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

Lassana Samassi; Rabah Tahraoui

ESAIM: Control, Optimisation and Calculus of Variations (2008)

  • Volume: 14, Issue: 2, page 381-409
  • ISSN: 1292-8119

Abstract

top
The aim of this paper is to give a general idea to state optimality conditions of control problems in the following form: inf ( u , v ) 𝒰 a d 0 1 f t , u ( θ v ( t ) ) , u ' ( t ) , v ( t ) d t , (1) where 𝒰 a d is a set of admissible controls and θ v is the solution of the following equation: { d θ ( t ) d t = g ( t , θ ( t ) , v ( t ) ) , t [ 0 , 1 ] ; θ ( 0 ) = θ 0 , θ ( t ) [ 0 , 1 ] t . (2). The results are nonlocal and new.

How to cite

top

Samassi, Lassana, and Tahraoui, Rabah. "How to state necessary optimality conditions for control problems with deviating arguments?." ESAIM: Control, Optimisation and Calculus of Variations 14.2 (2008): 381-409. <http://eudml.org/doc/250383>.

@article{Samassi2008,
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. },
author = {Samassi, Lassana, Tahraoui, Rabah},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Functionals with deviating arguments; optimal control; Euler-Lagrange equation; financial market; functionals with deviating arguments},
language = {eng},
month = {3},
number = {2},
pages = {381-409},
publisher = {EDP Sciences},
title = {How to state necessary optimality conditions for control problems with deviating arguments?},
url = {http://eudml.org/doc/250383},
volume = {14},
year = {2008},
}

TY - JOUR
AU - Samassi, Lassana
AU - Tahraoui, Rabah
TI - How to state necessary optimality conditions for control problems with deviating arguments?
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2008/3//
PB - EDP Sciences
VL - 14
IS - 2
SP - 381
EP - 409
AB - 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.
LA - eng
KW - Functionals with deviating arguments; optimal control; Euler-Lagrange equation; financial market; functionals with deviating arguments
UR - http://eudml.org/doc/250383
ER -

References

top
  1. G. Carlier and R. Tahraoui, On some optimal control problems governed by a state equation with memory. ESAIM: COCV (to appear) Zbl05359684
  2. M. Drakhlin, On the variational problem in the space of absolutely continuous functions. Nonlin. Anal. TMA23 (1994) 1345–1351.  Zbl0827.34072
  3. M. Drakhlin and E. Litsyn, On the variation problem for a family of functionals in the space of absolutly continuous functions. Nonlin. Anal. TMA26 (1996) 463–468.  Zbl0837.49003
  4. M.E. Drakhlin and E. Stepanov, On weak lower semi-continuity for a class of functionals with deviating argument. Nonlin. Anal. TMA28 (1997) 2005–2015.  Zbl0876.49015
  5. M.E. Drakhlin, E. Litsyn and E. Stepanov, Variational methods for a class of nonlocal functionals. Comput. Math. Appl37 (1999) 79–100.  Zbl0944.49013
  6. L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. CRC Press, Inc. (1992).  Zbl0804.28001
  7. 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.  Zbl0972.49003
  8. A.A. Gruzdev and S.A. Gusarenko, On reduction of variational problems to extremal problems without constraints. Russians mathematics38 (1994) 37–47.  Zbl0846.49006
  9. E. Jouini, P.F. Koehl and N. Touzi, Optimal investment with taxes: an optimal control problem with endogeneous delay. Nonlin. Anal. TMA37 (1999) 31–56.  Zbl1126.91370
  10. E. Jouini, P.F. Koehl and N.Touzi, Optimal investment with taxes: an existence result. J. Math. Economics33 (2000) 373–388.  Zbl0978.91043
  11. G.A. Kamenskii, Variational and boundary value problems with deviating argument. Diff. Equ6 (1970) 1349–1358.  
  12. G.A. Kamenskii, On some necessary conditions of functionals with deviating argument. Nonlin. Anal. TMA17 (1991) 457–464.  
  13. G.A. Kamenskii, Boundary value problems for differential-difference equations arising from variational problems. Nonlin. Anal. TMA18 (1992) 801–813.  Zbl0787.34054
  14. P.L. Lions and B. Larrouturou, Optimisation et commande optimale, méthodes mathématiques pour l'ingénieur, cours de l'École Polytechnique, Palaiseau, France.  
  15. L. Samassi, Calculus of variation for funtionals with deviating arguments. Ph.D. thesis, University 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. Acad. Sci. Paris Ser338 (2004) 611–616.  Zbl1095.49021
  17. J.A. Wheeler and R.P. Feynman, Classical electrodynamics in term of direct interparticle actions. Rev. Modern Phys21 (1949) 425–433.  Zbl0034.27801

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.