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September  2018, 8(3&4): 509-533. doi: 10.3934/mcrf.2018021

Value function for regional control problems via dynamic programming and Pontryagin maximum principle

Guy Barles 1, , Ariela Briani 1, and  Emmanuel Trélat 2,,

1. 

Institut Denis Poisson, Université de Tours, Faculté des Sciences et Techniques, Parc de Grandmont, 37200 Tours, France
2. 

Sorbonne Université, Université Paris-Diderot SPC, CNRS, Inria, Laboratoire Jacques-Louis Lions, équipe CAGE, F-75005 Paris, France

* Corresponding author: Emmanuel Trélat

Received  October 2017 Revised  June 2018 Published  September 2018

Fund Project: This work was partially supported by the ANR HJnet ANR-12-BS01-0008-01.

In this paper we consider regional deterministic finite-dimensional optimal control problems, where the dynamics and the cost functional depend on the region of the state space where one is and have discontinuities at their interface.

Under the assumption that optimal trajectories have a locally finite number of switchings (i.e., no Zeno phenomenon), we use the duplication technique to show that the value function of the regional optimal control problem is the minimum over all possible structures of trajectories of value functions associated with classical optimal control problems settled over fixed structures, each of them being the restriction to some submanifold of the value function of a classical optimal control problem in higher dimension.

The lifting duplication technique is thus seen as a kind of desingularization of the value function of the regional optimal control problem.

In turn, we establish sensitivity relations for regional optimal control problems and we prove that the regularity of the value function of such problems is the same (i.e., is not more degenerate) than the one of the higher-dimensional classical optimal control problem that lifts the problem.

Keywords: Regional optimal control, discontinuous dynamics, Pontryagin maximum principle, Hamilton-Jacobi-Bellman equation.
Mathematics Subject Classification: Primary: 49K20, 49K15; Secondary: 35F21.
Citation: Guy Barles, Ariela Briani, Emmanuel Trélat. Value function for regional control problems via dynamic programming and Pontryagin maximum principle. Mathematical Control & Related Fields, 2018, 8 (3&4) : 509-533. doi: 10.3934/mcrf.2018021
References:
[1]

A. Agrachev and Y. Sachkov, Control Theory from the Geometric Viewpoint, Encyclopaedia Math. Sci. 87, Control Theory and Optimization, Ⅱ, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-06404-7.  Google Scholar

[2]

A. Agrachev, On regularity properties of extremal controls, J. Dynam. Control Systems, 1 (1995), 319-324.  doi: 10.1007/BF02269372.  Google Scholar

[3]

A. D. Ames, A. Abate and S. Sastry, Sufficient conditions for the existence of Zeno behavior, Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC'05, (2007), 696-701.  doi: 10.1109/CDC.2005.1582237.  Google Scholar

[4]

J. P. Aubin and H. Frankowska, Set-valued Analysis, Systems & Control : Foundations & Applications, 2, 1990.  Google Scholar

[5]

M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi- Bellman equations, Systems & Control: Foundations & Applications, Birkhauser Boston Inc., Boston, MA, 1997. doi: 10.1007/978-0-8176-4755-1.  Google Scholar

[6]

G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi, Springer-Verlag, Paris, 1994.  Google Scholar

[7]

G. Barles, A. Briani and E. Chasseigne, A Bellman approach for two-domains optimal control problems in $\mathbb{R}^\mathit{N}$, ESAIM: Control, Optimisation and Calculus of Variations, 19 (2013), 710-739.  doi: 10.1051/cocv/2012030.  Google Scholar

[8]

G. Barles, A. Briani and E. Chasseigne, A Bellman approach for regional optimal control problems in $\mathbb{R}^\mathit{N}$, SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 52 (2014), 1712-1744.  doi: 10.1137/130922288.  Google Scholar

[9]

G. Barles and E. Chasseigne, (Almost) everything you always wanted to know about deterministic control problems in stratified domains, Networks and Heterogeneous Media (NHM), 10 (2015), 809-836.  doi: 10.3934/nhm.2015.10.809.  Google Scholar

[10]

M. S. Branicky, V. S. Borkar and S. K. Mitter, A unified framework for hybrid control: Model and optimal control theory, IEEE Trans. Autom. Control, 43 (1998), 31-45.  doi: 10.1109/9.654885.  Google Scholar

