American Institute of Mathematical Sciences

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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  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
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J. Ackermann, Robust Control: The Parameter Space Approach, Springer Science & Business Media, 2012. Google Scholar

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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

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

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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

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

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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

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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

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N. Khalil, Optimality Conditions for Optimal Control Problems and Applications, Ph.D thesis, Université de Bretagne occidentale-Brest, 2017. Google Scholar

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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

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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

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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

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