American Institute of Mathematical Sciences

Advanced
September  2012, 17(6): 2201-2223. doi: 10.3934/dcdsb.2012.17.2201

Lyapunov-Schmidt reduction for optimal control problems

Heinz Schättler 1, and  Urszula Ledzewicz 2,

1. 

Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Missouri, 63130-4899
2. 

Dept. of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, Illinois, 62026-1653

Received  August 2011 Revised  March 2012 Published  May 2012

In this paper, we use the method of characteristics to study singularities in the flow of a parameterized family of extremals for an optimal control problem. By means of the Lyapunov--Schmidt reduction a characterization of fold and cusp points is given. Examples illustrate the local behaviors of the flow near these singular points. Singularities of fold type correspond to the typical conjugate points as they arise for the classical problem of minimum surfaces of revolution in the calculus of variations and local optimality of trajectories ceases at fold points. Simple cusp points, on the other hand, generate a cut-locus that limits the optimality of close-by trajectories globally to times prior to the conjugate points.
Keywords: Optimal control, fold and simple cusp singularities., Lyapunov--Schmidt reduction, maximum principle.
Mathematics Subject Classification: Primary: 49J15; Secondary: 93C5.
Citation: Heinz Schättler, Urszula Ledzewicz. Lyapunov-Schmidt reduction for optimal control problems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2201-2223. doi: 10.3934/dcdsb.2012.17.2201
References:
[1]

L. Berkovitz, "Optimal Control Theory,'', Applied Mathematical Sciences, (1974).   Google Scholar

[2]

G. Bliss, "Calculus of Variations,'', The Mathematical Association of America, (1925).   Google Scholar

[3]

A. Bressan and B. Piccoli, "Introduction to the Mathematical Theory of Control,'', AIMS Series on Applied Mathematics, 2 (2007).   Google Scholar

[4]

A. E. Bryson, Jr. and Y. C. Ho, "Applied Optimal Control. Optimization, Estimation, and Control,'' Revised Printing,, Hemisphere Publishing Corp., (1975).   Google Scholar

[5]

C. I. Byrnes and H. Frankowska, Unicité des solutions optimales et absence de chocs pour les équations d'Hamilton-Jacobi-Bellman et de Riccati,, C. R. Acad. Sci. Paris Série I Math., 315 (1992), 427.   Google Scholar

[6]

C. I. Byrnes and A. Jhemi, Shock waves for Riccati partial differential equations arising in nonlinear optimal control,, in, 12 (1992), 211.   Google Scholar

[7]

M. Golubitsky and V. Guillemin, "Stable Mappings and their Singularities,'', Graduate Texts in Mathematics, (1973).   Google Scholar

[8]

M. Golubitsky and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory," Vol. I,, Applied Mathematical Sciences, 51 (1985).   Google Scholar

[9]

M. Kiefer and H. Schättler, Parametrized families of extremals and singularities in solutions to the Hamilton-Jacobi-Bellman equation,, SIAM J. on Control and Optimization, 37 (1999), 1346.  doi: 10.1137/S0363012997319139.  Google Scholar

[10]

J. Noble and H. Schättler, Sufficient conditions for relative minima of broken extremals in optimal control theory,, J. of Mathematical Analysis and Applications, 269 (2002), 98.  doi: 10.1016/S0022-247X(02)00008-2.  Google Scholar

[11]

U. Ledzewicz, A. Nowakowski and H. Schättler, Stratifiable families of extremals and sufficient conditions for optimality in optimal control problems,, J. of Optimization Theory and Applications (JOTA), 122 (2004), 345.  doi: 10.1023/B:JOTA.0000042525.50701.9a.  Google Scholar

[12]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes,'', Translated by D. E. Brown, (1964).   Google Scholar

[13]

H. Schättler and U. Ledzewicz, Synthesis of optimal controlled trajectories with chattering arcs,, Dynamics of Continuous, 19 (2012), 161.   Google Scholar

[14]

