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 & 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.  doi: 10.1007/BF03322019.  Google Scholar

[2]

R. P. Agarwal, M. Bohner and A. Peterson, Inequalities on time scales: A survey,, Math. Inequal. Appl., 4 (2001), 535.  doi: 10.7153/mia-04-48.  Google Scholar

[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).   Google Scholar

[4]

A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint, volume 87 of {Encyclopaedia of Mathematical Sciences,, Springer-Verlag, (2004).  doi: 10.1007/978-3-662-06404-7.  Google Scholar

[5]

B. Aulbach and L. Neidhart, Integration on measure chains,, In Proceedings of the Sixth International Conference on Difference Equations, (2004), 239.   Google Scholar

[6]

F. M. Atici, D. C. Biles and A. Lebedinsky, An application of time scales to economics,, Math. Comput. Modelling, 43 (2006), 718.  doi: 10.1016/j.mcm.2005.08.014.  Google Scholar

[7]

Z. Bartosiewicz and D. F. M. Torres, Noether's theorem on time scales,, J. Math. Anal. Appl., 342 (2008), 1220.  doi: 10.1016/j.jmaa.2008.01.018.  Google Scholar

[8]

M. Bohner, Calculus of variations on time scales,, Dynam. Systems Appl., 13 (2004), 339.   Google Scholar

[9]

M. Bohner and G. S. Guseinov, Double integral calculus of variations on time scales,, Comput. Math. Appl., 54 (2007), 45.  doi: 10.1016/j.camwa.2006.10.032.  Google Scholar

[10]

M. Bohner and A. Peterson, Dynamic Equations on Time Scales. An Introduction with Applications,, Birkhäuser Boston Inc., (2001).  doi: 10.1007/978-1-4612-0201-1.  Google Scholar

[11]

M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales,, Birkhäuser Boston Inc., (2003).  doi: 10.1007/978-0-8176-8230-9.  Google Scholar

[12]

V. G. Boltyanskiĭ, Optimal Control of Discrete Systems,, John Wiley & Sons, (1978).   Google Scholar

[13]

B. Bonnard and M. Chyba, The Role of Singular Trajectories in Control Theory,, Springer Verlag, (2003).   Google Scholar

[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, (2006).  doi: 10.1007/3-540-37640-2.  Google Scholar

[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.  doi: 10.1080/10236198.2013.862358.  Google Scholar

[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.  doi: 10.1137/130912219.  Google Scholar

[17]

L. Bourdin, Nonshifted calculus of variations on time scales with nabla-differentiable sigma,, J. Math. Anal. Appl., 411 (2014), 543.  doi: 10.1016/j.jmaa.2013.10.013.  Google Scholar

[18]

A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, volume 2 of AIMS Series on Applied Mathematics,, Springfield, (2007).   Google Scholar

[19]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Springer, (2011).   Google Scholar

[20]

J. A. E. Bryson and Y. C. Ho, Applied Optimal Control,, Hemisphere Publishing Corp. Washington, (1975).   Google Scholar

[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, (2005).  doi: 10.1007/978-1-4899-7276-7.  Google Scholar

[22]

A. Cabada and D. R. Vivero, Criterions for absolute continuity on time scales,, J. Difference Equ. Appl., 11 (2005), 1013.  doi: 10.1080/10236190500272830.  Google Scholar

[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.  doi: 10.1016/j.mcm.2005.09.028.  Google Scholar

[24]

M. D. Canon, J. C. D. Cullum and E. Polak, Theory of Optimal Control and Mathematical Programming,, McGraw-Hill Book Co., (1970).   Google Scholar

[25]

L. Cesari, Optimization - Theory and Applications. Problems with Ordinary Differential Equations,, Applications of Mathematics, (1983).  doi: 10.1007/978-1-4613-8165-5.  Google Scholar

[26]

I. Ekeland, On the variational principle,, J. Math. Anal. Appl., 47 (1974), 324.  doi: 10.1016/0022-247X(74)90025-0.  Google Scholar

[27]

L. Fan and C. Wang, The Discrete Maximum Principle: A Study of Multistage Systems Optimization,, John Wiley & Sons, (1964).   Google Scholar

[28]