[11]

A. Bressan and Y. Hong, Optimal control problems on stratified domains, Netw. Heterog. Media, 2 (2007), 313-331 (electronic) and Errata corrigendum: "Optimal control problems on stratified domains". Netw. Heterog. Media, 8 (2013), p625 doi: 10.3934/nhm.2007.2.313.  Google Scholar

[12]

P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Progress in Nonlinear Differential Equations and their Applications, 58, Birkhäuser Boston, Inc., Boston, MA, 2004.  Google Scholar

[13]

M. Caponigro, R. Ghezzi, B. Piccoli and E. Trélat, Regularization of chattering phenomena via bounded variation controls, to appear in IEEE Trans. Automat. Control, 14 pages. Google Scholar

[14]

Y. Chitour, F. Jean and E. Trélat, Genericity results for singular curves, J. Differential Geom., 73 (2006), 45-73.  doi: 10.4310/jdg/1146680512.  Google Scholar

[15]

Y. Chitour, F. Jean and E. Trélat, Singular trajectories of control-affine systems, SIAM J. Control Optim., 47 (2008), 1078-1095.  doi: 10.1137/060663003.  Google Scholar

[16]

F. H. Clarke and R. Vinter, The relationship between the maximum principle and dynamic programming, SIAM Journal on Control and Optimization, 25 (1987), 1291-1311.  doi: 10.1137/0325071.  Google Scholar

[17]

F. H. Clarke and R. Vinter, Optimal multiprocesses, SIAM Journal on Control and Optimization, 27 (1989), 1072-1091.  doi: 10.1137/0327057.  Google Scholar

[18]

F. H. Clarke and R. Vinter, Application of optimal multiprocesses, SIAM Journal on Control and Optimization, 27 (1989), 1047-1071.  doi: 10.1137/0327056.  Google Scholar

[19]

A. V. Dmitruk and A. M. Kaganovich, The hybrid maximum principle is a consequence of Pontryagin maximum principle, Systems Control Lett., 57 (2008), 964-970.  doi: 10.1016/j.sysconle.2008.05.006.  Google Scholar

[20]

M. Garavello and B. Piccoli, Hybrid necessary principle, SIAM J. Control Optim., 43 (2005), 1867-1887.  doi: 10.1137/S0363012903416219.  Google Scholar

[21]

H. Haberkorn and E. Trélat, Convergence result for smooth regularizations of hybrid nonlinear optimal control problems, SIAM J. Control and Optim., 49 (2011), 1498-1522.  doi: 10.1137/100809209.  Google Scholar

[22]

C. Hermosilla and H. Zidani, Infinite horizon problems on stratifiable state-constraints sets, Journal of Differential Equations, Elsevier, 258 (2015), 1430-1460.  doi: 10.1016/j.jde.2014.11.001.  Google Scholar

[23]

M. Heymann, F. Lin, G. Meyer and S. Resmerita, Stefan Analysis of Zeno behaviors in a class of hybrid systems, IEEE Trans. Automat. Control, 50 (2005), 376-383.  doi: 10.1109/TAC.2005.843874.  Google Scholar

[24]

C. Imbert, R. Monneau and H. Zidani, A Hamilton-Jacobi approach to junction problems and application to traffic flows, ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 19 (2013), 129-166.  doi: 10.1051/cocv/2012002.  Google Scholar

[25]

K. H. Johansson, M. Egerstedt, J. Lygeros and S. Sastry, On the regularization of Zeno hybrid automata, Systems Control Lett., 38 (1999), 141-150.  doi: 10.1016/S0167-6911(99)00059-6.  Google Scholar

[26]

S. Oudet, Hamilton-Jacobi equations for optimal control on heterogeneous structures with geometric singularity, Preprint, hal-01093112, 2014. Google Scholar

[27]

L. Pontryagin, V. Boltyanskii, R. Gramkrelidze and E. Mischenko, The Mathematical Theory of Optimal Processes, Wiley Interscience, 1962.  Google Scholar

[28]

Z. Rao, A. Siconolfi and H. Zidani, Transmission conditions on interfaces for Hamilton-Jacobi-bellman equations, J. Differential Equations, 257 (2014), 3978-4014.  doi: 10.1016/j.jde.2014.07.015.  Google Scholar