H. Schättler and U. Ledzewicz, Perturbation feedback control-a geometric interpretation,, Numerical Algebra, (2012).   Google Scholar

[15]

H. Schättler and U. Ledzewicz, "Geometric Optimal Control-Theory, Methods and Examples,'', Springer-Verlag, (2012).   Google Scholar

[16]

H. Whitney, Elementary structure of real algebraic varieties,, Ann. Math. (2), 66 (1957), 545.  doi: 10.2307/1969908.  Google Scholar

[17]

L. C. Young, "Lectures on the Calculus of Variations and Optimal Control Theory,'', Foreword by Wendell H. Fleming, (1969).   Google Scholar

show all references

References:
[1]

L. Berkovitz, "Optimal Control Theory,'', Applied Mathematical Sciences, (1974).   Google Scholar

[2]

G. Bliss, "Calculus of Variations,'', The Mathematical Association of America, (1925).   Google Scholar

[3]

A. Bressan and B. Piccoli, "Introduction to the Mathematical Theory of Control,'', AIMS Series on Applied Mathematics, 2 (2007).   Google Scholar

[4]

A. E. Bryson, Jr. and Y. C. Ho, "Applied Optimal Control. Optimization, Estimation, and Control,'' Revised Printing,, Hemisphere Publishing Corp., (1975).   Google Scholar

[5]

C. I. Byrnes and H. Frankowska, Unicité des solutions optimales et absence de chocs pour les équations d'Hamilton-Jacobi-Bellman et de Riccati,, C. R. Acad. Sci. Paris Série I Math., 315 (1992), 427.   Google Scholar

[6]

C. I. Byrnes and A. Jhemi, Shock waves for Riccati partial differential equations arising in nonlinear optimal control,, in, 12 (1992), 211.   Google Scholar

[7]

M. Golubitsky and V. Guillemin, "Stable Mappings and their Singularities,'', Graduate Texts in Mathematics, (1973).   Google Scholar

[8]

M. Golubitsky and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory," Vol. I,, Applied Mathematical Sciences, 51 (1985).   Google Scholar

[9]

M. Kiefer and H. Schättler, Parametrized families of extremals and singularities in solutions to the Hamilton-Jacobi-Bellman equation,, SIAM J. on Control and Optimization, 37 (1999), 1346.  doi: 10.1137/S0363012997319139.  Google Scholar

[10]

J. Noble and H. Schättler, Sufficient conditions for relative minima of broken extremals in optimal control theory,, J. of Mathematical Analysis and Applications, 269 (2002), 98.  doi: 10.1016/S0022-247X(02)00008-2.  Google Scholar

[11]

U. Ledzewicz, A. Nowakowski and H. Schättler, Stratifiable families of extremals and sufficient conditions for optimality in optimal control problems,, J. of Optimization Theory and Applications (JOTA), 122 (2004), 345.  doi: 10.1023/B:JOTA.0000042525.50701.9a.  Google Scholar

[12]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes,'', Translated by D. E. Brown, (1964).   Google Scholar

[13]

H. Schättler and U. Ledzewicz, Synthesis of optimal controlled trajectories with chattering arcs,, Dynamics of Continuous, 19 (2012), 161.   Google Scholar

[14]

H. Schättler and U. Ledzewicz, Perturbation feedback control-a geometric interpretation,, Numerical Algebra, (2012).   Google Scholar

[15]

H. Schättler and U. Ledzewicz, "Geometric Optimal Control-Theory, Methods and Examples,'', Springer-Verlag, (2012).   Google Scholar

[16]

H. Whitney, Elementary structure of real algebraic varieties,, Ann. Math. (2), 66 (1957), 545.  doi: 10.2307/1969908.  Google Scholar

[17]

L. C. Young, "Lectures on the Calculus of Variations and Optimal Control Theory,'', Foreword by Wendell H. Fleming, (1969).   Google Scholar

[1]

Zaidong Zhan, Shuping Chen, Wei Wei. A unified theory of maximum principle for continuous and discrete time optimal control problems. Mathematical Control & Related Fields, 2012, 2 (2) : 195-215. doi: 10.3934/mcrf.2012.2.195
[2]

