Existence and stability analysis for nonlinear optimal control problems with 1-mean equicontinuous controls

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2015, 11(4): 1409-1422. doi: 10.3934/jimo.2015.11.1409

Existence and stability analysis for nonlinear optimal control problems with $1$-mean equicontinuous controls

1.	Department of Mathematics, Guizhou University, Guizhou, 550025, China, China

Received February 2014 Revised October 2014 Published March 2015

Abstract
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In this paper, the existence and stability of solutions of nonlinear optimal control problems with $1$-mean equicontinuous controls are discussed. In particular, a new existence theorem is obtained without convexity assumption. We investigate the stability of the optimal control problem with respect to the right-hand side functions, which is important in computational methods for optimal control problems when the function is approximated by a new function. Due to lack of uniqueness of solutions for an optimal control problem, the stability results for a class of optimal control problems with the measurable admissible control set is given based on the theory of set-valued mappings and the definition of essential solutions for optimal control problems. We show that the optimal control problems, whose solutions are all essential, form a dense residual set, and so every optimal control problem can be closely approximated arbitrarily by an essential optimal control problem.

Keywords: Optimal control, stability, essential solution., existence, set-valued mapping.

Mathematics Subject Classification: Primary: 49J15, 49J53; Secondary: 93D0.

Citation: Hongyong Deng, Wei Wei. Existence and stability analysis for nonlinear optimal control problems with $1$-mean equicontinuous controls. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1409-1422. doi: 10.3934/jimo.2015.11.1409

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).

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).

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