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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, Vol. 12, Springer-Verlag, New York-Heidelberg, 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, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2007.  Google Scholar

[4]

A. E. Bryson, Jr. and Y. C. Ho, "Applied Optimal Control. Optimization, Estimation, and Control,'' Revised Printing, Hemisphere Publishing Corp., Washington, D. C., distributed by Halsted Press [John Wiley & Sons], New York-London-Sydney, 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), 427-431.  Google Scholar

[6]

C. I. Byrnes and A. Jhemi, Shock waves for Riccati partial differential equations arising in nonlinear optimal control, in "Systems, Models and Feedback: Theory and Applications'' (eds. A. Isidori and T. J. Tarn) (Capri, 1992), Progr. Systems Control Theory, 12, Birkhäuser Boston, Boston, MA, (1992), 211-227.  Google Scholar

[7]

M. Golubitsky and V. Guillemin, "Stable Mappings and their Singularities,'' Graduate Texts in Mathematics, Vol. 14, Springer-Verlag, New York-Heidelberg, 1973.  Google Scholar

[8]

M. Golubitsky and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory," Vol. I, Applied Mathematical Sciences, 51, Springer-Verlag, New York, 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), 1346-1371. 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), 98-128. 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), 345-370. 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, A Pergamon Press Book, The Macmillan Co., New York, 1964.  Google Scholar

[13]

H. Schättler and U. Ledzewicz, Synthesis of optimal controlled trajectories with chattering arcs, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications & Algorithms, 19 (2012), 161-186. Google Scholar

[14]

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

[15]

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

[16]

H. Whitney, Elementary structure of real algebraic varieties, Ann. Math. (2), 66 (1957), 545-556. 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, W. B. Saunders Co., Philadelphia-London-Toronto, Ont., 1969.  Google Scholar

show all references

References:
[1]

L. Berkovitz, "Optimal Control Theory,'' Applied Mathematical Sciences, Vol. 12, Springer-Verlag, New York-Heidelberg, 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, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2007.  Google Scholar

[4]

A. E. Bryson, Jr. and Y. C. Ho, "Applied Optimal Control. Optimization, Estimation, and Control,'' Revised Printing, Hemisphere Publishing Corp., Washington, D. C., distributed by Halsted Press [John Wiley & Sons], New York-London-Sydney, 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), 427-431.  Google Scholar

[6]

C. I. Byrnes and A. Jhemi, Shock waves for Riccati partial differential equations arising in nonlinear optimal control, in "Systems, Models and Feedback: Theory and Applications'' (eds. A. Isidori and T. J. Tarn) (Capri, 1992), Progr. Systems Control Theory, 12, Birkhäuser Boston, Boston, MA, (1992), 211-227.  Google Scholar

[7]

M. Golubitsky and V. Guillemin, "Stable Mappings and their Singularities,'' Graduate Texts in Mathematics, Vol. 14, Springer-Verlag, New York-Heidelberg, 1973.  Google Scholar

[8]

M. Golubitsky and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory," Vol. I, Applied Mathematical Sciences, 51, Springer-Verlag, New York, 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), 1346-1371. 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), 98-128. 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), 345-370. 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, A Pergamon Press Book, The Macmillan Co., New York, 1964.  Google Scholar

[13]

H. Schättler and U. Ledzewicz, Synthesis of optimal controlled trajectories with chattering arcs, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications & Algorithms, 19 (2012), 161-186. Google Scholar

[14]

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

[15]

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

[16]

H. Whitney, Elementary structure of real algebraic varieties, Ann. Math. (2), 66 (1957), 545-556. 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, W. B. Saunders Co., Philadelphia-London-Toronto, Ont., 1969.  Google Scholar

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