R. A. C. Ferreira and D. F. M. Torres, Higher-order calculus of variations on time scales,, In Mathematical control theory and finance, (2008), 149.  doi: 10.1007/978-3-540-69532-5_9.  Google Scholar

[29]

J. G. P. Gamarra and R. V. Solvé, Complex discrete dynamics from simple continuous population models,, Bull. Math. Biol., 64 (2002), 611.  doi: 10.1006/bulm.2002.0286.  Google Scholar

[30]

R. V. Gamkrelidze, Discovery of the maximum principle,, In Mathematical events of the twentieth century, (2006), 85.  doi: 10.1007/3-540-29462-7_5.  Google Scholar

[31]

G. S. Guseinov, Integration on time scales,, J. Math. Anal. Appl., 285 (2003), 107.  doi: 10.1016/S0022-247X(03)00361-5.  Google Scholar

[32]

H. Halkin, A maximum principle of the Pontryagin type for systems described by nonlinear difference equations,, SIAM J. Control, 4 (1966), 90.  doi: 10.1137/0304009.  Google Scholar

[33]

M. R. Hestenes, Calculus of Variations and Optimal Control Theory,, Robert E. Krieger Publishing Co. Inc., (1980).   Google Scholar

[34]

S. Hilger, Ein Maßkettenkalkül mit Anwendungen auf Zentrumsmannigfaltigkeiten,, PhD thesis, (1988).   Google Scholar

[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.  doi: 10.1016/j.jmaa.2003.09.031.  Google Scholar

[36]

R. Hilscher and V. Zeidan, First-order conditions for generalized variational problems over time scales,, Comput. Math. Appl., 62 (2011), 3490.  doi: 10.1016/j.camwa.2011.08.065.  Google Scholar

[37]

R. Hilscher and V. Zeidan, Weak maximum principle and accessory problem for control problems on time scales,, Nonlinear Anal., 70 (2009), 3209.  doi: 10.1016/j.na.2008.04.025.  Google Scholar

[38]

R. Hilscher and V. Zeidan, Time scale embedding theorem and coercivity of quadratic functionals,, Analysis (Munich), 28 (2008), 1.  doi: 10.1524/anly.2008.0900.  Google Scholar

[39]

J. M. Holtzman, Convexity and the maximum principle for discrete systems,, IEEE Trans. Automatic Control, AC-11 (1966), 30.   Google Scholar

[40]

J.-B. Hiriart-Urruty, La Commande Optimale Pour Les Débutants,, Opuscules. Ellipses, (2008).   Google Scholar

[41]

J. M. Holtzman and H. Halkin, Discretional convexity and the maximum principle for discrete systems,, SIAM J. Control, 4 (1966), 263.  doi: 10.1137/0304023.  Google Scholar

[42]

V. Jurdjevic, Geometric Control Theory,, Cambridge Studies in Advanced Mathematics, (1997).   Google Scholar

[43]

E. B. Lee and L. Markus, Foundations of Optimal Control Theory,, John Wiley, (1967).   Google Scholar

[44]

R. M. May, Simple mathematical models with very complicated dynamics,, Nature, 261 (1976), 459.   Google Scholar

[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, (2006).   Google Scholar

[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, (1962).   Google Scholar

[47]

H. Schättler and U. Ledzewicz, Geometric Optimal Control, Theory, Methods and Examples,, Interdisciplinary Applied Mathematics, (2012).  doi: 10.1007/978-1-4614-3834-2.  Google Scholar

[48]

A. Seierstad and K. Sydsæter, Optimal Control Theory with Economic Applications,, North-Holland Publishing Co., (1987).   Google Scholar

[49]

S. P. Sethi and G. L. Thompson, Optimal Control Theory. Applications to Management Science and Economics,, Kluwer Academic Publishers, (2000).   Google Scholar

[50]

E. Trélat, Contrôle Optimal, Théorie & Applications,, Mathématiques Concrètes. Vuibert, (2005).   Google Scholar

[51]

E. Trélat, Optimal control and applications to aerospace: Some results and challenges,, J. Optim. Theory Appl., 154 (2012), 713.  doi: 10.1007/s10957-012-0050-5.  Google Scholar

