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 and 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), 55-77. doi: 10.1007/s10957-004-6465-x.

[2]

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

[3]

G. Barles, "Solutions de viscosité des équations de Hamilton-Jacobi," Mathématiques et Applications, vol 17, Springer, Paris, 1994.

[4]

G. Barles, A new stability result for viscosity solutions of nonlinear parabolic equations with weak convergence in time, C.R. Acad. Sci. Paris, Ser. I, 343 (2006), 173-178.

[5]

G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Analysis, 4 (1991), 271-283.

[6]

E. N. Barron, Viscosity solutions and analysis in $L^\infty$, Nonlinear analysis, differential equations and control (Montreal, QC, 1998), 1-60, NATO Sci. Ser. C Math. Phys. Sci., 528, Kluwer Acad. Publ., Dordrecht, 1999.

[7]

E. N. Barron and R. Jensen, Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians, Comm. Partial Differential Equations, 15 (1990), 1713-1742. doi: 10.1080/03605309908820745.

[8]

J. Baumeister, On optimal control of a fishery, In "Proceedings of NOLCOS'01, volume 5th IFAC Symposium on Nonlinear Control Systems," St Petersburg, Russia, 2001.

[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/)., (). 

[10]

A. Bressan, On differential systems with impulsive controls, Rend. Sem. Mat. Univ. Padova, 78 (1987), 227-236.

[11]

A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Mat. Ital., 7 (1988), 641-656.

[12]

A. Bressan and F. Rampazzo, Impulsive control-systems with commutativity assumptions, Journal of Optimization Theory and Applications, 71 (1991), 67-83. doi: 10.1007/BF00940040.

[13]

A. Bressan and F. Rampazzo, Impulsive control-systems without commutativity assumptions, Journal of Optimization Theory and Applications, 81 (1994), 435-457. doi: 10.1007/BF02193094.

[14]

A. Briani, A Hamilton-Jacobi equation with measures arising in $\Gamma$-convergence of optimal control problems, Differential and Integral Equations, 12 (1999), 849-886.

[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), 71-92.

[16]

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

[17]

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

[18]

G. Dal Maso and F. Rampazzo, On systems of ordinary differential equations with measures as controls, Differential and Integral Equations, 4 (1991), 738-765.

[19]

V. Dykhta and O. N. Samsonyuk, "Optimal Impulse Control with Applications," Nauka, Moscow, Russia, 1991.

[20]

H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations, SIAM Journal on Control and Optimization, 31 (1993), 257-272. doi: 10.1137/0331016.

[21]

H. Frankowska and S. Plaskacz, A measurable upper semicontinuous viability theorem for tubes, J. of Nonlinear Analysis, TMA, 26 (1996), 565-582. doi: 10.1016/0362-546X(94)00299-W.

[22]

H. Frankowska, S. Plaskacz and T. Rzeuchowski, Measurable viability theorems and Hamilton-Jacobi-Bellman equation, J. Diff. Eqs, 116 (1995), 265-305. doi: 10.1006/jdeq.1995.1036.

[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), 2827-2856. doi: 10.1137/070695204.

[24]

H. Ishii, Hamilton-Jacobi equations with discontinuous Hamiltonians on arbitrary open sets, Bull. Facul. Sci. & Eng., 28 (1985), 33-77.

[25]

P. L. Lions and B. Perthame, Remarks on Hamilton-Jacobi equations with measurable time-dependent Hamiltonians, Nonlinear analysis, Theory, Methods & Applications, 11 (1987), 613-612.

[26]

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

[27]

A. Monteillet, Convergence of approximation schemes for nonolocal front propagation equations, Mathematics of Computation, 79 (2010), 125-146. doi: 10.1090/S0025-5718-09-02270-4.

[28]

D. Nunziante, Uniqueness of viscosity solutions of fully nonlinear second order parabolic equations with discontinuous time dependence, Differential Integral Equations, 3 (1990), 77-91.

[29]

D. Nunziante, Existence and uniqueness of unbounded viscosity solutions of parabolic equations with discontinuous time-dependence, Nonlinear Anal., 18 (1992), 1033-1062. doi: 10.1016/0362-546X(92)90194-J.

[30]

J.-P. Raymond, Optimal control problems in spaces of functions of bounded variation, Differential Integral Equations, 10 (1997), 105-136.

[31]

A. V. Sarychev, Nonlinear systems with impulsive and generalised functions controls, Proc. Conf. on NONlinear Synthesis, Sopron, Hungary, 1989.

[32]

H. J. Sussmann, On the gap between deterministic and stochastic ordinary differential equations, Ann. of Probability, 6 (1978), 17-41. doi: 10.1214/aop/1176995608.

