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Sufficient conditions for strong local optimality in optimal control problems with $L_{2}$-type objectives and control constraints
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 |
3. | Institut für Numerische und Angewandte Mathematik, Westfälische Wilhelms Universität Münster, D-48149 Münster, Germany |
References:
[1] |
B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory,, Series: Mathematics and Applications, (2003).
|
[2] |
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.
doi: 10.1137/0301011. |
[3] |
A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control,, American Institute of Mathematical Sciences, (2007).
|
[4] |
A. E. Bryson and Y. C. Ho, Applied Optimal Control,, Hemisphere Publishing, (1975).
|
[5] |
C. Büskens and H. Maurer, SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and realtime control,, J. of Computational and Applied Mathematics, 120 (2000), 85.
doi: 10.1016/S0377-0427(00)00305-8. |
[6] |
C. Büskens and H. Maurer, Sensitivity analysis and real-time optimization of parametric nonlinear programming problems,, in Online Optimization of Large Scale Systems (eds. M. Grötschel, (2001), 3.
|
[7] |
C. Büskens and H. Maurer, Sensitivity analysis and real-time control of parametric optimal control problems using nonlinear programming methods,, in Online Optimization of Large Scale Systems (eds. M. Grötschel, (2001), 57.
|
[8] |
N. Caroff and H. Frankowska, Conjugate points and shocks in nonlinear optimal control,, Trans. of the American Mathematical Society, 348 (1996), 3133.
doi: 10.1090/S0002-9947-96-01577-2. |
[9] |
J. H. Eschenburg and E. Heintze, Comparison theory for Riccati equations,, Manuscripta Matematicae, 68 (1990), 209.
doi: 10.1007/BF02568760. |
[10] |
R. Fourer, D.M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming,, Duxbury Press, (1993). Google Scholar |
[11] |
H. K. Khalil, Nonlinear Systems,, 3rd. ed., (2002). Google Scholar |
[12] |
M. Kimmel and A. Swierniak, An optimal control problem related to leukemia chemotherapy,, Scientific Bulletin of the Silesian Technical University, 65 (1983), 120. Google Scholar |
[13] |
H. W. Knobloch and H. Kwakernaak, Lineare Kontrolltheorie,, Springer Verlag, (1985).
doi: 10.1007/978-3-642-69884-2. |
[14] |
U. Ledzewicz and H. Schättler, Optimal bang-bang controls for a 2-compartment model in cancer chemotherapy,, J. of Optimization Theory and Applications - JOTA, 114 (2002), 609.
doi: 10.1023/A:1016027113579. |
[15] |
U. Ledzewicz and H. Schättler, Analysis of a cell-cycle specific model for cancer chemotherapy,, J. of Biological Systems, 10 (2002), 183.
doi: 10.1142/S0218339002000597. |
[16] |
U. Ledzewicz and H. Schättler, On optimal chemotherapy for heterogeneous tumors,, J. of Biological Systems, 22 (2014), 177.
doi: 10.1142/S0218339014400014. |
[17] |
U. Ledzewicz, H. Schättler, M. Reisi Gahrooi and S. Mahmoudian Dehkordi, On the MTD paradigm and optimal control for multi-drug cancer chemotherapy,, Mathematical Biosciences and Engineering (MBE), 10 (2013), 803.
doi: 10.3934/mbe.2013.10.803. |
[18] |
S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models,, Chapman & Hall/CRC Mathematical & Computational Biology, (2007).
|
[19] |
K. Malanowski and H. Maurer, Sensitivity analysis for parametric control problems with control-state constraints,, Computational Optimization and Applications, 5 (1996), 253.
doi: 10.1007/BF00248267. |
[20] |
H. Maurer, C. Büskens, J. H. Kim and Y. Kaja, Optimization techniques for the verification of second-order sufficient conditions for bang-bang controls,, Optimal Control, 26 (2005), 129.