[29]

Z. Rao and H. Zidani, Hamilton-Jacobi-Bellman equations on multi-domains, Control and Optimization with PDE Constraints, 164 (2010), 93-116.  doi: 10.1007/978-3-0348-0631-2_6.  Google Scholar

[30]

P. Riedinger, C. Iung and F. Kratz, An optimal control approach for hybrid systems, European Journal of Control, 9 (2003), 449-458.   Google Scholar

[31]

L. Rifford and E. Trélat, Morse-Sard type results in sub-Riemannian geometry, Math. Ann., 332 (2005), 145-159.  doi: 10.1007/s00208-004-0622-2.  Google Scholar

[32]

L. Rifford and E. Trélat, On the stabilization problem for nonholonomic distributions, J. Eur. Math. Soc., 11 (2009), 223-255.  doi: 10.4171/JEMS/148.  Google Scholar

[33]

M. S. Shaikh and P. E. Caines, On the hybrid optimal control problem: Theory and algorithms, IEEE Trans. Automat. Control, 52 (2007), 1587-1603.  doi: 10.1109/TAC.2007.904451.  Google Scholar

[34]

G. Stefani, Regularity properties of the minimum-time map, Nonlinear Synthesis (Sopron, 1989), Progr. Systems Control Theory, Birkhäuser Boston, Boston, MA, 9 (1991), 270-282. doi: 10.1007/978-1-4757-2135-5_21.  Google Scholar

[35]

H. J. Sussmann, A nonsmooth hybrid maximum principle, Stability and Stabilization of Nonlinear Systems (Ghent, 1999), Lecture Notes in Control and Inform. Sci., Springer, London, 246 (1999), 325-354. doi: 10.1007/1-84628-577-1_17.  Google Scholar

[36]

E. Trélat, Contrȏle Optimal: Théorie & Applications, Vuibert, Collection "Mathématiques Concrétes", 2005.  Google Scholar

[37]

E. Trélat, Global subanalytic solutions of Hamilton-Jacobi type equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 363-387.  doi: 10.1016/j.anihpc.2005.05.002.  Google Scholar

[38]

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.  Google Scholar

[39]

J. Zhang, K. H. Johansson, J. Lygeros and S. Sastry, Zeno hybrid systems, Internat. J. Robust Nonlinear Control, 11 (2001), 435-451.  doi: 10.1002/rnc.592.  Google Scholar

[40]

J. Zhu, E. Trélat and M. Cerf, Planar tilting maneuver of a spacecraft: Singular arcs in the minimum time problem and chattering, Discrete Cont. Dynam. Syst. Ser. B., 21 (2016), 1347-1388.  doi: 10.3934/dcdsb.2016.21.1347.  Google Scholar

[41]

J. Zhu, E. Trélat and M. Cerf, Minimum time control of the rocket attitude reorientation associated with orbit dynamics, SIAM J. Cont. Optim., 54 (2016), 391-422.  doi: 10.1137/15M1028716.  Google Scholar

show all references

References:
[1]

A. Agrachev and Y. Sachkov, Control Theory from the Geometric Viewpoint, Encyclopaedia Math. Sci. 87, Control Theory and Optimization, Ⅱ, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-06404-7.  Google Scholar

[2]

A. Agrachev, On regularity properties of extremal controls, J. Dynam. Control Systems, 1 (1995), 319-324.  doi: 10.1007/BF02269372.  Google Scholar

[3]

A. D. Ames, A. Abate and S. Sastry, Sufficient conditions for the existence of Zeno behavior, Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC'05, (2007), 696-701.  doi: 10.1109/CDC.2005.1582237.  Google Scholar

[4]

J. P. Aubin and H. Frankowska, Set-valued Analysis, Systems & Control : Foundations & Applications, 2, 1990.  Google Scholar

[5]

M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi- Bellman equations, Systems & Control: Foundations & Applications, Birkhauser Boston Inc., Boston, MA, 1997. doi: 10.1007/978-0-8176-4755-1.  Google Scholar

[6]

G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi, Springer-Verlag, Paris, 1994.  Google Scholar

[7]

