American Institute of Mathematical Sciences

Advanced
May  2019, 24(5): 2093-2124. doi: 10.3934/dcdsb.2019086

Necessary optimality conditions for average cost minimization problems

Piernicola Bettiol 1, and  Nathalie Khalil 2,

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

Dedicated to U. Ledzewicz, H. Maurer and H. Schättler

Received  January 2018 Revised  January 2019 Published  May 2019 Early access  March 2019

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).

Keywords: Optimal control, unknown parameters, average cost, necessary conditions.
Mathematics Subject Classification: Primary: 49J15.
Citation: Piernicola Bettiol, Nathalie Khalil. Necessary optimality conditions for average cost minimization problems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2093-2124. doi: 10.3934/dcdsb.2019086
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.  Google Scholar

[3] R. B. Ash, Measure, Integration, and Functional Analysis, Academic Press, New York-London, 1972.   Google Scholar
[4]

J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Springer Science & Business Media, 2009. doi: 10.1007/978-0-8176-4848-0.  Google Scholar

[5]

V. I. Bogachev, Measure Theory, Springer Science & Business Media, 2007. doi: 10.1007/978-3-540-34514-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[9]

F. H. Clarke, Optimization and Nonsmooth Analysis, SIAM, 1990. doi: 10.1137/1.9781611971309.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

K. R. Parthasarathy, Probability Measures on Metric Spaces, American Mathematical Soc., 2005. doi: 10.1090/chel/352.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

R. B. Vinter, Minimax optimal control, SIAM Journal on Control and Optimization, 44 (2005), 939-968.  doi: 10.1137/S0363012902415244.  Google Scholar

[19]

R. B. Vinter, Optimal Control, Springer Science & Business Media, 2010. doi: 10.1007/978-0-8176-8086-2.  Google Scholar

[20] J. Warga, Optimal Control of Differential and Functional Equations, Academic press, 1972.   Google Scholar
[21]

J. Warga, Nonsmooth problems with conflicting controls, SIAM journal on control and optimization, 29 (1991), 678-701.  doi: 10.1137/0329038.  Google Scholar

[22]

E. Zuazua, Averaged Control, Automatica, 50 (2014), 3077-3087.  doi: 10.1016/j.automatica.2014.10.054.  Google Scholar

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.  Google Scholar

[3] R. B. Ash, Measure, Integration, and Functional Analysis, Academic Press, New York-London, 1972.   Google Scholar
[4]

J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Springer Science & Business Media, 2009. doi: 10.1007/978-0-8176-4848-0.  Google Scholar

[5]

V. I. Bogachev, Measure Theory, Springer Science & Business Media, 2007. doi: 10.1007/978-3-540-34514-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[9]

F. H. Clarke, Optimization and Nonsmooth Analysis, SIAM, 1990. doi: 10.1137/1.9781611971309.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

K. R. Parthasarathy, Probability Measures on Metric Spaces, American Mathematical Soc., 2005. doi: 10.1090/chel/352.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

R. B. Vinter, Minimax optimal control, SIAM Journal on Control and Optimization, 44 (2005), 939-968.  doi: 10.1137/S0363012902415244.  Google Scholar

[19]

R. B. Vinter, Optimal Control, Springer Science & Business Media, 2010. doi: 10.1007/978-0-8176-8086-2.  Google Scholar

[20] J. Warga, Optimal Control of Differential and Functional Equations, Academic press, 1972.   Google Scholar
[21]

J. Warga, Nonsmooth problems with conflicting controls, SIAM journal on control and optimization, 29 (1991), 678-701.  doi: 10.1137/0329038.  Google Scholar

[22]

E. Zuazua, Averaged Control, Automatica, 50 (2014), 3077-3087.  doi: 10.1016/j.automatica.2014.10.054.  Google Scholar

[1]

Francis Clarke. A general theorem on necessary conditions in optimal control. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 485-503. doi: 10.3934/dcds.2011.29.485
[2]

Sofia O. Lopes, Fernando A. C. C. Fontes, Maria do Rosário de Pinho. On constraint qualifications for nondegenerate necessary conditions of optimality applied to optimal control problems. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 559-575. doi: 10.3934/dcds.2011.29.559
[3]