doi: 10.1002/oca.756. |
[21] |
N. P. Osmolovskii and H. Maurer, Applications to Regular and Bang-Bang Control: Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control,, SIAM Advances in Design and Control, (2012).
doi: 10.1137/1.9781611972368. |
[22] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes,, MacMillan, (1964).
|
[23] |
H. Schättler and U. Ledzewicz, Perturbation feedback control: A geometric interpretation,, Numerical Algebra, 2 (2012), 631.
doi: 10.3934/naco.2012.2.631. |
[24] |
H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples,, Springer Verlag, (2012).
doi: 10.1007/978-1-4614-3834-2. |
[25] |
H. Schättler, U. Ledzewicz, S. Mahmoudian Dehkordi and M. Reisi Gahrooi, A geometric analysis of bang-bang extremals in optimal control problems for combination cancer chemotherapy,, in Proc. of the 51st IEEE Conference on Decision and Control (Maui, (2012), 7691. Google Scholar |
[26] |
A. Swierniak, Optimal treatment protocols in leukemia - modelling the proliferation cycle,, IMACS Ann. Comput. Appl. Math., 5 (1989), 51.
|
[27] |
A. Swierniak, Cell cycle as an object of control,, J. of Biological Systems, 3 (1995), 41.
doi: 10.1142/S0218339095000058. |
[28] |
A. Swierniak, U. Ledzewicz and H. Schättler, Optimal control for a class of compartmental models in cancer chemotherapy,, Int. J. Applied Mathematics and Computer Science, 13 (2003), 357.
|
[29] |
A. Swierniak and J. Smieja, Cancer chemotherapy optimization under evolving drug resistance,, Nonlinear Analysis, 47 (2001), 375.
doi: 10.1016/S0362-546X(01)00184-5. |
[30] |
A. Swierniak, A. Polanski and M. Kimmel, Optimal control problems arising in cell-cycle-specific cancer chemotherapy,, Cell prolif., 29 (1996), 117.
doi: 10.1046/j.1365-2184.1996.00995.x. |
[31] |
A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming,, Mathematical Programming, 106 (2006), 25.
doi: 10.1007/s10107-004-0559-y. |
show all references
References:
[1] |
B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory,, Series: Mathematics and Applications, (2003).
|
[2] |
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.
doi: 10.1137/0301011. |
[3] |
A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control,, American Institute of Mathematical Sciences, (2007).
|
[4] |
A. E. Bryson and Y. C. Ho, Applied Optimal Control,, Hemisphere Publishing, (1975).
|
[5] |
C. Büskens and H. Maurer, SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and realtime control,, J. of Computational and Applied Mathematics, 120 (2000), 85.
doi: 10.1016/S0377-0427(00)00305-8. |
[6] |
C. Büskens and H. Maurer, Sensitivity analysis and real-time optimization of parametric nonlinear programming problems,, in Online Optimization of Large Scale Systems (eds. M. Grötschel, (2001), 3.
|
[7] |
C. Büskens and H. Maurer, Sensitivity analysis and real-time control of parametric optimal control problems using nonlinear programming methods,, in Online Optimization of Large Scale Systems (eds. M. Grötschel, (2001), 57.
|
[8] |
N. Caroff and H. Frankowska, Conjugate points and shocks in nonlinear optimal control,, Trans. of the American Mathematical Society, 348 (1996), 3133.
doi: 10.1090/S0002-9947-96-01577-2. |
[9] |
J. H. Eschenburg and E. Heintze, Comparison theory for Riccati equations,, Manuscripta Matematicae, 68 (1990), 209.
doi: 10.1007/BF02568760. |
[10] |
R. Fourer, D.M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming,, Duxbury Press, (1993). Google Scholar |
[11] |
H. K. Khalil, Nonlinear Systems,, 3rd. ed., (2002). Google Scholar |
[12] |
M. Kimmel and A. Swierniak, An optimal control problem related to leukemia chemotherapy,, Scientific Bulletin of the Silesian Technical University, 65 (1983), 120. Google Scholar |
[13] |
H. W. Knobloch and H. Kwakernaak, Lineare Kontrolltheorie,, Springer Verlag, (1985).
doi: 10.1007/978-3-642-69884-2. |
[14] |
U. Ledzewicz and H. Schättler, Optimal bang-bang controls for a 2-compartment model in cancer chemotherapy,, J. of Optimization Theory and Applications - JOTA, 114 (2002), 609.