G. Barles, A. Briani and E. Chasseigne, A Bellman approach for two-domains optimal control problems in $\mathbb{R}^\mathit{N}$, ESAIM: Control, Optimisation and Calculus of Variations, 19 (2013), 710-739.  doi: 10.1051/cocv/2012030.  Google Scholar

[8]

G. Barles, A. Briani and E. Chasseigne, A Bellman approach for regional optimal control problems in $\mathbb{R}^\mathit{N}$, SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 52 (2014), 1712-1744.  doi: 10.1137/130922288.  Google Scholar

[9]

G. Barles and E. Chasseigne, (Almost) everything you always wanted to know about deterministic control problems in stratified domains, Networks and Heterogeneous Media (NHM), 10 (2015), 809-836.  doi: 10.3934/nhm.2015.10.809.  Google Scholar

[10]

M. S. Branicky, V. S. Borkar and S. K. Mitter, A unified framework for hybrid control: Model and optimal control theory, IEEE Trans. Autom. Control, 43 (1998), 31-45.  doi: 10.1109/9.654885.  Google Scholar

[11]

A. Bressan and Y. Hong, Optimal control problems on stratified domains, Netw. Heterog. Media, 2 (2007), 313-331 (electronic) and Errata corrigendum: "Optimal control problems on stratified domains". Netw. Heterog. Media, 8 (2013), p625 doi: 10.3934/nhm.2007.2.313.  Google Scholar

[12]

P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Progress in Nonlinear Differential Equations and their Applications, 58, Birkhäuser Boston, Inc., Boston, MA, 2004.  Google Scholar

[13]

M. Caponigro, R. Ghezzi, B. Piccoli and E. Trélat, Regularization of chattering phenomena via bounded variation controls, to appear in IEEE Trans. Automat. Control, 14 pages. Google Scholar

[14]

Y. Chitour, F. Jean and E. Trélat, Genericity results for singular curves, J. Differential Geom., 73 (2006), 45-73.  doi: 10.4310/jdg/1146680512.  Google Scholar

[15]

Y. Chitour, F. Jean and E. Trélat, Singular trajectories of control-affine systems, SIAM J. Control Optim., 47 (2008), 1078-1095.  doi: 10.1137/060663003.  Google Scholar

[16]

F. H. Clarke and R. Vinter, The relationship between the maximum principle and dynamic programming, SIAM Journal on Control and Optimization, 25 (1987), 1291-1311.  doi: 10.1137/0325071.  Google Scholar

[17]

F. H. Clarke and R. Vinter, Optimal multiprocesses, SIAM Journal on Control and Optimization, 27 (1989), 1072-1091.  doi: 10.1137/0327057.  Google Scholar

[18]

F. H. Clarke and R. Vinter, Application of optimal multiprocesses, SIAM Journal on Control and Optimization, 27 (1989), 1047-1071.  doi: 10.1137/0327056.  Google Scholar

[19]

A. V. Dmitruk and A. M. Kaganovich, The hybrid maximum principle is a consequence of Pontryagin maximum principle, Systems Control Lett., 57 (2008), 964-970.  doi: 10.1016/j.sysconle.2008.05.006.  Google Scholar

[20]

M. Garavello and B. Piccoli, Hybrid necessary principle, SIAM J. Control Optim., 43 (2005), 1867-1887.  doi: 10.1137/S0363012903416219.  Google Scholar

[21]

H. Haberkorn and E. Trélat, Convergence result for smooth regularizations of hybrid nonlinear optimal control problems, SIAM J. Control and Optim., 49 (2011), 1498-1522.  doi: 10.1137/100809209.  Google Scholar

[22]

C. Hermosilla and H. Zidani, Infinite horizon problems on stratifiable state-constraints sets, Journal of Differential Equations, Elsevier, 258 (2015), 1430-1460.  doi: 10.1016/j.jde.2014.11.001.  Google Scholar

[23]

M. Heymann, F. Lin, G. Meyer and S. Resmerita, Stefan Analysis of Zeno behaviors in a class of hybrid systems, IEEE Trans. Automat. Control, 50 (2005), 376-383.  doi: 10.1109/TAC.2005.843874.  Google Scholar

[24]

C. Imbert, R. Monneau and H. Zidani, A Hamilton-Jacobi approach to junction problems and application to traffic flows, ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 19 (2013), 129-166.  doi: 10.1051/cocv/2012002.  Google Scholar

