Optimal sampled-data control, and generalizations on time scales

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March 2016, 6(1): 53-94. doi: 10.3934/mcrf.2016.6.53

Optimal sampled-data control, and generalizations on time scales

Loïc Bourdin ^1, and Emmanuel Trélat ^2,

1.	Université de Limoges, Institut de recherche XLIM, Département de Mathématiques et d'Informatique, CNRS UMR 7252, Limoges, France
2.	Sorbonne Universités, UPMC Univ. Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, Institut Universitaire de France, F-75005, Paris

Received January 2015 Revised October 2015 Published January 2016

Abstract
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In this paper, we derive a version of the Pontryagin maximum principle for general finite-dimensional nonlinear optimal sampled-data control problems. Our framework is actually much more general, and we treat optimal control problems for which the state variable evolves on a given time scale (arbitrary non-empty closed subset of $\mathbb{R}$), and the control variable evolves on a smaller time scale. Sampled-data systems are then a particular case. Our proof is based on the construction of appropriate needle-like variations and on the Ekeland variational principle.

Keywords: Optimal control, sampled-data, Pontryagin maximum principle, time scale..

Mathematics Subject Classification: 49J15, 93C57, 34N99, 39A1.

Citation: Loïc Bourdin, Emmanuel Trélat. Optimal sampled-data control, and generalizations on time scales. Mathematical Control and Related Fields, 2016, 6 (1) : 53-94. doi: 10.3934/mcrf.2016.6.53

References:

