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

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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

Abstract
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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 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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