November  2011, 10(6): 1567-1587. doi: 10.3934/cpaa.2011.10.1567

Characterization of the value function of final state constrained control problems with BV trajectories

1. 

Laboratoire de Mathématiques et Physique Théorique, Faculté de sciences et Techniques, Université Francois Rabelais, Parc de Grandmont, 37200 Tours, France

2. 

Equipe Commands, ENSTA ParisTech & INRIA Saclay, 32 Boulevard Victor, 75739 Paris cedex 15, France

Received  January 2010 Revised  May 2011 Published  May 2011

This paper aims to investigate a control problem governed by differential equations with Radon measure as data and with final state constraints. By using a known reparametrization method (by Dal Maso and Rampazzo [18]), we obtain that the value function can be characterized by means of an auxiliary control problem of absolutely continuous trajectories, involving time-measurable Hamiltonian. We study the characterization of the value function of this auxiliary problem and discuss its numerical approximations.
Citation: Ariela Briani, Hasnaa Zidani. Characterization of the value function of final state constrained control problems with BV trajectories. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1567-1587. doi: 10.3934/cpaa.2011.10.1567
References:
[1]

A. Arutyunov, V. Dykhta and L. Lobo Pereira, Necessary conditions for impulsive nonlinear optimal control problems without a priori normality assumptions,, Journal of Optimization Theory and applications, 124 (2005), 55. doi: 10.1007/s10957-004-6465-x. Google Scholar

[2]

M. Bardi and I. Capuzzo-Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations,", Systems and Control: Foundations and Applications, (1997). Google Scholar

[3]

G. Barles, "Solutions de viscosité des équations de Hamilton-Jacobi,", Math\'ematiques et Applications, (1994). Google Scholar

[4]

G. Barles, A new stability result for viscosity solutions of nonlinear parabolic equations with weak convergence in time,, C.R. Acad. Sci. Paris, 343 (2006), 173. Google Scholar

[5]

G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations,, Asymptotic Analysis, 4 (1991), 271. Google Scholar

[6]

E. N. Barron, Viscosity solutions and analysis in $L^\infty$,, Nonlinear analysis, (1998), 1. Google Scholar

[7]

E. N. Barron and R. Jensen, Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians,, Comm. Partial Differential Equations, 15 (1990), 1713. doi: 10.1080/03605309908820745. Google Scholar

[8]

J. Baumeister, On optimal control of a fishery,, In, (2001). Google Scholar

[9]

O. Bokanowski, E. Cristiani, J. Laurent-Varin and H. Zidani, Hamilton-Jacobi-Bellman approach for the climbing problem,, preprint submitted (http://hal.archives-ouvertes.fr/hal-00537649/fr/)., (). Google Scholar

[10]

A. Bressan, On differential systems with impulsive controls,, Rend. Sem. Mat. Univ. Padova, 78 (1987), 227. Google Scholar

[11]

A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls,, Boll. Un. Mat. Ital., 7 (1988), 641. Google Scholar

[12]

A. Bressan and F. Rampazzo, Impulsive control-systems with commutativity assumptions,, Journal of Optimization Theory and Applications, 71 (1991), 67. doi: 10.1007/BF00940040. Google Scholar

[13]

A. Bressan and F. Rampazzo, Impulsive control-systems without commutativity assumptions,, Journal of Optimization Theory and Applications, 81 (1994), 435. doi: 10.1007/BF02193094. Google Scholar

[14]

A. Briani, A Hamilton-Jacobi equation with measures arising in $\Gamma$-convergence of optimal control problems,, Differential and Integral Equations, 12 (1999), 849. Google Scholar

[15]

A. Briani and F. Rampazzo, A density approach to Hamilton-Jacobi equations with t-measurable Hamiltonians,, NoDEA-Nonlinear Differential Equations and Applications, 21 (2005), 71. Google Scholar

[16]

B. Brogliato, "Nonsmooth Impact Mechanics: Models, Dynamics and Control," volume 220 of Lecture Notes in Control and Information Sciences,, Springer Verlag, (1996). Google Scholar

[17]

C. Clark, F. Clarke and G. Munro, The optimal exploitation of renewable stocks,, Econometrica, 47 (1979), 25. doi: 10.2307/1912344. Google Scholar

[18]

G. Dal Maso and F. Rampazzo, On systems of ordinary differential equations with measures as controls,, Differential and Integral Equations, 4 (1991), 738. Google Scholar