[1]	R. P. Agarwal and M. Bohner, Basic calculus on time scales and some of its applications, Results Math., 35 (1999), 3-22. doi: 10.1007/BF03322019.
[2]	R. P. Agarwal, M. Bohner and A. Peterson, Inequalities on time scales: A survey, Math. Inequal. Appl., 4 (2001), 535-557. doi: 10.7153/mia-04-48.
[3]	R. P. Agarwal, V. Otero-Espinar, K. Perera and D. R. Vivero, Basic properties of Sobolev's spaces on time scales, Adv. Difference Equ., (2006), Art. ID 38121, 14pp.
[4]	A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint, volume 87 of {Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-06404-7.
[5]	B. Aulbach and L. Neidhart, Integration on measure chains, In Proceedings of the Sixth International Conference on Difference Equations, 239-252, CRC, Boca Raton, FL, 2004.
[6]	F. M. Atici, D. C. Biles and A. Lebedinsky, An application of time scales to economics, Math. Comput. Modelling, 43 (2006), 718-726. doi: 10.1016/j.mcm.2005.08.014.
[7]	Z. Bartosiewicz and D. F. M. Torres, Noether's theorem on time scales, J. Math. Anal. Appl., 342 (2008), 1220-1226. doi: 10.1016/j.jmaa.2008.01.018.
[8]	M. Bohner, Calculus of variations on time scales, Dynam. Systems Appl., 13 (2004), 339-349.
[9]	M. Bohner and G. S. Guseinov, Double integral calculus of variations on time scales, Comput. Math. Appl., 54 (2007), 45-57. doi: 10.1016/j.camwa.2006.10.032.
[10]	M. Bohner and A. Peterson, Dynamic Equations on Time Scales. An Introduction with Applications, Birkhäuser Boston Inc., Boston, MA, 2001. doi: 10.1007/978-1-4612-0201-1.
[11]	M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser Boston Inc., Boston, MA, 2003. doi: 10.1007/978-0-8176-8230-9.
[12]	V. G. Boltyanskiĭ, Optimal Control of Discrete Systems, John Wiley & Sons, New York-Toronto, Ont., 1978.
[13]	B. Bonnard and M. Chyba, The Role of Singular Trajectories in Control Theory, Springer Verlag, 2003.
[14]	B. Bonnard, L. Faubourg and E. Trélat, Mécanique Céleste et Contrôle des Véhicules Spatiaux, volume 51 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-37640-2.
[15]	L. Bourdin and E. Trélat, General Cauchy-Lipschitz theory for Delta-Cauchy problems with Carathéodory dynamics on time scales, J. Difference Equ. Appl., 20 (2014), 526-547. doi: 10.1080/10236198.2013.862358.
[16]	L. Bourdin and E. Trélat, Pontryagin maximum principle for finite dimensional nonlinear optimal control problems on time scales, SIAM J. Control Optim., 51 (2013), 3781-3813. doi: 10.1137/130912219.
[17]	L. Bourdin, Nonshifted calculus of variations on time scales with nabla-differentiable sigma, J. Math. Anal. Appl., 411 (2014), 543-554. doi: 10.1016/j.jmaa.2013.10.013.
[18]	A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, volume 2 of AIMS Series on Applied Mathematics, Springfield, MO, 2007.
[19]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.
[20]	J. A. E. Bryson and Y. C. Ho, Applied Optimal Control, Hemisphere Publishing Corp. Washington, D. C., 1975. Optimization, estimation, and control, Revised printing.
[21]	F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems. Modeling, Analysis, and Design for Simple Mechanical Control Systems, Texts in Applied Mathematics, 49, Springer-Verlag, New York, 2005. doi: 10.1007/978-1-4899-7276-7.
[22]	A. Cabada and D. R. Vivero, Criterions for absolute continuity on time scales, J. Difference Equ. Appl., 11 (2005), 1013-1028. doi: 10.1080/10236190500272830.
[23]	A. Cabada and D. R. Vivero, Expression of the Lebesgue $\Delta$-integral on time scales as a usual Lebesgue integral: application to the calculus of $\Delta$-antiderivatives, Math. Comput. Modelling, 43 (2006), 194-207. doi: 10.1016/j.mcm.2005.09.028.
[24]	M. D. Canon, J. C. D. Cullum and E. Polak, Theory of Optimal Control and Mathematical Programming, McGraw-Hill Book Co., New York, 1970.
[25]	L. Cesari, Optimization - Theory and Applications. Problems with Ordinary Differential Equations, Applications of Mathematics, 17, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4613-8165-5.
