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

Carlier, Guillaume; Tahraoui, Rabah

ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 4, pp. 725-743.

The aim of this paper is to study problems of the form: $in{f}_{(u\in V)}J\left(u\right)$ with $J\left(u\right):={\int }_0^{1}L(s,y_u\left(s\right),u\left(s\right))\mathrm{d}s+g\left(y_u\left(1\right)\right)$ where $V$ is a set of admissible controls and $y_u$ is the solution of the Cauchy problem: $\dot{x}\left(t\right)=\langle f(.,x(.)),{\nu }_t\rangle +u\left(t\right),t\in (0,1)$, $x\left(0\right)=x_0$ 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.

[1] L. Ambrosio, Lecture Notes on Optimal Transport Problems, Mathematical aspects of evolving interfaces, CIME Summer School in Madeira 1812. Springer (2003). | MR | Zbl

[2] R. Bellman and K.L. Cooke, Differential-difference equations, Mathematics in Science and Engineering. Academic Press, New York-London (1963). | MR | Zbl

[3] R. Boucekkine, O. Licandro, L. Puch and F. Del Rio, Vintage capital and the dynamics of the AK model. J. Economic Theory 120 (2005) 39-72. | MR | Zbl

[4] P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations and Optimal Control. Birkhäuser (2004). | MR | Zbl

[5] C. Dellacherie and P.-A. Meyer, Probabilities and Potential, Mathematical Studies 29. North-Holland (1978). | MR | Zbl

[6] M.E. Drakhlin and E. Stepanov, On weak lower-semi continuity for a class of functionals with deviating arguments. Nonlinear Anal. TMA 28 (1997) 2005-2015. | MR | Zbl

[7] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Classics in Mathematics. Society for Industrial and Applied Mathematics, Philadelphia (1999). | MR | Zbl

[8] I. Elsanosi, B. Øksendal and A. Sulem, Some solvable stochastic control problems with delay. Stoch. Stoch. Rep. 71 (2000) 69-89. | MR | Zbl

[9] L. El'Sgol'Ts, Introduction to the Theory of Differential Equations with Deviating Arguments. Holden-Day, San Francisco (1966). | MR | Zbl

[10] 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. | MR | Zbl

[11] 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. | MR | Zbl

[12] E. Jouini, P.-F. Koehl and N. Touzi, Optimal investment with taxes: an existence result. J. Math. Econom. 33 (2000) 373-388. | MR | Zbl

[13] M.N. Oguztöreli, Time-Lag Control Systems. Academic Press, New-York (1966). | MR | Zbl

[14] F.P. Ramsey, A mathematical theory of saving. Economic J. 38 (1928) 543-559.

[15] L. Samassi, Calcul des variations des fonctionelles à arguments déviés. Ph.D. thesis, University of Paris Dauphine, France (2004).

[16] 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. Paris 338 (2004) 611-616. | MR | Zbl

[17] 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. | Numdam | MR | Zbl