[19]

V. Dykhta and O. N. Samsonyuk, "Optimal Impulse Control with Applications,", Nauka, (1991). Google Scholar

[20]

H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations,, SIAM Journal on Control and Optimization, 31 (1993), 257. doi: 10.1137/0331016. Google Scholar

[21]

H. Frankowska and S. Plaskacz, A measurable upper semicontinuous viability theorem for tubes,, J. of Nonlinear Analysis, 26 (1996), 565. doi: 10.1016/0362-546X(94)00299-W. Google Scholar

[22]

H. Frankowska, S. Plaskacz and T. Rzeuchowski, Measurable viability theorems and Hamilton-Jacobi-Bellman equation,, J. Diff. Eqs, 116 (1995), 265. doi: 10.1006/jdeq.1995.1036. Google Scholar

[23]

P. Gajardo, C. Ramírez and A. Rappaport, Minimal time sequential batch reactors with bounded and impulse controls for one or more species,, SIAM J. Control and Optim., 47 (2008), 2827. doi: 10.1137/070695204. Google Scholar

[24]

H. Ishii, Hamilton-Jacobi equations with discontinuous Hamiltonians on arbitrary open sets,, Bull. Facul. Sci. & Eng., 28 (1985), 33. Google Scholar

[25]

P. L. Lions and B. Perthame, Remarks on Hamilton-Jacobi equations with measurable time-dependent Hamiltonians,, Nonlinear analysis, 11 (1987), 613. Google Scholar

[26]

B. M. Miller, Optimization of dynamic systems with a generalized control,, Automation and Remote Control, 50 (1989). Google Scholar

[27]

A. Monteillet, Convergence of approximation schemes for nonolocal front propagation equations,, Mathematics of Computation, 79 (2010), 125. doi: 10.1090/S0025-5718-09-02270-4. Google Scholar

[28]

D. Nunziante, Uniqueness of viscosity solutions of fully nonlinear second order parabolic equations with discontinuous time dependence,, Differential Integral Equations, 3 (1990), 77. Google Scholar

[29]

D. Nunziante, Existence and uniqueness of unbounded viscosity solutions of parabolic equations with discontinuous time-dependence,, Nonlinear Anal., 18 (1992), 1033. doi: 10.1016/0362-546X(92)90194-J. Google Scholar

[30]

J.-P. Raymond, Optimal control problems in spaces of functions of bounded variation,, Differential Integral Equations, 10 (1997), 105. Google Scholar

[31]

A. V. Sarychev, Nonlinear systems with impulsive and generalised functions controls,, Proc. Conf. on NONlinear Synthesis, (1989). Google Scholar

[32]

H. J. Sussmann, On the gap between deterministic and stochastic ordinary differential equations,, Ann. of Probability, 6 (1978), 17. doi: 10.1214/aop/1176995608. Google Scholar

show all references

References:
[1]

A. Arutyunov, V. Dykhta and L. Lobo Pereira, Necessary conditions for impulsive nonlinear optimal control problems without a priori normality assumptions,, Journal of Optimization Theory and applications, 124 (2005), 55. doi: 10.1007/s10957-004-6465-x. Google Scholar

[2]

M. Bardi and I. Capuzzo-Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations,", Systems and Control: Foundations and Applications, (1997). Google Scholar

[3]

G. Barles, "Solutions de viscosité des équations de Hamilton-Jacobi,", Math\'ematiques et Applications, (1994). Google Scholar

[4]

G. Barles, A new stability result for viscosity solutions of nonlinear parabolic equations with weak convergence in time,, C.R. Acad. Sci. Paris, 343 (2006), 173. Google Scholar

[5]

G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations,, Asymptotic Analysis, 4 (1991), 271. Google Scholar

[6]

E. N. Barron, Viscosity solutions and analysis in $L^\infty$,, Nonlinear analysis, (1998), 1. Google Scholar

[7]

E. N. Barron and R. Jensen, Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians,, Comm. Partial Differential Equations, 15 (1990), 1713. doi: 10.1080/03605309908820745. Google Scholar

[8]

J. Baumeister, On optimal control of a fishery,, In, (2001). Google Scholar

[9]

