-
Previous Article
Distributed delays in Hes1 gene expression model
- DCDS-B Home
- This Issue
-
Next Article
Delay reaction-diffusion equation for infection dynamics
Necessary optimality conditions for average cost minimization problems
1. | Laboratoire de Mathématiques, Université de Bretagne Occidentale, 6 Avenue Victor Le Gorgeu, 29200 Brest, France |
2. | MODAL'X, Université Paris Ouest Nanterre La Défense, 200 Avenue de la République, 92001 Paris Nanterre, France |
Control systems involving unknown parameters appear a natural framework for applications in which the model design has to take into account various uncertainties. In these circumstances the performance criterion can be given in terms of an average cost, providing a paradigm which differs from the more traditional minimax or robust optimization criteria. In this paper, we provide necessary optimality conditions for a nonrestrictive class of optimal control problems in which unknown parameters intervene in the dynamics, the cost function and the right end-point constraint. An important feature of our results is that we allow the unknown parameters belonging to a mere complete separable metric space (not necessarily compact).
References:
[1] |
J. Ackermann, Robust Control: The Parameter Space Approach, Springer Science & Business Media, 2012. Google Scholar |
[2] |
A. Agrachev, Y. Baryshnikov and A. Sarychev,
Ensemble controllability by Lie algebraic methods, ESAIM: Control, Optimisation and Calculus of Variations, 22 (2016), 921-938.
doi: 10.1051/cocv/2016029. |
[3] |
R. B. Ash, Measure, Integration, and Functional Analysis, Academic Press, New York-London, 1972.
![]() |
[4] |
J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Springer Science & Business Media, 2009.
doi: 10.1007/978-0-8176-4848-0. |
[5] |
V. I. Bogachev, Measure Theory, Springer Science & Business Media, 2007.
doi: 10.1007/978-3-540-34514-5. |
[6] |
V. G. Boltyanskii and A. S. Poznyak, The Robust Maximum Principle: Theory and Applications, Birkhauser. New York, 2012.
doi: 10.1007/978-0-8176-8152-4. |
[7] |
J-B. Caillau, M. Cerf, A. Sassi, E. Trélat and H. Zidani,
Solving chance constrained optimal control problems in aerospace via Kernel Density Estimation, Optimal Control Applications and Methods, 39 (2018), 1833-1858.
doi: 10.1002/oca.2445. |
[8] |
C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Springer-Verlag, Berlin-New York, 1977. |
[9] |
F. H. Clarke, Optimization and Nonsmooth Analysis, SIAM, 1990.
doi: 10.1137/1.9781611971309. |
[10] |
F. H. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, Springer Science & Business Media, 2013.
doi: 10.1007/978-1-4471-4820-3. |
[11] |
D. Karamzin, V. de Oliveira, F. Pereira and G. Silva,
Minimax optimal control problem with state constraints, European Journal of Control, 32 (2016), 24-31.
doi: 10.1016/j.ejcon.2016.06.002. |
[12] |
N. Khalil, Optimality Conditions for Optimal Control Problems and Applications, Ph.D thesis, Université de Bretagne occidentale-Brest, 2017. Google Scholar |
[13] |
M. Palladino,
Necessary Conditions for Adverse Control Problems Expressed by Relaxed Derivatives, Set-Valued and Variational Analysis, 24 (2016), 659-683.
doi: 10.1007/s11228-016-0364-9. |
[14] |
K. R. Parthasarathy, Probability Measures on Metric Spaces, American Mathematical Soc., 2005.
doi: 10.1090/chel/352. |
[15] |
I. M. Ross, M. Karpenko and R. J. Proulx, A Lebesgue-Stieltjes framework for optimal control and allocation, IEEE, American Control Conference (ACC), 346 (2015), 5599-5604. Google Scholar |
[16] |
I. M. Ross, R. J. Proulx, M. Karpenko and Q. Gong,
Riemann–Stieltjes optimal control problems for uncertain dynamic systems, Journal of Guidance, Control, and Dynamics, 38 (2015), 1251-1263.
doi: 10.2514/1.G000505. |
[17] |
V. M. Veliov,
Optimal control of heterogeneous systems: Basic theory, Journal of Mathematical Analysis and Applications, 346 (2008), 227-242.
doi: 10.1016/j.jmaa.2008.05.012. |
[18] |
R. B. Vinter,
Minimax optimal control, SIAM Journal on Control and Optimization, 44 (2005), 939-968.
doi: 10.1137/S0363012902415244. |
[19] |
R. B. Vinter, Optimal Control, Springer Science & Business Media, 2010.
doi: 10.1007/978-0-8176-8086-2. |
[20] |
J. Warga, Optimal Control of Differential and Functional Equations, Academic press, 1972.
![]() |
[21] |
J. Warga,
Nonsmooth problems with conflicting controls, SIAM journal on control and optimization, 29 (1991), 678-701.
doi: 10.1137/0329038. |
[22] |
E. Zuazua,
Averaged Control, Automatica, 50 (2014), 3077-3087.
doi: 10.1016/j.automatica.2014.10.054. |
show all references
References:
[1] |
J. Ackermann, Robust Control: The Parameter Space Approach, Springer Science & Business Media, 2012. Google Scholar |
[2] |
A. Agrachev, Y. Baryshnikov and A. Sarychev,
Ensemble controllability by Lie algebraic methods, ESAIM: Control, Optimisation and Calculus of Variations, 22 (2016), 921-938.
doi: 10.1051/cocv/2016029. |
[3] |
R. B. Ash, Measure, Integration, and Functional Analysis, Academic Press, New York-London, 1972.
![]() |
[4] |
J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Springer Science & Business Media, 2009.
