In this paper we consider optimal sampled-data control problems on time scales with inequality state constraints. A Pontryagin maximum principle is established, extending to the state constrained case existing results in the time scale literature. The proof is based on the Ekeland variational principle and on the concept of implicit spike variations adapted to the time scale setting. The main result is then applied to continuous-time min-max optimal sampled-data control problems, and a maximal velocity minimization problem for the harmonic oscillator with sampled-data control is numerically solved for illustration.
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Keywords: Optimal control, Pontryagin maximum principle, sampled-data control, time scales, state constraints, Ekeland variational principle, implicit spike variations, min-max optimal control problems
@article{COCV_2021__27_1_A53_0,
author = {Bettiol, Piernicola and Bourdin, Lo{\"\i}c},
title = {Pontryagin maximum principle for state constrained optimal sampled-data control problems on time scales},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021046},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021046/}
}
TY - JOUR AU - Bettiol, Piernicola AU - Bourdin, Loïc TI - Pontryagin maximum principle for state constrained optimal sampled-data control problems on time scales JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021046/ DO - 10.1051/cocv/2021046 LA - en ID - COCV_2021__27_1_A53_0 ER -
%0 Journal Article %A Bettiol, Piernicola %A Bourdin, Loïc %T Pontryagin maximum principle for state constrained optimal sampled-data control problems on time scales %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021046/ %R 10.1051/cocv/2021046 %G en %F COCV_2021__27_1_A53_0
Bettiol, Piernicola; Bourdin, Loïc. Pontryagin maximum principle for state constrained optimal sampled-data control problems on time scales. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 51. doi: 10.1051/cocv/2021046
[1] , Sampled-Data Control Systems: Analysis and Synthesis, Robust System Design. Springer-Verlag Berlin Heidelberg (1985).
[2] , , and , Basic properties of Sobolev’s spaces on time scales. Adv. Differ. Equ. 14 (2006) 38121.
[3] and , Basic calculus on time scales and some of its applications. Results Math. 35 (1999) 3–22.
[4] , On the choice of sampling rates in optimal linear systems. IBM Res.: Eng. Stud. (1963).
[5] , , and , Pontryagin-type conditions for optimal muscular force response to functional electric stimulations. J. Optim. Theory Appl. 184 (2020) 581–602.
[6] and , The ℌ2 problem for sampled-data systems. Syst. Control Lett. 19 (1992) 1–12.
[7] and , Noether’s theorem on time scales. J. Math. Anal. Appl. 342 (2008) 1220–1226.
[8] , Calculus of variations on time scales. Dyn. Syst. Appl. 13 (2004) 339–349.
[9] and , Dynamic equations on time scales. An introduction with applications. Birkhäuser Boston Inc., Boston, MA (2001).
[10] and , Advances in dynamic equations on time scales. Birkhäuser Boston Inc., Boston, MA (2003).
[11] , , and , Pontryaginś maximum principle for dynamic systems on time scales. J. Differ. Equ. Appl. 3 (2017) 1161–1189.
[12] , Optimal control of discrete systems. John Wiley & Sons, New York-Toronto, Ont. (1978).
[13] and , Optimal control of state constrained integral equations. Set-Valued Anal. 18 (2010) 307–326.
[14] , , and , Optimal control with state constraints and the space shuttle re-entry problem. J. Dynam. Control Syst. 9 (2003) 155–199.
[15] , Nonshifted calculus of variations on time scales with nabla-differentiable sigma. J. Math. Anal. Appl. 411 (2014) 543–554.
[16] , Note on Pontryagin maximum principle with running state constraints and smooth dynamics – proof based on the Ekeland variational principle. Res. Notes (2016).
[17] and , Continuity/constancy of the Hamiltonian function in a Pontryagin maximum principle for optimal sampled-data control problems with free sampling times. Math. Control Signals Syst. 31 (2019) 503–544.
[18] and , Optimal sampled-data controls with running inequality state constraints: Pontryagin maximum principle and bouncing trajectory phenomenon. To appear in: Math. Program., Ser. A (2020). | DOI
[19] and , General Cauchy-Lipschitz theory for Delta-Cauchy problems with Carathéodory dynamics on time scales. J. Differ. Equ. Appl. 20 (2014) 526–547.
[20] and , Pontryagin Maximum Principle for finite dimensional nonlinear optimal control problems on time scales. SIAM J. Control Optim. 51 (2013) 3781–3813.
[21] and , Optimal sampled-data control, and generalizations on time scales. Math. Control Relat. Fields 6 (2016) 53–94.
[22] and , Pontryagin Maximum Principle for optimal sampled-data control problems. In proceedings of the IFAC Workshop CAO (2015).
[23] , and , Addendum to “Pontryagin’s maximum principle for dynamic systems on time scales”. J. Differ. Equ. Appl. 23 (2017) 1760–1763.
[24] and , Introduction to the mathematical theory of control. Vol. 2 of AIMS Series on Applied Mathematics. American Institute of Mathematical Sciences (AIMS), Springfield, MO (2007).
[25] and , Criterions for absolute continuity on time scales. J. Differ. Equ. Appl. 11 (2005) 1013–1028.
[26] and , Expression of the Lebesgue Δ-integral on time scales as a usual Lebesgue integral: application to the calculus of Δ-antiderivatives. Math. Comput. Model. 43 (2006) 194–207.
[27] and , Optimal sampled-data control systems. Springer-Verlag London, Ltd., London (1996).
