# Linearization techniques for $\mathrm{See\; PDF}$-control problems and dynamic programming principles in classical and $\mathrm{See\; PDF}$-control problems

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

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

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topGoreac, 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

top- 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). Zbl0890.49011
- G. Barles, Solutions de viscosity des equations de Hamilton-Jacobi (Viscosity solutions of Hamilton-Jacobi equations), Mathematiques & Applications (Paris)17. Springer-Verlag, Paris (1994). Zbl0819.35002
- G. Barles and E.R. Jakobsen, On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations. ESAIM : M2AN36 (2002) 33–54. Zbl0998.65067
- E.N. Barron and H. Ishii, The bellman equation for minimizing the maximum cost. Nonlinear Anal.13 (1989) 1067–1090. Zbl0691.49030
- E.N. Barron and R. Jensen, Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians. Commun. Partial Differ. Equ.15 (1990) 1713–1742. Zbl0732.35014
- A.G. Bhatt and V.S. Borkar, Occupation measures for controlled markov processes : Characterization and optimality. Ann. Probab.24 (1996) 1531–1562. Zbl0863.93086
- V. Borkar and V. Gaitsgory, Averaging of singularly perturbed controlled stochastic differential equations. Appl. Math. Optim.56 (2007) 169–209. Zbl1139.93022
- R. Buckdahn, D. Goreac and M. Quincampoix, Stochastic optimal control and linear programming approach. Appl. Math. Optim.63 (2011) 257–276. Zbl1226.93137
- W.H. Fleming and D. Vermes, Convex duality approach to the optimal control of diffusions. SIAM J. Control Optim.27 (1989) 1136–1155. Zbl0693.93082
- H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim.31 (1993) 257–272. Zbl0796.49024
- 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. Zbl1201.49040
- 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. Zbl1109.93017
- 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. Zbl1195.49007
- D. Goreac and O.S. Serea, Mayer and optimal stopping stochastic control problems with discontinuous cost. J. Math. Anal. Appl.380 (2011) 327–342. Zbl1215.93150
- 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. Zbl0971.65081
- 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. Zbl0977.49018
- 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. Zbl1180.49025
- O.S. Serea, Discontinuous differential games and control systems with supremum cost. J. Math. Anal. Appl.270 (2002) 519–542. Zbl1011.91017
- O.S. Serea, On reflecting boundary problem for optimal control. SIAM J. Control Optim.42 (2003) 559–575. Zbl1041.49027
- A.I. Subbotin, Generalized solutions of first-order PDEs, The dynamical optimization perspective. Birkhäuser, Basel (1994). Zbl0820.35003
- C. Villani, Optimal Transport : Old and New. Springer (2009). Zbl1156.53003
- R. Vinter, Convex duality and nonlinear optimal control. SIAM J. Control Optim.31 (1993) 518–538. Zbl0781.49012

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