This article is devoted to the optimal control of state equations with memory of the form: with initial conditions . Denoting by the solution of the previous Cauchy problem and: where V is a class of admissible controls, we prove that v is the only viscosity solution of an Hamilton-Jacobi-Bellman equation of the form: in the sense of the theory of viscosity solutions in infinite-dimensions of Crandall and Lions.
Keywords: dynamic programming, state equations with memory, viscosity solutions, Hamilton-Jacobi-Bellman equations in infinite dimensions
@article{COCV_2010__16_3_744_0, author = {Carlier, Guillaume and Tahraoui, Rabah}, title = {Hamilton-Jacobi-Bellman equations for the optimal control of a state equation with memory}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {744--763}, publisher = {EDP-Sciences}, volume = {16}, number = {3}, year = {2010}, doi = {10.1051/cocv/2009024}, mrnumber = {2674635}, zbl = {1195.49032}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2009024/} }
TY - JOUR AU - Carlier, Guillaume AU - Tahraoui, Rabah TI - Hamilton-Jacobi-Bellman equations for the optimal control of a state equation with memory JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2010 SP - 744 EP - 763 VL - 16 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2009024/ DO - 10.1051/cocv/2009024 LA - en ID - COCV_2010__16_3_744_0 ER -
%0 Journal Article %A Carlier, Guillaume %A Tahraoui, Rabah %T Hamilton-Jacobi-Bellman equations for the optimal control of a state equation with memory %J ESAIM: Control, Optimisation and Calculus of Variations %D 2010 %P 744-763 %V 16 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2009024/ %R 10.1051/cocv/2009024 %G en %F COCV_2010__16_3_744_0
Carlier, Guillaume; Tahraoui, Rabah. Hamilton-Jacobi-Bellman equations for the optimal control of a state equation with memory. ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 3, pp. 744-763. doi : 10.1051/cocv/2009024. http://archive.numdam.org/articles/10.1051/cocv/2009024/
[1] A Report on the Use of Delay Differential Equations in Numerical Modelling in the Biosciences. Technical report, Manchester Centre for Computational Mathematics, UK (1999).
, and ,[2] Representation and control of infinite dimensional systems. Second Edition, Birkhäuser (2007). | Zbl
, , and ,[3] Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Boston (1997). | Zbl
and ,[4] Solutions de viscosité des équations de Hamilton-Jacobi, Mathematics and Applications 17. Springer-Verlag, Paris (1994). | Zbl
,[5] Vintage capital and the dynamics of the AK model. J. Econ. Theory 120 (2005) 39-72. | Zbl
, , and ,[6] Analyse fonctionnelle, théorie et applications. Masson, Paris (1983). | Zbl
,[7] On some optimal control problems governed by a state equation with memory. ESAIM: COCV 14 (2008) 725-743. | Numdam
and ,[8] Hamilton-Jacobi equations in infinite dimensions. I. Uniqueness of viscosity solutions. J. Funct. Anal. 62 (1985) 379-396. | Zbl
and ,[9] Hamilton-Jacobi equations in infinite dimensions. II. Existence of viscosity solutions. J. Funct. Anal. 65 (1986) 368-405. | Zbl
and ,[10] Hamilton-Jacobi equations in infinite dimensions. III. J. Funct. Anal. 68 (1986) 214-247. | Zbl
and ,[11] Viscosity solutions of Hamilton-Jacobi equations in infinite dimensions. IV. Hamiltonians with unbounded linear terms. J. Funct. Anal. 90 (1990) 237-283. | Zbl
and ,[12] Viscosity solutions of Hamilton-Jacobi equations in infinite dimensions. V. Unbounded linear terms and B-continuous solutions. J. Funct. Anal. 97 (1991) 417-465. | Zbl
and ,[13] Hamilton-Jacobi equations in infinite dimensions. VI. Nonlinear A and Tataru's method refined, in Evolution equations, control theory, and biomathematics, Lect. Notes Pure Appl. Math. 155, Dekker, New York (1994) 51-89. | Zbl
and ,[14] Some solvable stochastic control problems with delay. Stochast. Stochast. Rep. 71 (2000) 69-89. | Zbl
, and ,[15] Viscosity solutions to delay differential equations in demo-economy. Math. Popul. Stud. 15 (2008) 27-54. | Zbl
,[16] On dynamic programming in economic models governed by DDEs. Math. Popul. Stud. 15 (2008) 267-290. | Zbl
, and ,[17] On the dynamic programming approach for optimal control problems of PDE's with age structure. Math. Popul. Stud. 11 (2004) 233-270. | Zbl
and ,[18] 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. | Zbl
and ,[19] Control of systems with aftereffect, Translations of Mathematical Monographs. American Mathematical Society, Providence, USA (1996). | Zbl
and ,[20] When are HJB-equations in stochastic control of delay systems finite dimensional? Stochastic Anal. Appl. 21 (2003) 643-671. | Zbl
and ,[21] 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. | Zbl
and ,[22] How to state necessary optimality conditions for control problems with deviating arguments? ESAIM: COCV 14 (2008) 381-409. | Numdam | Zbl
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