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Existence and stability analysis for nonlinear optimal control problems with $1$-mean equicontinuous controls
1. | Department of Mathematics, Guizhou University, Guizhou, 550025, China, China |
References:
[1] |
N. U. Ahmed and K. L. Teo, Optimal Control of Distributed Parameter Systems, Elsevier North Holland, New York, 1981. |
[2] |
J. P. Aubin and H. Frankowska, Set-valued Analysis, Birkhauser, Boston, 1990. |
[3] |
V. I. Bogachev, Measure Theory , Springer-Verlag, Berlin, 2007.
doi: 10.1007/978-3-540-34514-5. |
[4] |
J. F. Bonnans and A. Hermant, Stability and sensitivity analysis for optimal control problems with a first-order state constraint and application to continuation methods, ESAIM Control Optim. Calc. Var., 14 (2008), 825-863.
doi: 10.1051/cocv:2008016. |
[5] |
A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, American Institute of Mathematical Sciences Press, 2007. |
[6] |
A. L. Dontchev and W. W. Hager, Lipschitzian stability for state constrained nonlinear optimal control, SIAM J. Control Optim., 36 (1998), 698-718.
doi: 10.1137/S0363012996299314. |
[7] |
A. L. Dontchev and W. W. Hager, The Euler approximation in state constrained optimal control, Math. Comp., 70 (2001), 173-203.
doi: 10.1090/S0025-5718-00-01184-4. |
[8] |
H. Hanche-Olsen and H. Holden, The Kolmogorov-Riesz compactness theorem, Expositiones Mathematicae, 28 (2010), 385-394.
doi: 10.1016/j.exmath.2010.03.001. |
[9] |
A. Hermant, Stability analysis of optimal control problems with a second-order constraint, SIAM J. Control Optim., 20 (2009), 104-129.
doi: 10.1137/070707993. |
[10] |
E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons Inc., New York, 1978. |
[11] |
X. J. Li and J. M. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhauser, Boston, 1995.
doi: 10.1007/978-1-4612-4260-4. |
[12] |
Q. Lin, R. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey, Journal of Industrial and Management Optimization, 10 (2014), 275-309.
doi: 10.3934/jimo.2014.10.275. |
[13] |
R. Loxton, Q. Lin, V. Rehbock and K. L. Teo, Control parameterization for optimal control problems with continuous inequality constrains: New convergence results, Numerical Algebra, Control and Optimization, 2 (2012), 571-599.
doi: 10.3934/naco.2012.2.571. |
[14] |
R. Loxton, Q. Lin and K. L. Teo, Minimizing control variation in nonlinear optimal control, Automatica, 49 (2013), 2652-2664.
doi: 10.1016/j.automatica.2013.05.027. |
[15] |
W. Rudin, Functional Analysis, $2^{nd}$ edition. McGraw-Hill, Inc., New York, 1991. |
[16] |
S. Sager, H. G. Bock and G. Reinelt, Direct methods with maximal lower bound for mixed-integer optimal control problems, Mathematical Programming, 118 (2009), 109-149.
doi: 10.1007/s10107-007-0185-6. |
[17] |
K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, New York: John Wiley & Sons Inc., 1991. |
[18] |
S. F. Woon, V. Rehbock and R. Loxton, Towards global solutions of optimal discrete-valued control problems, Optimal Control Applications and Methods, 33 (2012), 576-594.
doi: 10.1002/oca.1015. |
[19] |
C. Yu, B. Li, R. Loxton and K. L. Teo, Optimal discrete-valued control computation, Journal of Global Optimization, 56 (2013), 503-518.
doi: 10.1007/s10898-012-9858-7. |
[20] |
J. Yu, Z. X. Liu and D. T. Peng, Existence and stability analysis of optimal control, Optimal Control Applications and Methods, 35 (2014), 721-729.