[25]

K. H. Johansson, M. Egerstedt, J. Lygeros and S. Sastry, On the regularization of Zeno hybrid automata, Systems Control Lett., 38 (1999), 141-150.  doi: 10.1016/S0167-6911(99)00059-6.  Google Scholar

[26]

S. Oudet, Hamilton-Jacobi equations for optimal control on heterogeneous structures with geometric singularity, Preprint, hal-01093112, 2014. Google Scholar

[27]

L. Pontryagin, V. Boltyanskii, R. Gramkrelidze and E. Mischenko, The Mathematical Theory of Optimal Processes, Wiley Interscience, 1962.  Google Scholar

[28]

Z. Rao, A. Siconolfi and H. Zidani, Transmission conditions on interfaces for Hamilton-Jacobi-bellman equations, J. Differential Equations, 257 (2014), 3978-4014.  doi: 10.1016/j.jde.2014.07.015.  Google Scholar

[29]

Z. Rao and H. Zidani, Hamilton-Jacobi-Bellman equations on multi-domains, Control and Optimization with PDE Constraints, 164 (2010), 93-116.  doi: 10.1007/978-3-0348-0631-2_6.  Google Scholar

[30]

P. Riedinger, C. Iung and F. Kratz, An optimal control approach for hybrid systems, European Journal of Control, 9 (2003), 449-458.   Google Scholar

[31]

L. Rifford and E. Trélat, Morse-Sard type results in sub-Riemannian geometry, Math. Ann., 332 (2005), 145-159.  doi: 10.1007/s00208-004-0622-2.  Google Scholar

[32]

L. Rifford and E. Trélat, On the stabilization problem for nonholonomic distributions, J. Eur. Math. Soc., 11 (2009), 223-255.  doi: 10.4171/JEMS/148.  Google Scholar

[33]

M. S. Shaikh and P. E. Caines, On the hybrid optimal control problem: Theory and algorithms, IEEE Trans. Automat. Control, 52 (2007), 1587-1603.  doi: 10.1109/TAC.2007.904451.  Google Scholar

[34]

G. Stefani, Regularity properties of the minimum-time map, Nonlinear Synthesis (Sopron, 1989), Progr. Systems Control Theory, Birkhäuser Boston, Boston, MA, 9 (1991), 270-282. doi: 10.1007/978-1-4757-2135-5_21.  Google Scholar

[35]

H. J. Sussmann, A nonsmooth hybrid maximum principle, Stability and Stabilization of Nonlinear Systems (Ghent, 1999), Lecture Notes in Control and Inform. Sci., Springer, London, 246 (1999), 325-354. doi: 10.1007/1-84628-577-1_17.  Google Scholar

[36]

E. Trélat, Contrȏle Optimal: Théorie & Applications, Vuibert, Collection "Mathématiques Concrétes", 2005.  Google Scholar

[37]

E. Trélat, Global subanalytic solutions of Hamilton-Jacobi type equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 363-387.  doi: 10.1016/j.anihpc.2005.05.002.  Google Scholar

[38]

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.  Google Scholar

[39]

J. Zhang, K. H. Johansson, J. Lygeros and S. Sastry, Zeno hybrid systems, Internat. J. Robust Nonlinear Control, 11 (2001), 435-451.  doi: 10.1002/rnc.592.  Google Scholar

[40]

J. Zhu, E. Trélat and M. Cerf, Planar tilting maneuver of a spacecraft: Singular arcs in the minimum time problem and chattering, Discrete Cont. Dynam. Syst. Ser. B., 21 (2016), 1347-1388.  doi: 10.3934/dcdsb.2016.21.1347.  Google Scholar

[41]

J. Zhu, E. Trélat and M. Cerf, Minimum time control of the rocket attitude reorientation associated with orbit dynamics, SIAM J. Cont. Optim., 54 (2016), 391-422.  doi: 10.1137/15M1028716.  Google Scholar

Figure 1.  Structure 1-2
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Figure 2.  Structure 1-$\mathcal{H}$-2
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Figure 3.  Going "to the left" is not optimal
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Figure 4.  The trajectory 1-$\mathcal{H}$-2
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Figure 5.  The trajectory 1-2
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