American Institute of Mathematical Sciences

Advanced
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:
[1]

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

[2]

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

[3]

V. I. Bogachev, Measure Theory ,, Springer-Verlag, (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.  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.  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.  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.  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.  doi: 10.1137/070707993.  Google Scholar

[10]

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

[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.  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.  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, 2 (2012), 571.  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.  doi: 10.1016/j.automatica.2013.05.027.  Google Scholar

[15]

W. Rudin, Functional Analysis,, $2^{nd}$ edition. McGraw-Hill, (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.  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.  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.  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.  doi: 10.1002/oca.2096.  Google Scholar

[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).   Google Scholar

[2]

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

[3]

V. I. Bogachev, Measure Theory ,, Springer-Verlag, (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.  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.  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.  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.  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.  doi: 10.1137/070707993.  Google Scholar

[10]

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

[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.  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.  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, 2 (2012), 571.  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.  doi: 10.1016/j.automatica.2013.05.027.  Google Scholar

[15]

W. Rudin, Functional Analysis,, $2^{nd}$ edition. McGraw-Hill, (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.  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.  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.  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.  doi: 10.1002/oca.2096.  Google Scholar

[21]

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

[1]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521
[2]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[3]

Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183
[4]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399
[5]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
[6]

Guido De Philippis, Antonio De Rosa, Jonas Hirsch. The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7031-7056. doi: 10.3934/dcds.2019243
[7]

Yves Dumont, Frederic Chiroleu. Vector control for the Chikungunya disease. Mathematical Biosciences & Engineering, 2010, 7 (2) : 313-345. doi: 10.3934/mbe.2010.7.313
[8]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094
[9]

Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212
[10]

Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089
[11]

Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233
[12]

Hirofumi Notsu, Masato Kimura. Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity. Networks & Heterogeneous Media, 2014, 9 (4) : 617-634. doi: 10.3934/nhm.2014.9.617
[13]

Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810
[14]

Chin-Chin Wu. Existence of traveling wavefront for discrete bistable competition model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 973-984. doi: 10.3934/dcdsb.2011.16.973
[15]

Shu-Yu Hsu. Existence and properties of ancient solutions of the Yamabe flow. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 91-129. doi: 10.3934/dcds.2018005
[16]

Graziano Crasta, Philippe G. LeFloch. Existence result for a class of nonconservative and nonstrictly hyperbolic systems. Communications on Pure & Applied Analysis, 2002, 1 (4) : 513-530. doi: 10.3934/cpaa.2002.1.513
[17]

Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995
[18]

Martin Bohner, Sabrina Streipert. Optimal harvesting policy for the Beverton--Holt model. Mathematical Biosciences & Engineering, 2016, 13 (4) : 673-695. doi: 10.3934/mbe.2016014
[19]

Xingchun Wang, Yongjin Wang. Variance-optimal hedging for target volatility options. Journal of Industrial & Management Optimization, 2014, 10 (1) : 207-218. doi: 10.3934/jimo.2014.10.207
[20]

Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (89)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences