, Geometric control of mechanical systems, Vol. 49 of Texts in Applied Mathematics. Springer-Verlag, New York (2005). [CrossRef] [Google Scholar]
- L. Cesari, Optimization—theory and applications, Vol. 17 of Applications of Mathematics. Springer-Verlag, New York (1983). [Google Scholar]
- J. Cresson, Fractional embedding of differential operators and Lagrangian systems. J. Math. Phys. 48 (2007) 033504. [Google Scholar]
- K. Diethelm, The analysis of fractional differential equations, Vol. 2004 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (2010). [CrossRef] [Google Scholar]
- I. Ekeland, On the variational principle. J. Math. Anal. Appl. 47 (1974) 324–353. [Google Scholar]
- H.O. Fattorini, Infinite-dimensional optimization and control theory, Vol. 62 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1999). [Google Scholar]
- A.F. Filippov, On some questions in the theory of optimal regulation: existence of a solution of the problem of optimal regulation in the class of bounded measurable functions. Vestnik Moskov. Univ. Ser. Mat. Meh. Astr. Fiz. Him. 1959 (1959) 25–32. [MathSciNet] [Google Scholar]
- G.S.F. Frederico and D.F.M. Torres, Fractional optimal control in the sense of Caputo and the fractional Noether’s theorem. Int. Math. Forum 3 (2008) 479–493. [Google Scholar]
- R.V. Gamkrelidze, Discovery of the maximum principle, in Mathematical events of the twentieth century. Springer, Berlin (2006) 85–99. [CrossRef] [Google Scholar]
- T.L. Guo, The necessary conditions of fractional optimal control in the sense of Caputo. J. Optim. Theory Appl. 156 (2013) 115–126. [Google Scholar]
- M.R. Hestenes, Calculus of variations and optimal control theory. Corrected reprint ofthe 1966 original. Robert E. Krieger Publishing Co., Inc., Huntington, N.Y. (1980). [Google Scholar]
- R. Hilfer, Applications of Fractional Calculus in Physics, Applications of Fractional Calculus in Physics. World Scientific (2000). [Google Scholar]
- D.G. Hull, Optimal control theory for applications. Mechanical Engineering Series. Springer-Verlag, New York (2003). [CrossRef] [Google Scholar]
- Z.D. Jelicic and N. Petrovacki, Optimality conditions and a solution scheme for fractional optimal control problems. Struct. Multidiscip. Optim. 38 (2009) 571–581. [CrossRef] [Google Scholar]
- V. Jurdjevic, Geometric control theory, Vol. 52 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1997). [Google Scholar]
- R. Kamocki, On the existence of optimal solutions to fractional optimal control problems. Appl. Math. Comput. 235 (2014) 94–104. [Google Scholar]
- R. Kamocki, Pontryagin maximum principle for fractional ordinary optimal control problems. Math. Methods Appl. Sci. 37 (2014) 1668–1686. [Google Scholar]
- R. Kamocki and M. Majewski, Fractional linear control systems with Caputo derivative and their optimization. Optimal Control Appl. Methods 36 (2015) 953–967. [CrossRef] [Google Scholar]
- A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and applications of fractional differential equations, Vol. 204 of North-Holland Mathematics Studies. Elsevier Science B.V., Amsterdam (2006). [Google Scholar]
- E.B. Lee and L. Markus, Foundations of optimal control theory, 2nd edn. Robert E. Krieger Publishing Co., Inc., Melbourne, FL (1986). [Google Scholar]
- X.J. Li and J.M. Yong, Optimal control theory for infinite-dimensional systems. Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston, MA (1995). [Google Scholar]
- D. Liberzon, Calculus of variations and optimal control theory. Princeton University Press, Princeton, NJ (2012). [Google Scholar]
- A.B. Malinowska and D.F.M. Torres, Introduction to the fractional calculus of variations. Imperial College Press, London (2012). [CrossRef] [Google Scholar]
- I. Matychyn and V. Onyshchenko, Time-optimal control of fractional-order linear systems. Fract. Calc. Appl. Anal. 18 (2015) 687–696. [Google Scholar]
- T. Odzijewicz, A.B. Malinowska and D.F.M. Torres, Fractional variational calculus with classical and combined Caputo derivatives. Nonlinear Anal. 75 (2012) 1507–1515. [CrossRef] [MathSciNet] [Google Scholar]
- I. Podlubny, Fractional differential equations, vol. 198 of Mathematics in Science and Engineering. Academic Press, Inc., San Diego, CA (1999). [Google Scholar]
- 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, Inc., New York-London (1962). [Google Scholar]
- S. Pooseh, R. Almeida and D.F.M. Torres, Fractional order optimal control problems with free terminal time. J. Ind. Manag. Optim. 10 (2014) 363–381. [Google Scholar]
- F. Riewe, Mechanics with fractional derivatives. Phys. Rev. E 55 (1997) 3581–3592. [Google Scholar]
- S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional integrals and derivatives. Gordon and Breach Science Publishers, Yverdon (1993). Theory and applications, Edited and with a foreword by S. M. Nikolskiui, Translated from the 1987 Russian original, Revised by the authors.. [Google Scholar]
- H. Schättler and U. Ledzewicz, Geometric optimal control, Vol. 38 of Interdisciplinary Applied Mathematics. Springer, New York (2012). [Google Scholar]
- R. Scherer, S.L. Kalla, Y. Tang and J. Huang, The Grünwald-Letnikov method for fractional differential equations. Comput. Math. Appl. 62 (2011) 902–917. [Google Scholar]
- S.P. Sethi and G.L. Thompson, Optimal control theory, 2nd edn. Kluwer Academic Publishers, Boston, MA (2000). [Google Scholar]
- E. Trélat, Contrôle optimal, Mathématiques Concrètes [Concrete Mathematics]. Vuibert, Paris (2005). [Google Scholar]
- C. Tricaud and Y. Chen, Time-optimal control of systems with fractional dynamics. Int. J. Differ. Equ. 2010 (2010) 461048. [Google Scholar]
- R. Vinter, Optimal control, Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston, MA (2000). [Google Scholar]
- D.H. Wagner, Survey of measurable selection theorems. SIAM J. Control Optim. 15 (1977) 859–903. [Google Scholar]
- H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328 (2007) 1075–1081. [Google Scholar]
- S.A. Yousefi, M. Dehghan and A. Lotfi, Generalized Euler-Lagrange equations for fractional variational problems with free boundary conditions. Comput. Math. Appl. 62 (2011) 987–995. [Google Scholar]
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.