On some optimal control problems governed by a state equation with memory
Guillaume Carlier; Rabah Tahraoui
ESAIM: Control, Optimisation and Calculus of Variations (2008)
- Volume: 14, Issue: 4, page 725-743
- ISSN: 1292-8119
Access Full Article
topAbstract
topHow to cite
topCarlier, Guillaume, and Tahraoui, Rabah. "On some optimal control problems governed by a state equation with memory." ESAIM: Control, Optimisation and Calculus of Variations 14.4 (2008): 725-743. <http://eudml.org/doc/250320>.
@article{Carlier2008,
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 yu 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.
},
author = {Carlier, Guillaume, Tahraoui, Rabah},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Optimal control; memory; optimal control},
language = {eng},
month = {1},
number = {4},
pages = {725-743},
publisher = {EDP Sciences},
title = {On some optimal control problems governed by a state equation with memory},
url = {http://eudml.org/doc/250320},
volume = {14},
year = {2008},
}
TY - JOUR
AU - Carlier, Guillaume
AU - Tahraoui, Rabah
TI - On some optimal control problems governed by a state equation with memory
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2008/1//
PB - EDP Sciences
VL - 14
IS - 4
SP - 725
EP - 743
AB -
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 yu 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.
LA - eng
KW - Optimal control; memory; optimal control
UR - http://eudml.org/doc/250320
ER -
References
top- L. Ambrosio, Lecture Notes on Optimal Transport Problems, Mathematical aspects of evolving interfaces, CIME Summer School in Madeira1812. Springer (2003).
- R. Bellman and K.L. Cooke, Differential-difference equations, Mathematics in Science and Engineering. Academic Press, New York-London (1963).
- R. Boucekkine, O. Licandro, L. Puch and F. del Rio, Vintage capital and the dynamics of the AK model. J. Economic Theory120 (2005) 39–72.
- 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 Studies29. North-Holland (1978).
- M.E. Drakhlin and E. Stepanov, On weak lower-semi continuity for a class of functionals with deviating arguments. Nonlinear Anal. TMA28 (1997) 2005–2015.
- I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Classics in Mathematics. Society for Industrial and Applied Mathematics, Philadelphia (1999).
- I. Elsanosi, B. Øksendal and A. Sulem, Some solvable stochastic control problems with delay. Stoch. Stoch. Rep.71 (2000) 69–89.
- 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 applicationsVII, Chapman & Hall, Boca Raton, Lect. Notes Pure Appl. Math.245 (2006) 133–148.
- E. Jouini, P.-F. Koehl and N. Touzi, Optimal investment with taxes: an optimal control problem with endogenous delay. Nonlinear Anal. Theory Methods Appl.37 (1999) 31–56.
- E. Jouini, P.-F. Koehl and N. Touzi, Optimal investment with taxes: an existence result. J. Math. Econom.33 (2000) 373–388.
- M.N. Oguztöreli, Time-Lag Control Systems. Academic Press, New-York (1966).
- F.P. Ramsey, A mathematical theory of saving. Economic J.38 (1928) 543–559.
- L. Samassi, Calcul des variations des fonctionelles à arguments déviés. Ph.D. thesis, University of Paris Dauphine, France (2004).
- L. Samassi and R. Tahraoui, Comment établir des conditions nécessaires d'optimalité dans les problèmes de contrôle dont certains arguments sont déviés? C. R. Math. Acad. Sci. Paris338 (2004) 611–616.
- L. Samassi and R. Tahraoui, How to state necessary optimality conditions for control problems with deviating arguments? ESAIM: COCV (2007) e-first, doi: . DOI10.1051/cocv:2007058
Citations in EuDML Documents
topNotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.