Md. Haider Ali Biswas, Maria do Rosário de Pinho. A nonsmooth maximum principle for optimal control problems with state and mixed constraints - convex case. Conference Publications, 2011, 2011 (Special) : 174-183. doi: 10.3934/proc.2011.2011.174
[3]

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
[4]

H. O. Fattorini. The maximum principle for linear infinite dimensional control systems with state constraints. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 77-101. doi: 10.3934/dcds.1995.1.77
[5]

Shaolin Ji, Xiaole Xue. A stochastic maximum principle for linear quadratic problem with nonconvex control domain. Mathematical Control & Related Fields, 2019, 9 (3) : 495-507. doi: 10.3934/mcrf.2019022
[6]

Yan Wang, Yanxiang Zhao, Lei Wang, Aimin Song, Yanping Ma. Stochastic maximum principle for partial information optimal investment and dividend problem of an insurer. Journal of Industrial & Management Optimization, 2018, 14 (2) : 653-671. doi: 10.3934/jimo.2017067
[7]

Shanjian Tang. A second-order maximum principle for singular optimal stochastic controls. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1581-1599. doi: 10.3934/dcdsb.2010.14.1581
[8]

Huaying Guo, Jin Liang. An optimal control model of carbon reduction and trading. Mathematical Control & Related Fields, 2016, 6 (4) : 535-550. doi: 10.3934/mcrf.2016015
[9]

H. O. Fattorini. The maximum principle in infinite dimension. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 557-574. doi: 10.3934/dcds.2000.6.557
[10]

Hancheng Guo, Jie Xiong. A second-order stochastic maximum principle for generalized mean-field singular control problem. Mathematical Control & Related Fields, 2018, 8 (2) : 451-473. doi: 10.3934/mcrf.2018018
[11]

Guy Barles, Ariela Briani, Emmanuel Trélat. Value function for regional control problems via dynamic programming and Pontryagin maximum principle. Mathematical Control & Related Fields, 2018, 8 (3&4) : 509-533. doi: 10.3934/mcrf.2018021
[12]

José Godoy, Nolbert Morales, Manuel Zamora. Existence and multiplicity of periodic solutions to an indefinite singular equation with two singularities. The degenerate case. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4137-4156. doi: 10.3934/dcds.2019167
[13]

Christian Pötzsche. Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 739-776. doi: 10.3934/dcdsb.2010.14.739
[14]

Carlo Orrieri. A stochastic maximum principle with dissipativity conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5499-5519. doi: 10.3934/dcds.2015.35.5499
[15]

Djamila Moulay, M. A. Aziz-Alaoui, Hee-Dae Kwon. Optimal control of chikungunya disease: Larvae reduction, treatment and prevention. Mathematical Biosciences & Engineering, 2012, 9 (2) : 369-392. doi: 10.3934/mbe.2012.9.369
[16]

Shi'an Wang, N. U. Ahmed. Optimal control and stabilization of building maintenance units based on minimum principle. Journal of Industrial & Management Optimization, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2020041
[17]

Adel Chala, Dahbia Hafayed. On stochastic maximum principle for risk-sensitive of fully coupled forward-backward stochastic control of mean-field type with application. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020035
[18]

Torsten Lindström. Discrete models and Fisher's maximum principle in ecology. Conference Publications, 2003, 2003 (Special) : 571-579. doi: 10.3934/proc.2003.2003.571
[19]

Andrejs Reinfelds, Klara Janglajew. Reduction principle in the theory of stability of difference equations. Conference Publications, 2007, 2007 (Special) : 864-874. doi: 10.3934/proc.2007.2007.864
[20]

Jean-Pierre de la Croix, Magnus Egerstedt. Analyzing human-swarm interactions using control Lyapunov functions and optimal control. Networks & Heterogeneous Media, 2015, 10 (3) : 609-630. doi: 10.3934/nhm.2015.10.609

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences