Lyapunov-Schmidt reduction for optimal control problems

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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

Abstract
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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).
[2]	G. Bliss, "Calculus of Variations,'', The Mathematical Association of America, (1925).
[3]	A. Bressan and B. Piccoli, "Introduction to the Mathematical Theory of Control,'', AIMS Series on Applied Mathematics, 2 (2007).
[4]	A. E. Bryson, Jr. and Y. C. Ho, "Applied Optimal Control. Optimization, Estimation, and Control,'' Revised Printing,, Hemisphere Publishing Corp., (1975).
[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.
[6]	C. I. Byrnes and A. Jhemi, Shock waves for Riccati partial differential equations arising in nonlinear optimal control,, in, 12 (1992), 211.
[7]	M. Golubitsky and V. Guillemin, "Stable Mappings and their Singularities,'', Graduate Texts in Mathematics, (1973).
[8]	M. Golubitsky and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory," Vol. I,, Applied Mathematical Sciences, 51 (1985).
[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.
[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.
[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.
[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).
[13]	H. Schättler and U. Ledzewicz, Synthesis of optimal controlled trajectories with chattering arcs,, Dynamics of Continuous, 19 (2012), 161.
[14]	H. Schättler and U. Ledzewicz, Perturbation feedback control-a geometric interpretation,, Numerical Algebra, (2012).
[15]	H. Schättler and U. Ledzewicz, "Geometric Optimal Control-Theory, Methods and Examples,'', Springer-Verlag, (2012).
[16]	H. Whitney, Elementary structure of real algebraic varieties,, Ann. Math. (2), 66 (1957), 545. doi: 10.2307/1969908.
[17]	L. C. Young, "Lectures on the Calculus of Variations and Optimal Control Theory,'', Foreword by Wendell H. Fleming, (1969).

show all references

References:

[1]	L. Berkovitz, "Optimal Control Theory,'', Applied Mathematical Sciences, (1974).
[2]	G. Bliss, "Calculus of Variations,'', The Mathematical Association of America, (1925).
[3]	A. Bressan and B. Piccoli, "Introduction to the Mathematical Theory of Control,'', AIMS Series on Applied Mathematics, 2 (2007).
[4]	A. E. Bryson, Jr. and Y. C. Ho, "Applied Optimal Control. Optimization, Estimation, and Control,'' Revised Printing,, Hemisphere Publishing Corp., (1975).
[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.
[6]	C. I. Byrnes and A. Jhemi, Shock waves for Riccati partial differential equations arising in nonlinear optimal control,, in, 12 (1992), 211.
[7]	M. Golubitsky and V. Guillemin, "Stable Mappings and their Singularities,'', Graduate Texts in Mathematics, (1973).
[8]	M. Golubitsky and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory," Vol. I,, Applied Mathematical Sciences, 51 (1985).
[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.
[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.
[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.
[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).
[13]	H. Schättler and U. Ledzewicz, Synthesis of optimal controlled trajectories with chattering arcs,, Dynamics of Continuous, 19 (2012), 161.
[14]	H. Schättler and U. Ledzewicz, Perturbation feedback control-a geometric interpretation,, Numerical Algebra, (2012).
[15]	H. Schättler and U. Ledzewicz, "Geometric Optimal Control-Theory, Methods and Examples,'', Springer-Verlag, (2012).
[16]	H. Whitney, Elementary structure of real algebraic varieties,, Ann. Math. (2), 66 (1957), 545. doi: 10.2307/1969908.
[17]	L. C. Young, "Lectures on the Calculus of Variations and Optimal Control Theory,'', Foreword by Wendell H. Fleming, (1969).

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