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.  doi: 10.1007/BF03322019.  Google Scholar

[2]

R. P. Agarwal, M. Bohner and A. Peterson, Inequalities on time scales: A survey,, Math. Inequal. Appl., 4 (2001), 535.  doi: 10.7153/mia-04-48.  Google Scholar

[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).   Google Scholar

[4]

A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint, volume 87 of {Encyclopaedia of Mathematical Sciences,, Springer-Verlag, (2004).  doi: 10.1007/978-3-662-06404-7.  Google Scholar

[5]

B. Aulbach and L. Neidhart, Integration on measure chains,, In Proceedings of the Sixth International Conference on Difference Equations, (2004), 239.   Google Scholar

[6]

F. M. Atici, D. C. Biles and A. Lebedinsky, An application of time scales to economics,, Math. Comput. Modelling, 43 (2006), 718.  doi: 10.1016/j.mcm.2005.08.014.  Google Scholar

[7]

Z. Bartosiewicz and D. F. M. Torres, Noether's theorem on time scales,, J. Math. Anal. Appl., 342 (2008), 1220.  doi: 10.1016/j.jmaa.2008.01.018.  Google Scholar

[8]

M. Bohner, Calculus of variations on time scales,, Dynam. Systems Appl., 13 (2004), 339.   Google Scholar

[9]

M. Bohner and G. S. Guseinov, Double integral calculus of variations on time scales,, Comput. Math. Appl., 54 (2007), 45.  doi: 10.1016/j.camwa.2006.10.032.  Google Scholar

[10]

M. Bohner and A. Peterson, Dynamic Equations on Time Scales. An Introduction with Applications,, Birkhäuser Boston Inc., (2001).  doi: 10.1007/978-1-4612-0201-1.  Google Scholar

[11]

M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales,, Birkhäuser Boston Inc., (2003).  doi: 10.1007/978-0-8176-8230-9.  Google Scholar

[12]

V. G. Boltyanskiĭ, Optimal Control of Discrete Systems,, John Wiley & Sons, (1978).   Google Scholar

[13]

B. Bonnard and M. Chyba, The Role of Singular Trajectories in Control Theory,, Springer Verlag, (2003).   Google Scholar

[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, (2006).  doi: 10.1007/3-540-37640-2.  Google Scholar

[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.  doi: 10.1080/10236198.2013.862358.  Google Scholar

[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.  doi: 10.1137/130912219.  Google Scholar

[17]

L. Bourdin, Nonshifted calculus of variations on time scales with nabla-differentiable sigma,, J. Math. Anal. Appl., 411 (2014), 543.  doi: 10.1016/j.jmaa.2013.10.013.  Google Scholar

[18]

A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, volume 2 of AIMS Series on Applied Mathematics,, Springfield, (2007).   Google Scholar

[19]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Springer, (2011).   Google Scholar

[20]

J. A. E. Bryson and Y. C. Ho, Applied Optimal Control,, Hemisphere Publishing Corp. Washington, (1975).   Google Scholar

[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, (2005).  doi: 10.1007/978-1-4899-7276-7.  Google Scholar

[22]

A. Cabada and D. R. Vivero, Criterions for absolute continuity on time scales,, J. Difference Equ. Appl., 11 (2005), 1013.  doi: 10.1080/10236190500272830.  Google Scholar

[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.  doi: 10.1016/j.mcm.2005.09.028.  Google Scholar

[24]

M. D. Canon, J. C. D. Cullum and E. Polak, Theory of Optimal Control and Mathematical Programming,, McGraw-Hill Book Co., (1970).   Google Scholar

[25]

L. Cesari, Optimization - Theory and Applications. Problems with Ordinary Differential Equations,, Applications of Mathematics, (1983).  doi: 10.1007/978-1-4613-8165-5.  Google Scholar

[26]

I. Ekeland, On the variational principle,, J. Math. Anal. Appl., 47 (1974), 324.  doi: 10.1016/0022-247X(74)90025-0.  Google Scholar

[27]

L. Fan and C. Wang, The Discrete Maximum Principle: A Study of Multistage Systems Optimization,, John Wiley & Sons, (1964).   Google Scholar

[28]