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), 55-77. doi: 10.1007/s10957-004-6465-x.

[2]

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

[3]

G. Barles, "Solutions de viscosité des équations de Hamilton-Jacobi," Mathématiques et Applications, vol 17, Springer, Paris, 1994.

[4]

G. Barles, A new stability result for viscosity solutions of nonlinear parabolic equations with weak convergence in time, C.R. Acad. Sci. Paris, Ser. I, 343 (2006), 173-178.

[5]

G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Analysis, 4 (1991), 271-283.

[6]

E. N. Barron, Viscosity solutions and analysis in $L^\infty$, Nonlinear analysis, differential equations and control (Montreal, QC, 1998), 1-60, NATO Sci. Ser. C Math. Phys. Sci., 528, Kluwer Acad. Publ., Dordrecht, 1999.

[7]

E. N. Barron and R. Jensen, Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians, Comm. Partial Differential Equations, 15 (1990), 1713-1742. doi: 10.1080/03605309908820745.

[8]

J. Baumeister, On optimal control of a fishery, In "Proceedings of NOLCOS'01, volume 5th IFAC Symposium on Nonlinear Control Systems," St Petersburg, Russia, 2001.

[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/)., (). 

[10]

A. Bressan, On differential systems with impulsive controls, Rend. Sem. Mat. Univ. Padova, 78 (1987), 227-236.

[11]

A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Mat. Ital., 7 (1988), 641-656.

[12]

A. Bressan and F. Rampazzo, Impulsive control-systems with commutativity assumptions, Journal of Optimization Theory and Applications, 71 (1991), 67-83. doi: 10.1007/BF00940040.

[13]

A. Bressan and F. Rampazzo, Impulsive control-systems without commutativity assumptions, Journal of Optimization Theory and Applications, 81 (1994), 435-457. doi: 10.1007/BF02193094.

[14]

A. Briani, A Hamilton-Jacobi equation with measures arising in $\Gamma$-convergence of optimal control problems, Differential and Integral Equations, 12 (1999), 849-886.

[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), 71-92.

[16]

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

[17]

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

[18]

G. Dal Maso and F. Rampazzo, On systems of ordinary differential equations with measures as controls, Differential and Integral Equations, 4 (1991), 738-765.

[19]

V. Dykhta and O. N. Samsonyuk, "Optimal Impulse Control with Applications," Nauka, Moscow, Russia, 1991.

[20]

H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations, SIAM Journal on Control and Optimization, 31 (1993), 257-272. doi: 10.1137/0331016.

[21]

H. Frankowska and S. Plaskacz, A measurable upper semicontinuous viability theorem for tubes, J. of Nonlinear Analysis, TMA, 26 (1996), 565-582. doi: 10.1016/0362-546X(94)00299-W.

[22]

H. Frankowska, S. Plaskacz and T. Rzeuchowski, Measurable viability theorems and Hamilton-Jacobi-Bellman equation, J. Diff. Eqs, 116 (1995), 265-305. doi: 10.1006/jdeq.1995.1036.

[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), 2827-2856. doi: 10.1137/070695204.

[24]

H. Ishii, Hamilton-Jacobi equations with discontinuous Hamiltonians on arbitrary open sets, Bull. Facul. Sci. & Eng., 28 (1985), 33-77.

[25]

P. L. Lions and B. Perthame, Remarks on Hamilton-Jacobi equations with measurable time-dependent Hamiltonians, Nonlinear analysis, Theory, Methods & Applications, 11 (1987), 613-612.

[26]

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

[27]

A. Monteillet, Convergence of approximation schemes for nonolocal front propagation equations, Mathematics of Computation, 79 (2010), 125-146. doi: 10.1090/S0025-5718-09-02270-4.

[28]

D. Nunziante, Uniqueness of viscosity solutions of fully nonlinear second order parabolic equations with discontinuous time dependence, Differential Integral Equations, 3 (1990), 77-91.

[29]

D. Nunziante, Existence and uniqueness of unbounded viscosity solutions of parabolic equations with discontinuous time-dependence, Nonlinear Anal., 18 (1992), 1033-1062. doi: 10.1016/0362-546X(92)90194-J.

[30]

J.-P. Raymond, Optimal control problems in spaces of functions of bounded variation, Differential Integral Equations, 10 (1997), 105-136.

[31]

A. V. Sarychev, Nonlinear systems with impulsive and generalised functions controls, Proc. Conf. on NONlinear Synthesis, Sopron, Hungary, 1989.

[32]

H. J. Sussmann, On the gap between deterministic and stochastic ordinary differential equations, Ann. of Probability, 6 (1978), 17-41. doi: 10.1214/aop/1176995608.

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