O. Bokanowski, E. Cristiani, J. Laurent-Varin and H. Zidani, Hamilton-Jacobi-Bellman approach for the climbing problem,, preprint submitted (http://hal.archives-ouvertes.fr/hal-00537649/fr/)., (). Google Scholar

[10]

A. Bressan, On differential systems with impulsive controls,, Rend. Sem. Mat. Univ. Padova, 78 (1987), 227. Google Scholar

[11]

A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls,, Boll. Un. Mat. Ital., 7 (1988), 641. Google Scholar

[12]

A. Bressan and F. Rampazzo, Impulsive control-systems with commutativity assumptions,, Journal of Optimization Theory and Applications, 71 (1991), 67. doi: 10.1007/BF00940040. Google Scholar

[13]

A. Bressan and F. Rampazzo, Impulsive control-systems without commutativity assumptions,, Journal of Optimization Theory and Applications, 81 (1994), 435. doi: 10.1007/BF02193094. Google Scholar

[14]

A. Briani, A Hamilton-Jacobi equation with measures arising in $\Gamma$-convergence of optimal control problems,, Differential and Integral Equations, 12 (1999), 849. Google Scholar

[15]

A. Briani and F. Rampazzo, A density approach to Hamilton-Jacobi equations with t-measurable Hamiltonians,, NoDEA-Nonlinear Differential Equations and Applications, 21 (2005), 71. Google Scholar

[16]

B. Brogliato, "Nonsmooth Impact Mechanics: Models, Dynamics and Control," volume 220 of Lecture Notes in Control and Information Sciences,, Springer Verlag, (1996). Google Scholar

[17]

C. Clark, F. Clarke and G. Munro, The optimal exploitation of renewable stocks,, Econometrica, 47 (1979), 25. doi: 10.2307/1912344. Google Scholar

[18]

G. Dal Maso and F. Rampazzo, On systems of ordinary differential equations with measures as controls,, Differential and Integral Equations, 4 (1991), 738. Google Scholar

[19]

V. Dykhta and O. N. Samsonyuk, "Optimal Impulse Control with Applications,", Nauka, (1991). Google Scholar

[20]

H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations,, SIAM Journal on Control and Optimization, 31 (1993), 257. doi: 10.1137/0331016. Google Scholar

[21]

H. Frankowska and S. Plaskacz, A measurable upper semicontinuous viability theorem for tubes,, J. of Nonlinear Analysis, 26 (1996), 565. doi: 10.1016/0362-546X(94)00299-W. Google Scholar

[22]

H. Frankowska, S. Plaskacz and T. Rzeuchowski, Measurable viability theorems and Hamilton-Jacobi-Bellman equation,, J. Diff. Eqs, 116 (1995), 265. doi: 10.1006/jdeq.1995.1036. Google Scholar

[23]

P. Gajardo, C. Ramírez and A. Rappaport, Minimal time sequential batch reactors with bounded and impulse controls for one or more species,, SIAM J. Control and Optim., 47 (2008), 2827. doi: 10.1137/070695204. Google Scholar

[24]

H. Ishii, Hamilton-Jacobi equations with discontinuous Hamiltonians on arbitrary open sets,, Bull. Facul. Sci. & Eng., 28 (1985), 33. Google Scholar

[25]

P. L. Lions and B. Perthame, Remarks on Hamilton-Jacobi equations with measurable time-dependent Hamiltonians,, Nonlinear analysis, 11 (1987), 613. Google Scholar

[26]

B. M. Miller, Optimization of dynamic systems with a generalized control,, Automation and Remote Control, 50 (1989). Google Scholar

[27]

A. Monteillet, Convergence of approximation schemes for nonolocal front propagation equations,, Mathematics of Computation, 79 (2010), 125. doi: 10.1090/S0025-5718-09-02270-4. Google Scholar

[28]

D. Nunziante, Uniqueness of viscosity solutions of fully nonlinear second order parabolic equations with discontinuous time dependence,, Differential Integral Equations, 3 (1990), 77. Google Scholar

[29]

D. Nunziante, Existence and uniqueness of unbounded viscosity solutions of parabolic equations with discontinuous time-dependence,, Nonlinear Anal., 18 (1992), 1033. doi: 10.1016/0362-546X(92)90194-J. Google Scholar

[30]

J.-P. Raymond, Optimal control problems in spaces of functions of bounded variation,, Differential Integral Equations, 10 (1997), 105. Google Scholar

[31]

