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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, (1981).
|
[2] |
J. P. Aubin and H. Frankowska, Set-valued Analysis,, Birkhauser, (1990).
|
[3] |
V. I. Bogachev, Measure Theory ,, Springer-Verlag, (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.
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.
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.
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.
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.
doi: 10.1137/070707993. |
[10] |
E. Kreyszig, Introductory Functional Analysis with Applications,, John Wiley & Sons Inc., (1978).
|
[11] |
X. J. Li and J. M. Yong, Optimal Control Theory for Infinite Dimensional Systems,, Birkhauser, (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.
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, 2 (2012), 571.
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.
doi: 10.1016/j.automatica.2013.05.027. |
[15] |
W. Rudin, Functional Analysis,, $2^{nd}$ edition. McGraw-Hill, (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.
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.
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.
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.
doi: 10.1002/oca.2096. |
[21] |
E. Zeidler, Functional and Its Applications II/B,, Springer-Verlag, (1990). Google Scholar |
show all references
References:
[1] |
N. U. Ahmed and K. L. Teo, Optimal Control of Distributed Parameter Systems,, Elsevier North Holland, (1981).
|
[2] |
J. P. Aubin and H. Frankowska, Set-valued Analysis,, Birkhauser, (1990).
|
[3] |
V. I. Bogachev, Measure Theory ,, Springer-Verlag, (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.
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.
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.
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.
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.
doi: 10.1137/070707993. |
[10] |
E. Kreyszig, Introductory Functional Analysis with Applications,, John Wiley & Sons Inc., (1978).
|
[11] |
X. J. Li and J. M. Yong, Optimal Control Theory for Infinite Dimensional Systems,, Birkhauser, (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.
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, 2 (2012), 571.
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.
doi: 10.1016/j.automatica.2013.05.027. |
[15] |
W. Rudin, Functional Analysis,, $2^{nd}$ edition. McGraw-Hill, (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.
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.
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.
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.
doi: 10.1002/oca.2096. |
[21] |
E. Zeidler, Functional and Its Applications II/B,, Springer-Verlag, (1990). Google Scholar |
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