[26]	I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0.
[27]	L. Fan and C. Wang, The Discrete Maximum Principle: A Study of Multistage Systems Optimization, John Wiley & Sons, New York, 1964.
[28]	R. A. C. Ferreira and D. F. M. Torres, Higher-order calculus of variations on time scales, In Mathematical control theory and finance, pages 149-159. Springer, Berlin, 2008. doi: 10.1007/978-3-540-69532-5_9.
[29]	J. G. P. Gamarra and R. V. Solvé, Complex discrete dynamics from simple continuous population models, Bull. Math. Biol., 64 (2002), 611-620. doi: 10.1006/bulm.2002.0286.
[30]	R. V. Gamkrelidze, Discovery of the maximum principle, In Mathematical events of the twentieth century, pages 85-99. Springer, Berlin, 2006. doi: 10.1007/3-540-29462-7_5.
[31]	G. S. Guseinov, Integration on time scales, J. Math. Anal. Appl., 285 (2003), 107-127. doi: 10.1016/S0022-247X(03)00361-5.
[32]	H. Halkin, A maximum principle of the Pontryagin type for systems described by nonlinear difference equations, SIAM J. Control, 4 (1966), 90-111. doi: 10.1137/0304009.
[33]	M. R. Hestenes, Calculus of Variations and Optimal Control Theory, Robert E. Krieger Publishing Co. Inc., Huntington, N.Y., 1980. Corrected reprint of the 1966 original.
[34]	S. Hilger, Ein Maßkettenkalkül mit Anwendungen auf Zentrumsmannigfaltigkeiten, PhD thesis, Universität Würzburg, 1988.
[35]	R. Hilscher and V. Zeidan, Calculus of variations on time scales: Weak local piecewise $C_{rd}^1$ solutions with variable endpoints, J. Math. Anal. Appl., 289 (2004), 143-166. doi: 10.1016/j.jmaa.2003.09.031.
[36]	R. Hilscher and V. Zeidan, First-order conditions for generalized variational problems over time scales, Comput. Math. Appl., 62 (2011), 3490-3503. doi: 10.1016/j.camwa.2011.08.065.
[37]	R. Hilscher and V. Zeidan, Weak maximum principle and accessory problem for control problems on time scales, Nonlinear Anal., 70 (2009), 3209-3226. doi: 10.1016/j.na.2008.04.025.
[38]	R. Hilscher and V. Zeidan, Time scale embedding theorem and coercivity of quadratic functionals, Analysis (Munich), 28 (2008), 1-28. doi: 10.1524/anly.2008.0900.
[39]	J. M. Holtzman, Convexity and the maximum principle for discrete systems, IEEE Trans. Automatic Control, AC-11 (1966), 30-35.
[40]	J.-B. Hiriart-Urruty, La Commande Optimale Pour Les Débutants, Opuscules. Ellipses, Paris, 2008.
[41]	J. M. Holtzman and H. Halkin, Discretional convexity and the maximum principle for discrete systems, SIAM J. Control, 4 (1966), 263-275. doi: 10.1137/0304023.
[42]	V. Jurdjevic, Geometric Control Theory, Cambridge Studies in Advanced Mathematics, 52, Cambridge University Press, 1997.
[43]	E. B. Lee and L. Markus, Foundations of Optimal Control Theory, John Wiley, New York, 1967.
[44]	R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459-467.
[45]	B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic theory, II: Applications, Volumes 330 and 331 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2006.
[46]	L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience Publishers John Wiley & Sons, Inc. New York-London, 1962.
[47]	H. Schättler and U. Ledzewicz, Geometric Optimal Control, Theory, Methods and Examples, Interdisciplinary Applied Mathematics, Vol. 38, Springer, 2012. doi: 10.1007/978-1-4614-3834-2.
[48]	A. Seierstad and K. Sydsæter, Optimal Control Theory with Economic Applications, North-Holland Publishing Co., Amsterdam, Volume 24, 1987.
[49]	S. P. Sethi and G. L. Thompson, Optimal Control Theory. Applications to Management Science and Economics, Kluwer Academic Publishers, Boston, MA, second edition, 2000.
[50]	E. Trélat, Contrôle Optimal, Théorie & Applications, Mathématiques Concrètes. Vuibert, Paris, 2005.
[51]	E. Trélat, Optimal control and applications to aerospace: Some results and challenges, J. Optim. Theory Appl., 154 (2012), 713-758. doi: 10.1007/s10957-012-0050-5.