doi: 10.1007/978-0-8176-4848-0. |
[5] |
V. I. Bogachev, Measure Theory, Springer Science & Business Media, 2007.
doi: 10.1007/978-3-540-34514-5. |
[6] |
V. G. Boltyanskii and A. S. Poznyak, The Robust Maximum Principle: Theory and Applications, Birkhauser. New York, 2012.
doi: 10.1007/978-0-8176-8152-4. |
[7] |
J-B. Caillau, M. Cerf, A. Sassi, E. Trélat and H. Zidani,
Solving chance constrained optimal control problems in aerospace via Kernel Density Estimation, Optimal Control Applications and Methods, 39 (2018), 1833-1858.
doi: 10.1002/oca.2445. |
[8] |
C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Springer-Verlag, Berlin-New York, 1977. |
[9] |
F. H. Clarke, Optimization and Nonsmooth Analysis, SIAM, 1990.
doi: 10.1137/1.9781611971309. |
[10] |
F. H. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, Springer Science & Business Media, 2013.
doi: 10.1007/978-1-4471-4820-3. |
[11] |
D. Karamzin, V. de Oliveira, F. Pereira and G. Silva,
Minimax optimal control problem with state constraints, European Journal of Control, 32 (2016), 24-31.
doi: 10.1016/j.ejcon.2016.06.002. |
[12] |
N. Khalil, Optimality Conditions for Optimal Control Problems and Applications, Ph.D thesis, Université de Bretagne occidentale-Brest, 2017. Google Scholar |
[13] |
M. Palladino,
Necessary Conditions for Adverse Control Problems Expressed by Relaxed Derivatives, Set-Valued and Variational Analysis, 24 (2016), 659-683.
doi: 10.1007/s11228-016-0364-9. |
[14] |
K. R. Parthasarathy, Probability Measures on Metric Spaces, American Mathematical Soc., 2005.
doi: 10.1090/chel/352. |
[15] |
I. M. Ross, M. Karpenko and R. J. Proulx, A Lebesgue-Stieltjes framework for optimal control and allocation, IEEE, American Control Conference (ACC), 346 (2015), 5599-5604. Google Scholar |
[16] |
I. M. Ross, R. J. Proulx, M. Karpenko and Q. Gong,
Riemann–Stieltjes optimal control problems for uncertain dynamic systems, Journal of Guidance, Control, and Dynamics, 38 (2015), 1251-1263.
doi: 10.2514/1.G000505. |
[17] |
V. M. Veliov,
Optimal control of heterogeneous systems: Basic theory, Journal of Mathematical Analysis and Applications, 346 (2008), 227-242.
doi: 10.1016/j.jmaa.2008.05.012. |
[18] |
R. B. Vinter,
Minimax optimal control, SIAM Journal on Control and Optimization, 44 (2005), 939-968.
doi: 10.1137/S0363012902415244. |
[19] |
R. B. Vinter, Optimal Control, Springer Science & Business Media, 2010.
doi: 10.1007/978-0-8176-8086-2. |
[20] |
J. Warga, Optimal Control of Differential and Functional Equations, Academic press, 1972.
![]() |
[21] |
J. Warga,
Nonsmooth problems with conflicting controls, SIAM journal on control and optimization, 29 (1991), 678-701.
doi: 10.1137/0329038. |
[22] |
E. Zuazua,
Averaged Control, Automatica, 50 (2014), 3077-3087.
doi: 10.1016/j.automatica.2014.10.054. |
[1] |
Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020045 |
[2] |
Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019 |
[3] |
Tien-Yu Lin, Bhaba R. Sarker, Chien-Jui Lin. An optimal setup cost reduction and lot size for economic production quantity model with imperfect quality and quantity discounts. Journal of Industrial & Management Optimization, 2021, 17 (1) : 467-484. doi: 10.3934/jimo.2020043 |
[4] |
Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020046 |
[5] |
Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020107 |
[6] |
Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033 |
[7] |
Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020052 |
[8] |
Hongbo Guan, Yong Yang, Huiqing Zhu. A nonuniform anisotropic FEM for elliptic boundary layer optimal control problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1711-1722. doi: 10.3934/dcdsb.2020179 |
[9] |
Shengxin Zhu, Tongxiang Gu, Xingping Liu. AIMS: Average information matrix splitting. Mathematical Foundations of Computing, 2020, 3 (4) : 301-308. doi: 10.3934/mfc.2020012 |
[10] |
Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020347 |
[11] |
Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213 |
[12] |
Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020110 |
[13] |
Elimhan N. Mahmudov. Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions. Evolution Equations & Control Theory, 2021, 10 (1) : 37-59. doi: 10.3934/eect.2020051 |
[14] |
Lars Grüne, Roberto Guglielmi. On the relation between turnpike properties and dissipativity for continuous time linear quadratic optimal control problems. Mathematical Control & Related Fields, 2021, 11 (1) : 169-188. doi: 10.3934/mcrf.2020032 |
[15] |
Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control & Related Fields, 2021, 11 (1) : 47-71. doi: 10.3934/mcrf.2020026 |
[16] |
Arthur Fleig, Lars Grüne. Strict dissipativity analysis for classes of optimal control problems involving probability density functions. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020053 |
[17] |
Onur Şimşek, O. Erhun Kundakcioglu. Cost of fairness in agent scheduling for contact centers. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2021001 |
[18] |
Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020 doi: 10.3934/jgm.2020032 |
[19] |
Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077 |
[20] |
Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure & Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257 |
2019 Impact Factor: 1.27
Tools
Metrics
Other articles
by authors
[Back to Top]