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October  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

Hongyong Deng 1, and  Wei Wei 1,

1. 

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

Received  February 2014 Revised  October 2014 Published  March 2015

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:
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N. U. Ahmed and K. L. Teo, Optimal Control of Distributed Parameter Systems, Elsevier North Holland, New York, 1981.  Google Scholar

[2]

J. P. Aubin and H. Frankowska, Set-valued Analysis, Birkhauser, Boston, 1990.  Google Scholar

[3]

V. I. Bogachev, Measure Theory , Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.  Google Scholar

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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.  Google Scholar

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A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, American Institute of Mathematical Sciences Press, 2007.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons Inc., New York, 1978.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

W. Rudin, Functional Analysis, $2^{nd}$ edition. McGraw-Hill, Inc., New York, 1991.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

E. Zeidler, Functional and Its Applications II/B, Springer-Verlag, New York, 1990. Google Scholar

show all references

References:
[1]

N. U. Ahmed and K. L. Teo, Optimal Control of Distributed Parameter Systems, Elsevier North Holland, New York, 1981.  Google Scholar

[2]

J. P. Aubin and H. Frankowska, Set-valued Analysis, Birkhauser, Boston, 1990.  Google Scholar

[3]

V. I. Bogachev, Measure Theory , Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.  Google Scholar

[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.  Google Scholar

[5]

A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, American Institute of Mathematical Sciences Press, 2007.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons Inc., New York, 1978.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

W. Rudin, Functional Analysis, $2^{nd}$ edition. McGraw-Hill, Inc., New York, 1991.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

E. Zeidler, Functional and Its Applications II/B, Springer-Verlag, New York, 1990. Google Scholar

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