Linearization techniques for See PDF -control problems and dynamic programming principles in classical and See PDF -control problems

Dan Goreac; Oana-Silvia Serea

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 3, page 836-855
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Goreac, Dan, and Serea, Oana-Silvia. "Linearization techniques for $\mathbb{L}^{\infty}$-control problems and dynamic programming principles in classical and $\mathbb{L}^{\infty}$-control problems." ESAIM: Control, Optimisation and Calculus of Variations 18.3 (2012): 836-855. <http://eudml.org/doc/244349>.

@article{Goreac2012,
abstract = {The aim of the paper is to provide a linearization approach to the $\mathbb\{L\}^\{\infty\}$-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 $\mathbb\{L\}^\{p\}$ approach and the associated linear formulations. This seems to be the most appropriate tool for treating $\mathbb\{L\}^\{\infty\}$ problems in continuous and lower semicontinuous setting. },
author = {Goreac, Dan, Serea, Oana-Silvia},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Dynamic programming principle; essential supremum; HJ equations; occupational measures; $\mathbb\{L\}^\{p\}$ approximations; dynamic programming principle; -approximations},
language = {eng},
month = {11},
number = {3},
pages = {836-855},
publisher = {EDP Sciences},
title = {Linearization techniques for $\mathbb\{L\}^\{\infty\}$-control problems and dynamic programming principles in classical and $\mathbb\{L\}^\{\infty\}$-control problems},
url = {http://eudml.org/doc/244349},
volume = {18},
year = {2012},
}

TY - JOUR
AU - Goreac, Dan
AU - Serea, Oana-Silvia
TI - Linearization techniques for $\mathbb{L}^{\infty}$-control problems and dynamic programming principles in classical and $\mathbb{L}^{\infty}$-control problems
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2012/11//
PB - EDP Sciences
VL - 18
IS - 3
SP - 836
EP - 855
AB - The aim of the paper is to provide a linearization approach to the $\mathbb{L}^{\infty}$-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 $\mathbb{L}^{p}$ approach and the associated linear formulations. This seems to be the most appropriate tool for treating $\mathbb{L}^{\infty}$ problems in continuous and lower semicontinuous setting.
LA - eng
KW - Dynamic programming principle; essential supremum; HJ equations; occupational measures; $\mathbb{L}^{p}$ approximations; dynamic programming principle; -approximations
UR - http://eudml.org/doc/244349
ER -

References

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  1. M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Systems and Control : Foundations and Applications, Birkhäuser, Boston (1997).  
  2. G. Barles, Solutions de viscosity des equations de Hamilton-Jacobi (Viscosity solutions of Hamilton-Jacobi equations), Mathematiques & Applications (Paris)17. Springer-Verlag, Paris (1994).  
  3. G. Barles and E.R. Jakobsen, On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations. ESAIM : M2AN36 (2002) 33–54.  
  4. E.N. Barron and H. Ishii, The bellman equation for minimizing the maximum cost. Nonlinear Anal.13 (1989) 1067–1090.  
  5. E.N. Barron and R. Jensen, Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians. Commun. Partial Differ. Equ.15 (1990) 1713–1742.  
  6. A.G. Bhatt and V.S. Borkar, Occupation measures for controlled markov processes : Characterization and optimality. Ann. Probab.24 (1996) 1531–1562.  
  7. V. Borkar and V. Gaitsgory, Averaging of singularly perturbed controlled stochastic differential equations. Appl. Math. Optim.56 (2007) 169–209.  
  8. R. Buckdahn, D. Goreac and M. Quincampoix, Stochastic optimal control and linear programming approach. Appl. Math. Optim.63 (2011) 257–276.  
  9. W.H. Fleming and D. Vermes, Convex duality approach to the optimal control of diffusions. SIAM J. Control Optim.27 (1989) 1136–1155.  
  10. H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim.31 (1993) 257–272.  
  11. V. Gaitsgory and M. Quincampoix, Linear programming approach to deterministic infinite horizon optimal control problems with discounting. SIAM J. Control Optim.48 (2009) 2480–2512.  
  12. V. Gaitsgory and S. Rossomakhine, Linear programming approach to deterministic long run average problems of optimal control. SIAM J. Control Optim.44 (2006) 2006–2037.  
  13. D. Goreac and O.S. Serea, Discontinuous control problems for non-convex dynamics and near viability for singularly perturbed control systems. Nonlinear Anal.73 (2010) 2699–2713.  
  14. D. Goreac and O.S. Serea, Mayer and optimal stopping stochastic control problems with discontinuous cost. J. Math. Anal. Appl.380 (2011) 327–342.  
  15. N.V. Krylov, On the rate of convergence of finte-difference approximations for bellman’s equations with variable coefficients. Probab. Theory Relat. Fields117 (2000) 1–16.  
  16. S. Plaskacz and M. Quincampoix, Value-functions for differential games and control systems with discontinuous terminal cost. SIAM J. Control Optim.39 (2001) 1485–1498.  
  17. M. Quincampoix and O.S. Serea, The problem of optimal control with reflection studied through a linear optimization problem stated on occupational measures. Nonlinear Anal.72 (2010) 2803–2815.  
  18. O.S. Serea, Discontinuous differential games and control systems with supremum cost. J. Math. Anal. Appl.270 (2002) 519–542.  
  19. O.S. Serea, On reflecting boundary problem for optimal control. SIAM J. Control Optim.42 (2003) 559–575.  
  20. A.I. Subbotin, Generalized solutions of first-order PDEs, The dynamical optimization perspective. Birkhäuser, Basel (1994).  
  21. C. Villani, Optimal Transport : Old and New. Springer (2009).  
  22. R. Vinter, Convex duality and nonlinear optimal control. SIAM J. Control Optim.31 (1993) 518–538.  

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