doi: 10.1002/oca.2096. |
[21] |
E. Zeidler, Functional and Its Applications II/B, Springer-Verlag, New York, 1990. |
show all references
References:
[1] |
N. U. Ahmed and K. L. Teo, Optimal Control of Distributed Parameter Systems, Elsevier North Holland, New York, 1981. |
[2] |
J. P. Aubin and H. Frankowska, Set-valued Analysis, Birkhauser, Boston, 1990. |
[3] |
V. I. Bogachev, Measure Theory , Springer-Verlag, Berlin, 2007.
doi: 10.1007/978-3-540-34514-5. |
[4] |
J. F. Bonnans and A. Hermant, Stability and sensitivity analysis for optimal control problems with a first-order state constraint and application to continuation methods, ESAIM Control Optim. Calc. Var., 14 (2008), 825-863.
doi: 10.1051/cocv:2008016. |
[5] |
A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, American Institute of Mathematical Sciences Press, 2007. |
[6] |
A. L. Dontchev and W. W. Hager, Lipschitzian stability for state constrained nonlinear optimal control, SIAM J. Control Optim., 36 (1998), 698-718.
doi: 10.1137/S0363012996299314. |
[7] |
A. L. Dontchev and W. W. Hager, The Euler approximation in state constrained optimal control, Math. Comp., 70 (2001), 173-203.
doi: 10.1090/S0025-5718-00-01184-4. |
[8] |
H. Hanche-Olsen and H. Holden, The Kolmogorov-Riesz compactness theorem, Expositiones Mathematicae, 28 (2010), 385-394.
doi: 10.1016/j.exmath.2010.03.001. |
[9] |
A. Hermant, Stability analysis of optimal control problems with a second-order constraint, SIAM J. Control Optim., 20 (2009), 104-129.
doi: 10.1137/070707993. |
[10] |
E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons Inc., New York, 1978. |
[11] |
X. J. Li and J. M. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhauser, Boston, 1995.
doi: 10.1007/978-1-4612-4260-4. |
[12] |
Q. Lin, R. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey, Journal of Industrial and Management Optimization, 10 (2014), 275-309.
doi: 10.3934/jimo.2014.10.275. |
[13] |
R. Loxton, Q. Lin, V. Rehbock and K. L. Teo, Control parameterization for optimal control problems with continuous inequality constrains: New convergence results, Numerical Algebra, Control and Optimization, 2 (2012), 571-599.
doi: 10.3934/naco.2012.2.571. |
[14] |
R. Loxton, Q. Lin and K. L. Teo, Minimizing control variation in nonlinear optimal control, Automatica, 49 (2013), 2652-2664.
doi: 10.1016/j.automatica.2013.05.027. |
[15] |
W. Rudin, Functional Analysis, $2^{nd}$ edition. McGraw-Hill, Inc., New York, 1991. |
[16] |
S. Sager, H. G. Bock and G. Reinelt, Direct methods with maximal lower bound for mixed-integer optimal control problems, Mathematical Programming, 118 (2009), 109-149.
doi: 10.1007/s10107-007-0185-6. |
[17] |
K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, New York: John Wiley & Sons Inc., 1991. |
[18] |
S. F. Woon, V. Rehbock and R. Loxton, Towards global solutions of optimal discrete-valued control problems, Optimal Control Applications and Methods, 33 (2012), 576-594.
doi: 10.1002/oca.1015. |
[19] |
C. Yu, B. Li, R. Loxton and K. L. Teo, Optimal discrete-valued control computation, Journal of Global Optimization, 56 (2013), 503-518.
doi: 10.1007/s10898-012-9858-7. |
[20] |
J. Yu, Z. X. Liu and D. T. Peng, Existence and stability analysis of optimal control, Optimal Control Applications and Methods, 35 (2014), 721-729.
doi: 10.1002/oca.2096. |
[21] |
E. Zeidler, Functional and Its Applications II/B, Springer-Verlag, New York, 1990. |
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