A. V. Sarychev, Nonlinear systems with impulsive and generalised functions controls,, Proc. Conf. on NONlinear Synthesis, (1989). Google Scholar

[32]

H. J. Sussmann, On the gap between deterministic and stochastic ordinary differential equations,, Ann. of Probability, 6 (1978), 17. doi: 10.1214/aop/1176995608. Google Scholar

[1]

Fabio Camilli, Paola Loreti, Naoki Yamada. Systems of convex Hamilton-Jacobi equations with implicit obstacles and the obstacle problem. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1291-1302. doi: 10.3934/cpaa.2009.8.1291

[2]

Xifeng Su, Lin Wang, Jun Yan. Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6487-6522. doi: 10.3934/dcds.2016080

[3]

Martino Bardi, Yoshikazu Giga. Right accessibility of semicontinuous initial data for Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2003, 2 (4) : 447-459. doi: 10.3934/cpaa.2003.2.447

[4]

Larry M. Bates, Francesco Fassò, Nicola Sansonetto. The Hamilton-Jacobi equation, integrability, and nonholonomic systems. Journal of Geometric Mechanics, 2014, 6 (4) : 441-449. doi: 10.3934/jgm.2014.6.441

[5]

Claudio Marchi. On the convergence of singular perturbations of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1363-1377. doi: 10.3934/cpaa.2010.9.1363

[6]

Isabeau Birindelli, J. Wigniolle. Homogenization of Hamilton-Jacobi equations in the Heisenberg group. Communications on Pure & Applied Analysis, 2003, 2 (4) : 461-479. doi: 10.3934/cpaa.2003.2.461

[7]

Xia Li. Long-time asymptotic solutions of convex hamilton-jacobi equations depending on unknown functions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5151-5162. doi: 10.3934/dcds.2017223

[8]

Kai Zhao, Wei Cheng. On the vanishing contact structure for viscosity solutions of contact type Hamilton-Jacobi equations I: Cauchy problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4345-4358. doi: 10.3934/dcds.2019176

[9]

Laura Caravenna, Annalisa Cesaroni, Hung Vinh Tran. Preface: Recent developments related to conservation laws and Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : ⅰ-ⅲ. doi: 10.3934/dcdss.201805i

[10]

Yasuhiro Fujita, Katsushi Ohmori. Inequalities and the Aubry-Mather theory of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2009, 8 (2) : 683-688. doi: 10.3934/cpaa.2009.8.683

[11]

Olga Bernardi, Franco Cardin. On $C^0$-variational solutions for Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 385-406. doi: 10.3934/dcds.2011.31.385

[12]

Emeric Bouin. A Hamilton-Jacobi approach for front propagation in kinetic equations. Kinetic & Related Models, 2015, 8 (2) : 255-280. doi: 10.3934/krm.2015.8.255

[13]

Gawtum Namah, Mohammed Sbihi. A notion of extremal solutions for time periodic Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 647-664. doi: 10.3934/dcdsb.2010.13.647

[14]

Antonio Avantaggiati, Paola Loreti, Cristina Pocci. Mixed norms, functional Inequalities, and Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1855-1867. doi: 10.3934/dcdsb.2014.19.1855

[15]

Gui-Qiang Chen, Bo Su. Discontinuous solutions for Hamilton-Jacobi equations: Uniqueness and regularity. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 167-192. doi: 10.3934/dcds.2003.9.167

[16]

David McCaffrey. A representational formula for variational solutions to Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1205-1215. doi: 10.3934/cpaa.2012.11.1205

[17]

Mihai Bostan, Gawtum Namah. Time periodic viscosity solutions of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2007, 6 (2) : 389-410. doi: 10.3934/cpaa.2007.6.389

[18]

Piermarco Cannarsa, Marco Mazzola, Carlo Sinestrari. Global propagation of singularities for time dependent Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4225-4239. doi: 10.3934/dcds.2015.35.4225

[19]

Olga Bernardi, Franco Cardin. Minimax and viscosity solutions of Hamilton-Jacobi equations in the convex case. Communications on Pure & Applied Analysis, 2006, 5 (4) : 793-812. doi: 10.3934/cpaa.2006.5.793

[20]

Kaizhi Wang, Jun Yan. Lipschitz dependence of viscosity solutions of Hamilton-Jacobi equations with respect to the parameter. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1649-1659. doi: 10.3934/dcds.2016.36.1649

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]