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

Optimal sampled-data control, and generalizations on time scales

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

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.
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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