The aim of the paper is to provide a linearization approach to the See PDF-control problems. We begin by proving a semigroup-type behaviour of the set of constraints appearing in the linearized formulation of (standard) control problems. As a byproduct we obtain a linear formulation of the dynamic programming principle. Then, we use the See PDF approach and the associated linear formulations. This seems to be the most appropriate tool for treating See PDF problems in continuous and lower semicontinuous setting.
Mots clés : dynamic programming principle, essential supremum, hj equations, occupational measures, $\mathbb {L}^{p}$See pdf approximations
@article{COCV_2012__18_3_836_0,
author = {Goreac, Dan and Serea, Oana-Silvia},
title = {Linearization techniques for $\mathbb {L}^{\infty }${See} {PDF-control} problems and dynamic programming principles in classical and $\mathbb {L}^{\infty }${See} {PDF-control} problems},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {836--855},
publisher = {EDP-Sciences},
volume = {18},
number = {3},
year = {2012},
doi = {10.1051/cocv/2011183},
mrnumber = {3041666},
zbl = {1262.49030},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2011183/}
}
TY - JOUR
AU - Goreac, Dan
AU - Serea, Oana-Silvia
TI - Linearization techniques for $\mathbb {L}^{\infty }$See PDF-control problems and dynamic programming principles in classical and $\mathbb {L}^{\infty }$See PDF-control problems
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2012
SP - 836
EP - 855
VL - 18
IS - 3
PB - EDP-Sciences
UR - http://www.numdam.org/articles/10.1051/cocv/2011183/
DO - 10.1051/cocv/2011183
LA - en
ID - COCV_2012__18_3_836_0
ER -
%0 Journal Article
%A Goreac, Dan
%A Serea, Oana-Silvia
%T Linearization techniques for $\mathbb {L}^{\infty }$See PDF-control problems and dynamic programming principles in classical and $\mathbb {L}^{\infty }$See PDF-control problems
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2012
%P 836-855
%V 18
%N 3
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2011183/
%R 10.1051/cocv/2011183
%G en
%F COCV_2012__18_3_836_0
Goreac, Dan; Serea, Oana-Silvia. Linearization techniques for $\mathbb {L}^{\infty }$See PDF-control problems and dynamic programming principles in classical and $\mathbb {L}^{\infty }$See PDF-control problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 3, pp. 836-855. doi : 10.1051/cocv/2011183. http://www.numdam.org/articles/10.1051/cocv/2011183/
[1] and , Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Systems and Control : Foundations and Applications, Birkhäuser, Boston (1997). | MR | Zbl
[2] , Solutions de viscosity des equations de Hamilton-Jacobi (Viscosity solutions of Hamilton-Jacobi equations), Mathematiques & Applications (Paris) 17. Springer-Verlag, Paris (1994). | MR | Zbl
[3] and , On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations. ESAIM : M2AN 36 (2002) 33-54. | Numdam | MR | Zbl
[4] and , The bellman equation for minimizing the maximum cost. Nonlinear Anal. 13 (1989) 1067-1090. | MR | Zbl
[5] and , Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians. Commun. Partial Differ. Equ. 15 (1990) 1713-1742. | MR | Zbl
[6] and , Occupation measures for controlled markov processes : Characterization and optimality. Ann. Probab. 24 (1996) 1531-1562. | MR | Zbl
[7] and , Averaging of singularly perturbed controlled stochastic differential equations. Appl. Math. Optim. 56 (2007) 169-209. | MR | Zbl
[8] , and , Stochastic optimal control and linear programming approach. Appl. Math. Optim. 63 (2011) 257-276. | MR | Zbl
[9] and , Convex duality approach to the optimal control of diffusions. SIAM J. Control Optim. 27 (1989) 1136-1155. | MR | Zbl
[10] , Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 31 (1993) 257-272. | MR | Zbl
[11] and , Linear programming approach to deterministic infinite horizon optimal control problems with discounting. SIAM J. Control Optim. 48 (2009) 2480-2512. | MR | Zbl
[12] and , Linear programming approach to deterministic long run average problems of optimal control. SIAM J. Control Optim. 44 (2006) 2006-2037. | MR | Zbl
[13] and , Discontinuous control problems for non-convex dynamics and near viability for singularly perturbed control systems. Nonlinear Anal. 73 (2010) 2699-2713. | MR | Zbl
[14] and , Mayer and optimal stopping stochastic control problems with discontinuous cost. J. Math. Anal. Appl. 380 (2011) 327-342. | MR | Zbl
[15] , On the rate of convergence of finte-difference approximations for bellman's equations with variable coefficients. Probab. Theory Relat. Fields 117 (2000) 1-16. | MR | Zbl
[16] and , Value-functions for differential games and control systems with discontinuous terminal cost. SIAM J. Control Optim. 39 (2001) 1485-1498. | MR | Zbl
[17] and , The problem of optimal control with reflection studied through a linear optimization problem stated on occupational measures. Nonlinear Anal. 72 (2010) 2803-2815. | MR | Zbl
[18] , Discontinuous differential games and control systems with supremum cost. J. Math. Anal. Appl. 270 (2002) 519-542. | MR | Zbl
[19] , On reflecting boundary problem for optimal control. SIAM J. Control Optim. 42 (2003) 559-575. | MR | Zbl
[20] , Generalized solutions of first-order PDEs, The dynamical optimization perspective. Birkhäuser, Basel (1994). | MR | Zbl
[21] , Optimal Transport : Old and New. Springer (2009). | MR | Zbl
[22] , Convex duality and nonlinear optimal control. SIAM J. Control Optim. 31 (1993) 518-538. | MR | Zbl
Cité par Sources :
