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Lyapunov-Schmidt reduction for optimal control problems
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 |
References:
[1] |
L. Berkovitz, "Optimal Control Theory,'' Applied Mathematical Sciences, Vol. 12, Springer-Verlag, New York-Heidelberg, 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, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2007. |
[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. |
[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. |
[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. |
[7] |
M. Golubitsky and V. Guillemin, "Stable Mappings and their Singularities,'' Graduate Texts in Mathematics, Vol. 14, Springer-Verlag, New York-Heidelberg, 1973. |
[8] |
M. Golubitsky and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory," Vol. I, Applied Mathematical Sciences, 51, Springer-Verlag, New York, 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), 1346-1371.
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), 98-128.
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), 345-370.
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, A Pergamon Press Book, The Macmillan Co., New York, 1964. |
[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. |
[14] |
H. Schättler and U. Ledzewicz, Perturbation feedback control-a geometric interpretation, Numerical Algebra, Control and Applications, (2012), to appear. |
[15] |
H. Schättler and U. Ledzewicz, "Geometric Optimal Control-Theory, Methods and Examples,'' Springer-Verlag, New York, 2012. |
[16] |
H. Whitney, Elementary structure of real algebraic varieties, Ann. Math. (2), 66 (1957), 545-556.
doi: 10.2307/1969908. |
[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. |
show all references
References:
[1] |
L. Berkovitz, "Optimal Control Theory,'' Applied Mathematical Sciences, Vol. 12, Springer-Verlag, New York-Heidelberg, 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, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2007. |
[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. |
[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. |
[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. |
[7] |
M. Golubitsky and V. Guillemin, "Stable Mappings and their Singularities,'' Graduate Texts in Mathematics, Vol. 14, Springer-Verlag, New York-Heidelberg, 1973. |
[8] |
M. Golubitsky and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory," Vol. I, Applied Mathematical Sciences, 51, Springer-Verlag, New York, 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), 1346-1371.
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), 98-128.
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), 345-370.
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, A Pergamon Press Book, The Macmillan Co., New York, 1964. |
[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. |
[14] |
H. Schättler and U. Ledzewicz, Perturbation feedback control-a geometric interpretation, Numerical Algebra, Control and Applications, (2012), to appear. |
[15] |
H. Schättler and U. Ledzewicz, "Geometric Optimal Control-Theory, Methods and Examples,'' Springer-Verlag, New York, 2012. |
[16] |
H. Whitney, Elementary structure of real algebraic varieties, Ann. Math. (2), 66 (1957), 545-556.
doi: 10.2307/1969908. |
[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. |
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