doi: 10.1023/A:1016027113579. |
[15] |
U. Ledzewicz and H. Schättler, Analysis of a cell-cycle specific model for cancer chemotherapy,, J. of Biological Systems, 10 (2002), 183.
doi: 10.1142/S0218339002000597. |
[16] |
U. Ledzewicz and H. Schättler, On optimal chemotherapy for heterogeneous tumors,, J. of Biological Systems, 22 (2014), 177.
doi: 10.1142/S0218339014400014. |
[17] |
U. Ledzewicz, H. Schättler, M. Reisi Gahrooi and S. Mahmoudian Dehkordi, On the MTD paradigm and optimal control for multi-drug cancer chemotherapy,, Mathematical Biosciences and Engineering (MBE), 10 (2013), 803.
doi: 10.3934/mbe.2013.10.803. |
[18] |
S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models,, Chapman & Hall/CRC Mathematical & Computational Biology, (2007).
|
[19] |
K. Malanowski and H. Maurer, Sensitivity analysis for parametric control problems with control-state constraints,, Computational Optimization and Applications, 5 (1996), 253.
doi: 10.1007/BF00248267. |
[20] |
H. Maurer, C. Büskens, J. H. Kim and Y. Kaja, Optimization techniques for the verification of second-order sufficient conditions for bang-bang controls,, Optimal Control, 26 (2005), 129.
doi: 10.1002/oca.756. |
[21] |
N. P. Osmolovskii and H. Maurer, Applications to Regular and Bang-Bang Control: Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control,, SIAM Advances in Design and Control, (2012).
doi: 10.1137/1.9781611972368. |
[22] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes,, MacMillan, (1964).
|
[23] |
H. Schättler and U. Ledzewicz, Perturbation feedback control: A geometric interpretation,, Numerical Algebra, 2 (2012), 631.
doi: 10.3934/naco.2012.2.631. |
[24] |
H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples,, Springer Verlag, (2012).
doi: 10.1007/978-1-4614-3834-2. |
[25] |
H. Schättler, U. Ledzewicz, S. Mahmoudian Dehkordi and M. Reisi Gahrooi, A geometric analysis of bang-bang extremals in optimal control problems for combination cancer chemotherapy,, in Proc. of the 51st IEEE Conference on Decision and Control (Maui, (2012), 7691. Google Scholar |
[26] |
A. Swierniak, Optimal treatment protocols in leukemia - modelling the proliferation cycle,, IMACS Ann. Comput. Appl. Math., 5 (1989), 51.
|
[27] |
A. Swierniak, Cell cycle as an object of control,, J. of Biological Systems, 3 (1995), 41.
doi: 10.1142/S0218339095000058. |
[28] |
A. Swierniak, U. Ledzewicz and H. Schättler, Optimal control for a class of compartmental models in cancer chemotherapy,, Int. J. Applied Mathematics and Computer Science, 13 (2003), 357.
|
[29] |
A. Swierniak and J. Smieja, Cancer chemotherapy optimization under evolving drug resistance,, Nonlinear Analysis, 47 (2001), 375.
doi: 10.1016/S0362-546X(01)00184-5. |
[30] |
A. Swierniak, A. Polanski and M. Kimmel, Optimal control problems arising in cell-cycle-specific cancer chemotherapy,, Cell prolif., 29 (1996), 117.
doi: 10.1046/j.1365-2184.1996.00995.x. |
[31] |
A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming,, Mathematical Programming, 106 (2006), 25.
doi: 10.1007/s10107-004-0559-y. |
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