[28] , Functional analysis, calculus of variations and optimal control. Vol. 264 of Graduate Texts in Mathematics. Springer Science & Business Media (2013).
[29] , Geometric and numerical methods for a state constrained minimum time control problem of an electric vehicle. ESAIM: COCV 23 (2017) 1715–1749.
[30] , and , Direct and indirect methods in optimal control with state constraints and the climbing trajectory of an aircraft. Opt. Control Appl. Methods 39 (2018) 281–301.
[31] , On the development of Pontryagin’s maximum principle in the works of A. Ya. Dubovitskii and A. A. Milyutin. Control Cybern. 4A (2009) 923–957.
[32] and , Proof of the maximum principle for a problem with state constraints by the V-change of time variable. Discrete Contin. Dyn. Syst. Ser. B 24 (2019) 2189–2204.
[33] and , Extremum problems in the presence of restrictions. USSR Comput. Math. Math. Phys. 5 (1965) 1–80.
[34] , On the variational principle. J. Math. Anal. Appl. 47 (1974) 324–353.
[35] , Infinite-dimensional optimization and control theory. Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge (1999).
[36] and , Modern methods in the calculus of variations: L$$ spaces. Springer (2007).
[37] , Fixed point theory for decomposable sets. Springer Netherlands (2004).
[38] , Optimal control processes for bounded phase coordinates. Izv. Akad. Nauk SSSR. Ser. Mat. 24 (1960) 315–356.
[39] and , On an LMI approach to optimal sampled-data state feedback control design. Internat. J. Control 88 (2015) 2369–2379.
[40] , Lectures on mathematical theory of extremum problems. Vol. 67 of Lecture Notes in Economics and Mathematical Systems, edited by . Translated from the Russian by . Springer-Verlag, Berlin-New York (1972).
[41] , Integration on time scales. J. Math. Anal. Appl. 285 (2003) 107–127.
[42] , Ein Maßkettenkalkül mit Anwendungen auf Zentrumsmannigfaltigkeiten. PhD thesis, Universität Würzburg (1988).
[43] and , Calculus of variations on time scales: weak local piecewise Crd1 solutions with variable endpoints. J. Math. Anal. Appl. 289 (2004) 143–166.
[44] and , First-order conditions for generalized variational problems over time scales. Comput. Math. Appl. 62 (2011) 3490–3503.
[45] and , Weak maximum principle and accessory problem for control problems on time scales. Nonlinear Anal. 70 (2009) 3209–3226.
[46] and , Time scale embedding theorem and coercivity of quadratic functionals. Analysis (Munich) 28 (2008) 1–28.
[47] , and , A survey of the maximum principles for optimal control problems with state constraints. SIAM Rev. 37 (1995) 181–218.
[48] and , Directional convexity and the maximum principle for discrete systems. SIAM J. Control 4 (1966) 263–275.
[49] , The Riesz representation theorem on time scales. Math. Comput. Model. 55 (2012) 1570–1579.
[50] , and , New necessary conditions of optimality for control problems with state-variable inequality constraints. J. Math. Anal. Appl. 35 (1971) 255–284.
[51] , Digital Control Systems. Springer (2006).
[52] and , Foundations of optimal control theory. Robert E. Krieger Publishing Co., Inc., Melbourne, FL, second edition (1986).
[53] , and , On the behavior of optimal linear sampled-data regulators. Int. J. Control 13 (1971) 343–361.
[54] and , Optimal control theory for infinite dimensional systems. Birkhäuser Boston (1995).
[55] , On optimal control problems with bounded state variables and control appearing linearly. SIAM J. Control Optim. 15 (1977) 345–362.
[56] , and , On a state-constrained control problem in optimal production and maintenance. In Optimal Control and Dynamic Games. Springer (2005) 289–308.
[57] and , Sampling period sensitivity of the optimal sampled data linear regulator. Automatica J. IFAC 7 (1971) 367–370.
[58] and , Digital control and estimation: A unifiedapproach (1990).
[59] , Variational Analysis and Generalized Differentiation I. Springer-Verlag, Berlin Heidelberg (2006).
[60] and , Sampled-data control of nonlinear systems: an overview of recent results. Perspect. Robust Control 268 (2001) 221–239.
[61] , , and , The mathematical theory of optimal processes. John Wiley & Sons, Inc. (1962).
[62] , Necessary conditions for an extremum. Marcel Dekker, Inc., New York (1971).
[63] , and , Closed-form solutions of a fishery harvesting model with state constraint. Optimal Control Appl. Methods 35 (2014) 395–411.
[64] , Real and complex analysis. 3rd ed. McGraw-Hill, New York, NY. (1987).
[65] , and , Connection between continuous and discrete Riccati equations with applications to Kalman filtering. Proc. IEE-D 135 (1988) 28–34.
[66] and , Optimal Control Theory. Applications to Management Science and Economics. Kluwer Academic Publishers, Boston, MA, second edition (2000).
[67] , Sur les fonctions d’ensemble additives et continues. Fundam. Math. 3 (1922) 240–246.
[68] , and , Optimal sampled data state feedback control of linear systems. In Proceedings of the 19th World Congress The International Federation of Automatic Control (2014).
[69] , Contrôle optimal : théorie & applications. Vuibert Paris (2005).
[70] , , and , Solution for state constrained optimal control problems applied to power split control for hybrid vehicles. Automatica J. IFAC 50 (2014) 187–192.
[71] , Optimal Control. Birkhaüser, Boston (2000).
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