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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

Ariela Briani 1, and  Hasnaa Zidani 2,

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.
Keywords: differential systems with measures as data, measurable functions, Optimal control problem, Hamilton-Jacobi equations..
Mathematics Subject Classification: Primary: 49J15, 35F21; Secondary: 34A3.
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), 55-77. 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, Birkhäuser, Boston, 1997.  Google Scholar

[3]

G. Barles, "Solutions de viscosité des équations de Hamilton-Jacobi," Mathématiques et Applications, vol 17, Springer, Paris, 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, Ser. I, 343 (2006), 173-178.  Google Scholar

[5]

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

[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.  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), 1713-1742. doi: 10.1080/03605309908820745.  Google Scholar

[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. 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), 227-236.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  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), 849-886.  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), 71-92.  Google Scholar

[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.  Google Scholar

[17]

C. Clark, F. Clarke and G. Munro, The optimal exploitation of renewable stocks, Econometrica, 47 (1979), 25-47. 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), 738-765.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  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), 2827-2856. 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), 33-77.  Google Scholar

[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.  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), 125-146. 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), 77-91.  Google Scholar

[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.  Google Scholar

[30]

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

[31]

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

[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.  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), 55-77. 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, Birkhäuser, Boston, 1997.  Google Scholar

[3]

G. Barles, "Solutions de viscosité des équations de Hamilton-Jacobi," Mathématiques et Applications, vol 17, Springer, Paris, 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, Ser. I, 343 (2006), 173-178.  Google Scholar

[5]

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

[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.  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), 1713-1742. doi: 10.1080/03605309908820745.  Google Scholar

[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. 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), 227-236.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  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), 849-886.  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), 71-92.  Google Scholar

[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.  Google Scholar

[17]

C. Clark, F. Clarke and G. Munro, The optimal exploitation of renewable stocks, Econometrica, 47 (1979), 25-47. 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), 738-765.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  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), 2827-2856. 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), 33-77.  Google Scholar

[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.  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), 125-146. 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), 77-91.  Google Scholar

[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.  Google Scholar

[30]

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

[31]

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

[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.  Google Scholar

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