show all references

References:

[1]	R. P. Agarwal and M. Bohner, Basic calculus on time scales and some of its applications, Results Math., 35 (1999), 3-22. doi: 10.1007/BF03322019.
[2]	R. P. Agarwal, M. Bohner and A. Peterson, Inequalities on time scales: A survey, Math. Inequal. Appl., 4 (2001), 535-557. doi: 10.7153/mia-04-48.
[3]	R. P. Agarwal, V. Otero-Espinar, K. Perera and D. R. Vivero, Basic properties of Sobolev's spaces on time scales, Adv. Difference Equ., (2006), Art. ID 38121, 14pp.
[4]	A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint, volume 87 of {Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-06404-7.
[5]	B. Aulbach and L. Neidhart, Integration on measure chains, In Proceedings of the Sixth International Conference on Difference Equations, 239-252, CRC, Boca Raton, FL, 2004.
[6]	F. M. Atici, D. C. Biles and A. Lebedinsky, An application of time scales to economics, Math. Comput. Modelling, 43 (2006), 718-726. doi: 10.1016/j.mcm.2005.08.014.
[7]	Z. Bartosiewicz and D. F. M. Torres, Noether's theorem on time scales, J. Math. Anal. Appl., 342 (2008), 1220-1226. doi: 10.1016/j.jmaa.2008.01.018.
[8]	M. Bohner, Calculus of variations on time scales, Dynam. Systems Appl., 13 (2004), 339-349.
[9]	M. Bohner and G. S. Guseinov, Double integral calculus of variations on time scales, Comput. Math. Appl., 54 (2007), 45-57. doi: 10.1016/j.camwa.2006.10.032.
[10]	M. Bohner and A. Peterson, Dynamic Equations on Time Scales. An Introduction with Applications, Birkhäuser Boston Inc., Boston, MA, 2001. doi: 10.1007/978-1-4612-0201-1.
[11]	M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser Boston Inc., Boston, MA, 2003. doi: 10.1007/978-0-8176-8230-9.
[12]	V. G. Boltyanskiĭ, Optimal Control of Discrete Systems, John Wiley & Sons, New York-Toronto, Ont., 1978.
[13]	B. Bonnard and M. Chyba, The Role of Singular Trajectories in Control Theory, Springer Verlag, 2003.
[14]	B. Bonnard, L. Faubourg and E. Trélat, Mécanique Céleste et Contrôle des Véhicules Spatiaux, volume 51 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-37640-2.
[15]	L. Bourdin and E. Trélat, General Cauchy-Lipschitz theory for Delta-Cauchy problems with Carathéodory dynamics on time scales, J. Difference Equ. Appl., 20 (2014), 526-547. doi: 10.1080/10236198.2013.862358.
[16]	L. Bourdin and E. Trélat, Pontryagin maximum principle for finite dimensional nonlinear optimal control problems on time scales, SIAM J. Control Optim., 51 (2013), 3781-3813. doi: 10.1137/130912219.
[17]	L. Bourdin, Nonshifted calculus of variations on time scales with nabla-differentiable sigma, J. Math. Anal. Appl., 411 (2014), 543-554. doi: 10.1016/j.jmaa.2013.10.013.
[18]	A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, volume 2 of AIMS Series on Applied Mathematics, Springfield, MO, 2007.
[19]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.
[20]	J. A. E. Bryson and Y. C. Ho, Applied Optimal Control, Hemisphere Publishing Corp. Washington, D. C., 1975. Optimization, estimation, and control, Revised printing.
[21]	F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems. Modeling, Analysis, and Design for Simple Mechanical Control Systems, Texts in Applied Mathematics, 49, Springer-Verlag, New York, 2005. doi: 10.1007/978-1-4899-7276-7.
[22]	A. Cabada and D. R. Vivero, Criterions for absolute continuity on time scales, J. Difference Equ. Appl., 11 (2005), 1013-1028. doi: 10.1080/10236190500272830.
[23]	A. Cabada and D. R. Vivero, Expression of the Lebesgue $\Delta$-integral on time scales as a usual Lebesgue integral: application to the calculus of $\Delta$-antiderivatives, Math. Comput. Modelling, 43 (2006), 194-207. doi: 10.1016/j.mcm.2005.09.028.
[24]	M. D. Canon, J. C. D. Cullum and E. Polak, Theory of Optimal Control and Mathematical Programming, McGraw-Hill Book Co., New York, 1970.
[25]	L. Cesari, Optimization - Theory and Applications. Problems with Ordinary Differential Equations, Applications of Mathematics, 17, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4613-8165-5.
