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

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.
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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