The aim of this paper is to study problems of the form: with where is a set of admissible controls and is the solution of the Cauchy problem: , and each is a nonnegative measure with support in . 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.
@article{COCV_2008__14_4_725_0, author = {Carlier, Guillaume and Tahraoui, Rabah}, title = {On some optimal control problems governed by a state equation with memory}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {725--743}, publisher = {EDP-Sciences}, volume = {14}, number = {4}, year = {2008}, doi = {10.1051/cocv:2008005}, mrnumber = {2451792}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2008005/} }
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 PY - 2008 SP - 725 EP - 743 VL - 14 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2008005/ DO - 10.1051/cocv:2008005 LA - en ID - COCV_2008__14_4_725_0 ER -
%0 Journal Article %A Carlier, Guillaume %A Tahraoui, Rabah %T On some optimal control problems governed by a state equation with memory %J ESAIM: Control, Optimisation and Calculus of Variations %D 2008 %P 725-743 %V 14 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2008005/ %R 10.1051/cocv:2008005 %G en %F COCV_2008__14_4_725_0
Carlier, Guillaume; Tahraoui, Rabah. On some optimal control problems governed by a state equation with memory. ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 4, pp. 725-743. doi : 10.1051/cocv:2008005. http://www.numdam.org/articles/10.1051/cocv:2008005/
[1] Lecture Notes on Optimal Transport Problems, Mathematical aspects of evolving interfaces, CIME Summer School in Madeira 1812. Springer (2003). | MR | Zbl
,[2] Differential-difference equations, Mathematics in Science and Engineering. Academic Press, New York-London (1963). | MR | Zbl
and ,[3] Vintage capital and the dynamics of the AK model. J. Economic Theory 120 (2005) 39-72. | MR | Zbl
, , and ,[4] Semiconcave Functions, Hamilton-Jacobi Equations and Optimal Control. Birkhäuser (2004). | MR | Zbl
and ,[5] Probabilities and Potential, Mathematical Studies 29. North-Holland (1978). | MR | Zbl
and ,[6] On weak lower-semi continuity for a class of functionals with deviating arguments. Nonlinear Anal. TMA 28 (1997) 2005-2015. | MR | Zbl
and ,[7] Convex Analysis and Variational Problems, Classics in Mathematics. Society for Industrial and Applied Mathematics, Philadelphia (1999). | MR | Zbl
and ,[8] Some solvable stochastic control problems with delay. Stoch. Stoch. Rep. 71 (2000) 69-89. | MR | Zbl
, and ,[9] Introduction to the Theory of Differential Equations with Deviating Arguments. Holden-Day, San Francisco (1966). | MR | Zbl
,[10] 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. | MR | Zbl
and ,[11] Optimal investment with taxes: an optimal control problem with endogenous delay. Nonlinear Anal. Theory Methods Appl. 37 (1999) 31-56. | MR | Zbl
, and ,[12] Optimal investment with taxes: an existence result. J. Math. Econom. 33 (2000) 373-388. | MR | Zbl
, and ,[13] Time-Lag Control Systems. Academic Press, New-York (1966). | MR | Zbl
,[14] A mathematical theory of saving. Economic J. 38 (1928) 543-559.
,[15] Calcul des variations des fonctionelles à arguments déviés. Ph.D. thesis, University of Paris Dauphine, France (2004).
,[16] 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. | MR | Zbl
and ,[17] How to state necessary optimality conditions for control problems with deviating arguments? ESAIM: COCV (2007) e-first, doi: 10.1051/cocv:2007058. | Numdam | MR | Zbl
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