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Noether's symmetry Theorem for variational and optimal control problems with time delay
Perturbation feedback control: A geometric interpretation
1. | Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Missouri, 63130-4899 |
2. | Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL 62026 |
References:
[1] |
A. A. Agrachev and Y. Sachkov, "Control Theory from the Geometric Viewpoint,", Springer Verlag, (2004).
|
[2] |
A. Agrachev, G. Stefani and P. L. Zezza, A Hamiltonian approach to strong minima in optimal control,, in, (1999), 11.
|
[3] |
A. Agrachev, G. Stefani and P. L. Zezza, Strong optimality for a bang-bang trajectory,, SIAM J. Control and Optimization, 41 (2002), 991.
doi: 10.1137/S036301290138866X. |
[4] |
B. Bonnard and M. Chyba, "Singular Trajectories and their Role in Control Theory,", Mathématiques & Applications, (2003).
|
[5] |
J. V. Breakwell, J. L. Speyer and A. E. Bryson, jr., Optimization and control of nonlinear systems using the second variation,, SIAM J. Control, 1 (1963), 193. Google Scholar |
[6] |
A. Bressan and B. Piccoli, "Introduction to the Mathematical Theory of Control,", American Institute of Mathematical Sciences (AIMS), (2007).
|
[7] |
A. E. Bryson, jr. and Y. C. Ho, "Applied Optimal Control,", Revised Printing, (1975).
|
[8] |
M. E. Fisher, W. J. Grantham and K. L. Teo, Neighbouring extremals for nonlinear systems with control constraints,, Dynamics and Control, 5 (1995), 225.
doi: 10.1007/BF01968675. |
[9] |
M. Golubitsky and V. Guillemin, "Stable Mappings and their Singularities,", Springer-Verlag, (1973).
doi: 10.1007/978-1-4615-7904-5. |
[10] |
D. H. Jacobson, D. H. Martin, M. Pachter and T. Geveci, "Extensions of Linear-Quadratic Control Theory,", Lecture Notes in Control and Information Sciences, (1980).
|
[11] |
T. Kailath, "Linear Systems,", Prentice Hall, (1980).
|
[12] |
H. W. Knobloch and H. Kwakernaak, "Lineare Kontrolltheorie,", Springer Verlag, (1985).
doi: 10.1007/978-3-642-69884-2. |
[13] |
H. Maurer and H. J. Oberle, Second order sufficient conditions for optimal control problems with free final time: the Riccati approach,, SIAM J. on Control and Optimization, 41 (2002), 380.
doi: 10.1137/S0363012900377419. |
[14] |
J. Noble and H. Schättler, Sufficient conditions for relative minima of broken extremals,, J. of Mathematical Analysis and Applications, 269 (2002), 98.
doi: 10.1016/S0022-247X(02)00008-2. |
[15] |
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), 105.
doi: 10.1023/B:JOTA.0000042525.50701.9a. |
[16] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes,", MacMillan, (1964).
|
[17] |
H. Schättler, On classical envelopes in optimal control theory,, in, (2010), 1879. Google Scholar |
[18] |
H. Schättler and U. Ledzewicz, "Geometric Optimal Control,", Springer Verlag, (2012). Google Scholar |
[19] |
H. Schättler and U. Ledzewicz, Synthesis of optimal controlled trajectories with chattering arcs,, Dynamics of Continuous, 19 (2012), 161.
|
[20] |
H. Schättler and U. Ledzewicz, Lyapunov-Schmidt reduction for optimal control problems,, Discrete and Continuous Dynamical Systems, 17 (2012), 2201. Google Scholar |
[21] |
H. J. Sussmann, Envelopes, high-order optimality conditions and Lie brackets,, in, (1989), 1107.
doi: 10.1109/CDC.1989.70305. |
show all references
References:
[1] |
A. A. Agrachev and Y. Sachkov, "Control Theory from the Geometric Viewpoint,", Springer Verlag, (2004).
|
[2] |
A. Agrachev, G. Stefani and P. L. Zezza, A Hamiltonian approach to strong minima in optimal control,, in, (1999), 11.
|
[3] |
A. Agrachev, G. Stefani and P. L. Zezza, Strong optimality for a bang-bang trajectory,, SIAM J. Control and Optimization, 41 (2002), 991.
doi: 10.1137/S036301290138866X. |
[4] |
B. Bonnard and M. Chyba, "Singular Trajectories and their Role in Control Theory,", Mathématiques & Applications, (2003).
|
[5] |
J. V. Breakwell, J. L. Speyer and A. E. Bryson, jr., Optimization and control of nonlinear systems using the second variation,, SIAM J. Control, 1 (1963), 193. Google Scholar |
[6] |
A. Bressan and B. Piccoli, "Introduction to the Mathematical Theory of Control,", American Institute of Mathematical Sciences (AIMS), (2007).
|
[7] |
A. E. Bryson, jr. and Y. C. Ho, "Applied Optimal Control,", Revised Printing, (1975).
|
[8] |
M. E. Fisher, W. J. Grantham and K. L. Teo, Neighbouring extremals for nonlinear systems with control constraints,, Dynamics and Control, 5 (1995), 225.
doi: 10.1007/BF01968675. |
[9] |
M. Golubitsky and V. Guillemin, "Stable Mappings and their Singularities,", Springer-Verlag, (1973).
doi: 10.1007/978-1-4615-7904-5. |
[10] |
D. H. Jacobson, D. H. Martin, M. Pachter and T. Geveci, "Extensions of Linear-Quadratic Control Theory,", Lecture Notes in Control and Information Sciences, (1980).
|
[11] |
T. Kailath, "Linear Systems,", Prentice Hall, (1980).
|
[12] |
H. W. Knobloch and H. Kwakernaak, "Lineare Kontrolltheorie,", Springer Verlag, (1985).
doi: 10.1007/978-3-642-69884-2. |
[13] |
H. Maurer and H. J. Oberle, Second order sufficient conditions for optimal control problems with free final time: the Riccati approach,, SIAM J. on Control and Optimization, 41 (2002), 380.
doi: 10.1137/S0363012900377419. |
[14] |
J. Noble and H. Schättler, Sufficient conditions for relative minima of broken extremals,, J. of Mathematical Analysis and Applications, 269 (2002), 98.
doi: 10.1016/S0022-247X(02)00008-2. |
[15] |
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), 105.
doi: 10.1023/B:JOTA.0000042525.50701.9a. |
[16] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes,", MacMillan, (1964).
|
[17] |
H. Schättler, On classical envelopes in optimal control theory,, in, (2010), 1879. Google Scholar |
[18] |
H. Schättler and U. Ledzewicz, "Geometric Optimal Control,", Springer Verlag, (2012). Google Scholar |
[19] |
H. Schättler and U. Ledzewicz, Synthesis of optimal controlled trajectories with chattering arcs,, Dynamics of Continuous, 19 (2012), 161.
|
[20] |
H. Schättler and U. Ledzewicz, Lyapunov-Schmidt reduction for optimal control problems,, Discrete and Continuous Dynamical Systems, 17 (2012), 2201. Google Scholar |
[21] |
H. J. Sussmann, Envelopes, high-order optimality conditions and Lie brackets,, in, (1989), 1107.
doi: 10.1109/CDC.1989.70305. |
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