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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

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.

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.

[2]

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

[3]

A. D. AmesA. 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.

[4]

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

[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.

[6]

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

[7]

G. BarlesA. 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.

[8]

G. BarlesA. 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.

[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.

[10]

M. S. BranickyV. 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.

[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.

[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.

[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.

[14]

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

[15]

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

[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.

[17]

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

[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.

[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.

[20]

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

[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.

[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.

[23]

M. HeymannF. LinG. 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.

[24]

C. ImbertR. 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.

[25]

K. H. JohanssonM. EgerstedtJ. 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.

[26]

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

[27]

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

[28]

Z. RaoA. 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.

[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.

[30]

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

[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.

[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.

[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.

[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.

[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.

[36]

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

[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.

[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.

[39]

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

[40]

J. ZhuE. 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.

[41]

J. ZhuE. 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.

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.

[2]

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

[3]

A. D. AmesA. 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.

[4]

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

[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.

[6]

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

[7]

G. BarlesA. 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.

[8]

G. BarlesA. 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.

[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.

[10]

M. S. BranickyV. 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.

[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.

[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.

[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.

[14]

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

[15]

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

[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.

[17]

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

[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.

[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.

[20]

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

[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.

[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.

[23]

M. HeymannF. LinG. 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.

[24]

C. ImbertR. 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.

[25]

K. H. JohanssonM. EgerstedtJ. 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.

[26]

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

[27]

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

[28]

Z. RaoA. 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.

[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.

[30]

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

[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.

[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.

[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.

[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.

[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.

[36]

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

[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.

[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.

[39]

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

[40]

J. ZhuE. 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.

[41]

J. ZhuE. 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.

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