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Approximation of controlproblems involving ordinary and impulsive controls

Published online by Cambridge University Press:  15 August 2002

Fabio Camilli  and
Maurizio Falcone
Fabio Camilli
Affiliation:
Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy; [email protected].
Maurizio Falcone
Affiliation:
Dipartimento di Matematica, Università di Roma “La Sapienza", P.le Aldo Moro 2, 00185 Roma, Italy; [email protected].
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Abstract

In this paper we study an approximation scheme for a class of control problems involving an ordinary control v, an impulsive control u and its derivative $\dot u$. Adopting a space-time reparametrization of the problem which adds one variable to the state space we overcome some difficulties connected to the presence of $\dot u$. We construct an approximation scheme for that augmented system, prove that it converges to the value function of the augmented problem and establish an error estimates in L∞ for this approximation. Moreover, a characterization of the limit of the discrete optimal controls is given showing that it converges (in a suitable sense) to an optimal control for the continuous problem.

Keywords

Impulsive controlapproximation schemedynamic programmingviscosity solution
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 4 , 1999 , pp. 159 - 176
Copyright
© EDP Sciences, SMAI, 1999

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References

M. Bardi and I. Capuzzo Dolcetta, Viscosity solutions of Bellman equation and optimal deterministic control theory. Birkhäuser, Boston (1997).
Bardi, M. and Falcone, M., An approximation scheme for the minimum time function. SIAM J. Control Optim. 28 (1990) 950-965. CrossRef
Barles, G., Deterministic Impulse control problems. SIAM J. Control Optim. 23 (1985) 419-432. CrossRef
Barles, G. and Souganidis, P., Convergence of approximation scheme for fully nonlinear second order equations. Asymptotic Anal. 4 (1991) 271-283.
E. Barron, R. Jensen and J.L. Menaldi, Optimal control and differential games with measures. Nonlinear Anal. TMA 21 (1993) 241-268.
A. Bensoussan and J.L. Lions, Impulse control and quasi-variational inequalities. Gauthier-Villars, Paris (1984).
Aldo Bressan, Hyperimpulsive motions and controllizable coordinates for Lagrangean systems. Atti Accad. Naz. Lincei, Mem Cl. Sc. Fis. Mat. Natur. 19 (1991).
Bressan, A. and Rampazzo, F., Impulsive control systems with commutative vector fields. J. Optim. Th. & Appl. 71 (1991) 67-83. CrossRef
F. Camilli and M. Falcone, Approximation of optimal control problems with state constraints: estimates and applications, in Nonsmooth analysis and geometric methods in deterministic optimal control (Minneapolis, MN, 1993) Springer, New York (1996) 23-57.
Capuzzo Dolcetta, I. and Falcone, M., Discrete dynamic programming and viscosity solutions of the Bellman equation. Ann. Inst. H.Poincaré Anal. Nonlin. 6 (1989) 161-184. CrossRef
Capuzzo Dolcetta, I. and Ishii, H., Approximate solutions of Bellman equation of deterministic control theory. Appl. Math. Optim. 11 (1984) 161-181. CrossRef
Crandall, M.G., Evans, L.C. and Lions, P.L., Some properties of viscosity solutions of Hamilton-Jacobi equation. Trans. Amer. Math. Soc. 282 (1984) 487-502. CrossRef
Clark, C.W., Clarke, F.H. and Munro, G.R., The optimal exploitation of renewable resource stocks. Econometrica 48 (1979) 25-47. CrossRef
Dorroh, J.R. and Ferreyra, G., Optimal advertising in exponentially decaying markets. J. Optim. Th. & Appl. 79 (1993) 219-236. CrossRef
_____, A multistate multicontrol problem with unbounded controls. SIAM J. Control Optim. 32 (1994) 1322-1331. CrossRef
Falcone, M., A numerical approach to the infinite horizon problem. Appl. Math. & Optim. 15 (1987) 1-13 and 23 (1991) 213-214. CrossRef
M. Falcone, Numerical solution of Dynamic Programming equations, Appendix to M. Bardi and I. Capuzzo Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser, Boston (1997).
W. Fleming and H.M. Soner, Controlled Markov processes and viscosity solutions. Springer-Verlag (1992).
H. Kushner and P. Dupuis, Numerical methods for stochastic control problems in continuous time. Springer-Verlag (1992).
J.P. Marec, Optimal space trajectories. Elsevier (1979).
B.M. Miller, Generalized solutions of nonlinear optimization problems with impulse control I, II. Automat. Remote Control 55 (1995).
, Dynamic, programming for nonlinear systems driven by ordinary and impulsive controls. SIAM J. Control Optim. 34 (1996) 199-225.
Motta, M. and Rampazzo, F., Space-time trajectories of nonlinear system driven by ordinary and impulsive controls. Differential and Integral Equations 8 (1995) 269-288.
F. Rampazzo, On the Riemannian Structure of a Lagrangian system and the problem of adding time-dependent constraints as controls. Eur. J. Mech. A/Solids 10 (1991) 405-431.
Rouy, E., Numerical approximation of viscosity solutions of first-order Hamilton-Jacobi equations with Neumann type boundary conditions. Math. Meth. Appl. Sci. 2 (1992) 357-374. CrossRef
P. Souganidis, Approximation schemes for viscosity solutions of Hamilton-Jacobi equations. J. Diff. Eq. 57 1-43.