Vincenzo Basco, Piermarco Cannarsa, Hélène Frankowska. Necessary conditions for infinite horizon optimal control problems with state constraints. Mathematical Control & Related Fields, 2018, 8 (3&4) : 535-555. doi: 10.3934/mcrf.2018022
[4]

Jianxiong Ye, An Li. Necessary optimality conditions for nonautonomous optimal control problems and its applications to bilevel optimal control. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1399-1419. doi: 10.3934/jimo.2018101
[5]

Andrei V. Dmitruk, Nikolai P. Osmolovskii. Necessary conditions for a weak minimum in optimal control problems with integral equations on a variable time interval. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4323-4343. doi: 10.3934/dcds.2015.35.4323
[6]

Hongwei Lou, Jiongmin Yong. Second-order necessary conditions for optimal control of semilinear elliptic equations with leading term containing controls. Mathematical Control & Related Fields, 2018, 8 (1) : 57-88. doi: 10.3934/mcrf.2018003
[7]

Hongwei Lou. Second-order necessary/sufficient conditions for optimal control problems in the absence of linear structure. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1445-1464. doi: 10.3934/dcdsb.2010.14.1445
[8]

Andrei V. Dmitruk, Nikolai P. Osmolovski. Necessary conditions for a weak minimum in a general optimal control problem with integral equations on a variable time interval. Mathematical Control & Related Fields, 2017, 7 (4) : 507-535. doi: 10.3934/mcrf.2017019
[9]

Shahlar F. Maharramov. Necessary optimality conditions for switching control problems. Journal of Industrial & Management Optimization, 2010, 6 (1) : 47-55. doi: 10.3934/jimo.2010.6.47
[10]

Liangquan Zhang, Qing Zhou, Juan Yang. Necessary condition for optimal control of doubly stochastic systems. Mathematical Control & Related Fields, 2020, 10 (2) : 379-403. doi: 10.3934/mcrf.2020002
[11]

R. Enkhbat , N. Tungalag, A. S. Strekalovsky. Pseudoconvexity properties of average cost functions. Numerical Algebra, Control & Optimization, 2015, 5 (3) : 233-236. doi: 10.3934/naco.2015.5.233
[12]

Omid S. Fard, Javad Soolaki, Delfim F. M. Torres. A necessary condition of Pontryagin type for fuzzy fractional optimal control problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 59-76. doi: 10.3934/dcdss.2018004
[13]

Yujing Wang, Changjun Yu, Kok Lay Teo. A new computational strategy for optimal control problem with a cost on changing control. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 339-364. doi: 10.3934/naco.2016016
[14]

Shuren Liu, Qiying Hu, Yifan Xu. Optimal inventory control with fixed ordering cost for selling by internet auctions. Journal of Industrial & Management Optimization, 2012, 8 (1) : 19-40. doi: 10.3934/jimo.2012.8.19
[15]

Jésus Ildefonso Díaz, Tommaso Mingazzini, Ángel Manuel Ramos. On the optimal control for a semilinear equation with cost depending on the free boundary. Networks & Heterogeneous Media, 2012, 7 (4) : 605-615. doi: 10.3934/nhm.2012.7.605
[16]

Stepan Sorokin, Maxim Staritsyn. Feedback necessary optimality conditions for a class of terminally constrained state-linear variational problems inspired by impulsive control. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 201-210. doi: 10.3934/naco.2017014
[17]

M. Soledad Aronna. Second order necessary and sufficient optimality conditions for singular solutions of partially-affine control problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1233-1258. doi: 10.3934/dcdss.2018070
[18]

M. M. El-Dessoky, Muhammad Altaf Khan. Application of Caputo-Fabrizio derivative to a cancer model with unknown parameters. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3557-3575. doi: 10.3934/dcdss.2020429
[19]

María Teresa V. Martínez-Palacios, Adrián Hernández-Del-Valle, Ambrosio Ortiz-Ramírez. On the pricing of Asian options with geometric average of American type with stochastic interest rate: A stochastic optimal control approach. Journal of Dynamics & Games, 2019, 6 (1) : 53-64. doi: 10.3934/jdg.2019004
[20]

Zhaohua Gong, Chongyang Liu, Yujing Wang. Optimal control of switched systems with multiple time-delays and a cost on changing control. Journal of Industrial & Management Optimization, 2018, 14 (1) : 183-198. doi: 10.3934/jimo.2017042

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (166)
  • HTML views (114)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy