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On some optimal control problems governed by a state equation with memory

Published online by Cambridge University Press:  18 January 2008

Guillaume Carlier  and
Rabah Tahraoui
Guillaume Carlier
Affiliation:
Université Paris Dauphine, CEREMADE, Pl. de Lattre de Tassigny, 75775 Paris Cedex 16, France; carlier@ceremade.dauphine.fr; tahraoui@ceremade.dauphine.fr
Rabah Tahraoui
Affiliation:
Université Paris Dauphine, CEREMADE, Pl. de Lattre de Tassigny, 75775 Paris Cedex 16, France; carlier@ceremade.dauphine.fr; tahraoui@ceremade.dauphine.fr
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Abstract

The aim of this paper is to study problems of the form: $inf_{(u\in V)}J(u)$ with $J(u):=\int_0^1 L(s,y_u(s),u(s)){\rm d}s+g(y_u(1))$ where V is a set of admissible controls and y u is the solution of the Cauchy problem: $\dot{x}(t) =\langle f(.,x(.)), \nu_t \rangle + u(t), t \in (0,1)$ , $x(0) = x_{\rm 0}$ and each $\nu_t$ is a nonnegative measure with support in [0,t]. After studying the Cauchy problem, we establish existence of minimizers, optimality conditions (in particular in the form of a nonlocal version of the Pontryagin principle) and prove some regularity results. We also consider the more general case where the control also enters the dynamics in a nonlocal way.

Keywords

Optimal controlmemory

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 4 , October 2008 , pp. 725 - 743
Copyright
© EDP Sciences, SMAI, 2008

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References

L. Ambrosio, Lecture Notes on Optimal Transport Problems, Mathematical aspects of evolving interfaces, CIME Summer School in Madeira 1812. Springer (2003).
R. Bellman and K.L. Cooke, Differential-difference equations, Mathematics in Science and Engineering. Academic Press, New York-London (1963).
Boucekkine, R., Licandro, O., Puch, L. and del Rio, F., Vintage capital and the dynamics of the AK model. J. Economic Theory 120 (2005) 39–72. CrossRef
P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations and Optimal Control. Birkhäuser (2004).
C. Dellacherie and P.-A. Meyer, Probabilities and Potential, Mathematical Studies 29. North-Holland (1978).
Drakhlin, M.E. and Stepanov, E., On weak lower-semi continuity for a class of functionals with deviating arguments. Nonlinear Anal. TMA 28 (1997) 2005–2015. CrossRef
I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Classics in Mathematics. Society for Industrial and Applied Mathematics, Philadelphia (1999).
Elsanosi, I., Øksendal, B. and Sulem, A., Some solvable stochastic control problems with delay. Stoch. Stoch. Rep. 71 (2000) 69–89. CrossRef
L. El'sgol'ts, Introduction to the Theory of Differential Equations with Deviating Arguments. Holden-Day, San Francisco (1966).
F. Gozzi and C. Marinelli, Stochastic optimal control of delay equations arising in advertising models, in Stochastic partial differential equations and applications VII, Chapman & Hall, Boca Raton, Lect. Notes Pure Appl. Math. 245 (2006) 133–148.
Jouini, E., Koehl, P.-F. and Touzi, N., Optimal investment with taxes: an optimal control problem with endogenous delay. Nonlinear Anal. Theory Methods Appl. 37 (1999) 31–56. CrossRef
Jouini, E., Koehl, P.-F. and Touzi, N., Optimal investment with taxes: an existence result. J. Math. Econom. 33 (2000) 373–388. CrossRef
M.N. Oguztöreli, Time-Lag Control Systems. Academic Press, New-York (1966).
Ramsey, F.P., A mathematical theory of saving. Economic J. 38 (1928) 543–559. CrossRef
L. Samassi, Calcul des variations des fonctionelles à arguments déviés. Ph.D. thesis, University of Paris Dauphine, France (2004).
Samassi, L. and Tahraoui, R., Comment établir des conditions nécessaires d'optimalité dans les problèmes de contrôle dont certains arguments sont déviés? C. R. Math. Acad. Sci. Paris 338 (2004) 611–616. CrossRef
L. Samassi and R. Tahraoui, How to state necessary optimality conditions for control problems with deviating arguments? ESAIM: COCV (2007) e-first, doi: 10.1051/cocv:2007058.