[26]	I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0.
[27]	L. Fan and C. Wang, The Discrete Maximum Principle: A Study of Multistage Systems Optimization, John Wiley & Sons, New York, 1964.
[28]	R. A. C. Ferreira and D. F. M. Torres, Higher-order calculus of variations on time scales, In Mathematical control theory and finance, pages 149-159. Springer, Berlin, 2008. doi: 10.1007/978-3-540-69532-5_9.
[29]	J. G. P. Gamarra and R. V. Solvé, Complex discrete dynamics from simple continuous population models, Bull. Math. Biol., 64 (2002), 611-620. doi: 10.1006/bulm.2002.0286.
[30]	R. V. Gamkrelidze, Discovery of the maximum principle, In Mathematical events of the twentieth century, pages 85-99. Springer, Berlin, 2006. doi: 10.1007/3-540-29462-7_5.
[31]	G. S. Guseinov, Integration on time scales, J. Math. Anal. Appl., 285 (2003), 107-127. doi: 10.1016/S0022-247X(03)00361-5.
[32]	H. Halkin, A maximum principle of the Pontryagin type for systems described by nonlinear difference equations, SIAM J. Control, 4 (1966), 90-111. doi: 10.1137/0304009.
[33]	M. R. Hestenes, Calculus of Variations and Optimal Control Theory, Robert E. Krieger Publishing Co. Inc., Huntington, N.Y., 1980. Corrected reprint of the 1966 original.
[34]	S. Hilger, Ein Maßkettenkalkül mit Anwendungen auf Zentrumsmannigfaltigkeiten, PhD thesis, Universität Würzburg, 1988.
[35]	R. Hilscher and V. Zeidan, Calculus of variations on time scales: Weak local piecewise $C_{rd}^1$ solutions with variable endpoints, J. Math. Anal. Appl., 289 (2004), 143-166. doi: 10.1016/j.jmaa.2003.09.031.
[36]	R. Hilscher and V. Zeidan, First-order conditions for generalized variational problems over time scales, Comput. Math. Appl., 62 (2011), 3490-3503. doi: 10.1016/j.camwa.2011.08.065.
[37]	R. Hilscher and V. Zeidan, Weak maximum principle and accessory problem for control problems on time scales, Nonlinear Anal., 70 (2009), 3209-3226. doi: 10.1016/j.na.2008.04.025.
[38]	R. Hilscher and V. Zeidan, Time scale embedding theorem and coercivity of quadratic functionals, Analysis (Munich), 28 (2008), 1-28. doi: 10.1524/anly.2008.0900.
[39]	J. M. Holtzman, Convexity and the maximum principle for discrete systems, IEEE Trans. Automatic Control, AC-11 (1966), 30-35.
[40]	J.-B. Hiriart-Urruty, La Commande Optimale Pour Les Débutants, Opuscules. Ellipses, Paris, 2008.
[41]	J. M. Holtzman and H. Halkin, Discretional convexity and the maximum principle for discrete systems, SIAM J. Control, 4 (1966), 263-275. doi: 10.1137/0304023.
[42]	V. Jurdjevic, Geometric Control Theory, Cambridge Studies in Advanced Mathematics, 52, Cambridge University Press, 1997.
[43]	E. B. Lee and L. Markus, Foundations of Optimal Control Theory, John Wiley, New York, 1967.
[44]	R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459-467.
[45]	B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic theory, II: Applications, Volumes 330 and 331 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2006.
[46]	L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience Publishers John Wiley & Sons, Inc. New York-London, 1962.
[47]	H. Schättler and U. Ledzewicz, Geometric Optimal Control, Theory, Methods and Examples, Interdisciplinary Applied Mathematics, Vol. 38, Springer, 2012. doi: 10.1007/978-1-4614-3834-2.
[48]	A. Seierstad and K. Sydsæter, Optimal Control Theory with Economic Applications, North-Holland Publishing Co., Amsterdam, Volume 24, 1987.
[49]	S. P. Sethi and G. L. Thompson, Optimal Control Theory. Applications to Management Science and Economics, Kluwer Academic Publishers, Boston, MA, second edition, 2000.
[50]	E. Trélat, Contrôle Optimal, Théorie & Applications, Mathématiques Concrètes. Vuibert, Paris, 2005.
[51]	E. Trélat, Optimal control and applications to aerospace: Some results and challenges, J. Optim. Theory Appl., 154 (2012), 713-758. doi: 10.1007/s10957-012-0050-5.

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