R. A. C. Ferreira and D. F. M. Torres, Higher-order calculus of variations on time scales,, In Mathematical control theory and finance, (2008), 149.  doi: 10.1007/978-3-540-69532-5_9.  Google Scholar

[29]

J. G. P. Gamarra and R. V. Solvé, Complex discrete dynamics from simple continuous population models,, Bull. Math. Biol., 64 (2002), 611.  doi: 10.1006/bulm.2002.0286.  Google Scholar

[30]

R. V. Gamkrelidze, Discovery of the maximum principle,, In Mathematical events of the twentieth century, (2006), 85.  doi: 10.1007/3-540-29462-7_5.  Google Scholar

[31]

G. S. Guseinov, Integration on time scales,, J. Math. Anal. Appl., 285 (2003), 107.  doi: 10.1016/S0022-247X(03)00361-5.  Google Scholar

[32]

H. Halkin, A maximum principle of the Pontryagin type for systems described by nonlinear difference equations,, SIAM J. Control, 4 (1966), 90.  doi: 10.1137/0304009.  Google Scholar

[33]

M. R. Hestenes, Calculus of Variations and Optimal Control Theory,, Robert E. Krieger Publishing Co. Inc., (1980).   Google Scholar

[34]

S. Hilger, Ein Maßkettenkalkül mit Anwendungen auf Zentrumsmannigfaltigkeiten,, PhD thesis, (1988).   Google Scholar

[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.  doi: 10.1016/j.jmaa.2003.09.031.  Google Scholar

[36]

R. Hilscher and V. Zeidan, First-order conditions for generalized variational problems over time scales,, Comput. Math. Appl., 62 (2011), 3490.  doi: 10.1016/j.camwa.2011.08.065.  Google Scholar

[37]

R. Hilscher and V. Zeidan, Weak maximum principle and accessory problem for control problems on time scales,, Nonlinear Anal., 70 (2009), 3209.  doi: 10.1016/j.na.2008.04.025.  Google Scholar

[38]

R. Hilscher and V. Zeidan, Time scale embedding theorem and coercivity of quadratic functionals,, Analysis (Munich), 28 (2008), 1.  doi: 10.1524/anly.2008.0900.  Google Scholar

[39]

J. M. Holtzman, Convexity and the maximum principle for discrete systems,, IEEE Trans. Automatic Control, AC-11 (1966), 30.   Google Scholar

[40]

J.-B. Hiriart-Urruty, La Commande Optimale Pour Les Débutants,, Opuscules. Ellipses, (2008).   Google Scholar

[41]

J. M. Holtzman and H. Halkin, Discretional convexity and the maximum principle for discrete systems,, SIAM J. Control, 4 (1966), 263.  doi: 10.1137/0304023.  Google Scholar

[42]

V. Jurdjevic, Geometric Control Theory,, Cambridge Studies in Advanced Mathematics, (1997).   Google Scholar

[43]

E. B. Lee and L. Markus, Foundations of Optimal Control Theory,, John Wiley, (1967).   Google Scholar

[44]

R. M. May, Simple mathematical models with very complicated dynamics,, Nature, 261 (1976), 459.   Google Scholar

[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, (2006).   Google Scholar

[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, (1962).   Google Scholar

[47]

H. Schättler and U. Ledzewicz, Geometric Optimal Control, Theory, Methods and Examples,, Interdisciplinary Applied Mathematics, (2012).  doi: 10.1007/978-1-4614-3834-2.  Google Scholar

[48]

A. Seierstad and K. Sydsæter, Optimal Control Theory with Economic Applications,, North-Holland Publishing Co., (1987).   Google Scholar

[49]

S. P. Sethi and G. L. Thompson, Optimal Control Theory. Applications to Management Science and Economics,, Kluwer Academic Publishers, (2000).   Google Scholar

[50]

E. Trélat, Contrôle Optimal, Théorie & Applications,, Mathématiques Concrètes. Vuibert, (2005).   Google Scholar

[51]

E. Trélat, Optimal control and applications to aerospace: Some results and challenges,, J. Optim. Theory Appl., 154 (2012), 713.  doi: 10.1007/s10957-012-0050-5.  Google Scholar

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