Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-23T05:20:32.788Z Has data issue: false hasContentIssue false
  • English
  • Français

How to state necessary optimality conditions for control problems with deviating arguments?

Published online by Cambridge University Press:  20 March 2008

Lassana Samassi  and
Rabah Tahraoui
Lassana Samassi
Affiliation:
Ceremade, Université Paris IX-Dauphine, France; [email protected]; [email protected]
Rabah Tahraoui
Affiliation:
Ceremade, Université Paris IX-Dauphine, France; [email protected]; [email protected]
Get access

Abstract

The aim of this paper is to give a general idea to state optimality conditions of control problems in the following form: ${\displaystyle\inf_{{\displaystyle(u,v)\in {\cal U}_{ad}}}\int_{0}^{1} f\left(t, u(\theta_v(t)),u^{\prime}(t),v(t)\right){\rm d}t}$ , (1)where ${\cal U}_{ad} $ is a set of admissible controls and $\theta_v$ is the solution of the following equation: $\{ \frac{{\rm d}\theta(t)}{{\rm d}t}=g(t,\theta(t),v(t)), t\in [0,1]$ ; $\displaystyle\theta(0)=\theta_0, \theta(t)\in [0,1]\forall t$ . (2).The results are nonlocal and new.

Keywords

Functionals with deviating argumentsoptimal controlEuler-Lagrange equationfinancial market
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 2 , April 2008 , pp. 381 - 409
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

G. Carlier and R. Tahraoui, On some optimal control problems governed by a state equation with memory. ESAIM: COCV (to appear)
Drakhlin, M., On the variational problem in the space of absolutely continuous functions. Nonlin. Anal. TMA 23 (1994) 1345–1351. CrossRef
Drakhlin, M. and Litsyn, E., On the variation problem for a family of functionals in the space of absolutly continuous functions. Nonlin. Anal. TMA 26 (1996) 463–468. CrossRef
Drakhlin, M.E. and Stepanov, E., On weak lower semi-continuity for a class of functionals with deviating argument. Nonlin. Anal. TMA 28 (1997) 2005–2015. CrossRef
Drakhlin, M.E., Litsyn, E. and Stepanov, E., Variational methods for a class of nonlocal functionals. Comput. Math. Appl 37 (1999) 79–100. CrossRef
L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. CRC Press, Inc. (1992).
L. Freddi, Limits of control problems with weakly converging nonlocal input operators. Calculus of variations and optimal control (Haifa, 1998), Math. 411, Chapman Hall/CRC, Boca Raton, FL (2000) 117–140.
Gruzdev, A.A. and Gusarenko, S.A., On reduction of variational problems to extremal problems without constraints. Russians mathematics 38 (1994) 37–47.
Jouini, E., Koehl, P.F. and Touzi, N., Optimal investment with taxes: an optimal control problem with endogeneous delay. Nonlin. Anal. TMA 37 (1999) 31–56. CrossRef
Jouini, E., Koehl, P.F. and N.Touzi, Optimal investment with taxes: an existence result. J. Math. Economics 33 (2000) 373–388. CrossRef
Kamenskii, G.A., Variational and boundary value problems with deviating argument. Diff. Equ 6 (1970) 1349–1358.
Kamenskii, G.A., On some necessary conditions of functionals with deviating argument. Nonlin. Anal. TMA 17 (1991) 457–464. CrossRef
Kamenskii, G.A., Boundary value problems for differential-difference equations arising from variational problems. Nonlin. Anal. TMA 18 (1992) 801–813. CrossRef
P.L. Lions and B. Larrouturou, Optimisation et commande optimale, méthodes mathématiques pour l'ingénieur, cours de l'École Polytechnique, Palaiseau, France.
L. Samassi, Calculus of variation for funtionals with deviating arguments. Ph.D. thesis, University 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. Acad. Sci. Paris Ser 338 (2004) 611–616. CrossRef
Wheeler, J.A. and Feynman, R.P., Classical electrodynamics in term of direct interparticle actions. Rev. Modern Phys 21 (1949) 425–433. CrossRef