For a class of Bellman equations in bounded domains we prove that sub- and supersolutions whose growth at the boundary is suitably controlled must be constant. The ellipticity of the operator is assumed to degenerate at the boundary and a condition involving also the drift is further imposed. We apply this result to stochastic control problems, in particular to an exit problem and to the small discount limit related with ergodic control with state constraints. In this context, our condition on the behavior of the operator near the boundary ensures some invariance property of the domain for the associated controlled diffusion process.

DOI: 10.1051/cocv/2015033

Keywords: Hamilton-Jacobi-Bellman equations, degenerate elliptic PDEs, stochastic control, exit-time problems, ergodic control with state constraints, viscosity solutions

^{1}; Cesaroni, Annalisa

^{1, 2}; Rossi, Luca

^{1}

@article{COCV_2016__22_3_842_0, author = {Bardi, Martino and Cesaroni, Annalisa and Rossi, Luca}, title = {Nonexistence of nonconstant solutions of some degenerate {Bellman} equations and applications to stochastic control}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {842--861}, publisher = {EDP-Sciences}, volume = {22}, number = {3}, year = {2016}, doi = {10.1051/cocv/2015033}, mrnumber = {3527947}, zbl = {1346.35067}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2015033/} }

TY - JOUR AU - Bardi, Martino AU - Cesaroni, Annalisa AU - Rossi, Luca TI - Nonexistence of nonconstant solutions of some degenerate Bellman equations and applications to stochastic control JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2016 SP - 842 EP - 861 VL - 22 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2015033/ DO - 10.1051/cocv/2015033 LA - en ID - COCV_2016__22_3_842_0 ER -

%0 Journal Article %A Bardi, Martino %A Cesaroni, Annalisa %A Rossi, Luca %T Nonexistence of nonconstant solutions of some degenerate Bellman equations and applications to stochastic control %J ESAIM: Control, Optimisation and Calculus of Variations %D 2016 %P 842-861 %V 22 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2015033/ %R 10.1051/cocv/2015033 %G en %F COCV_2016__22_3_842_0

Bardi, Martino; Cesaroni, Annalisa; Rossi, Luca. Nonexistence of nonconstant solutions of some degenerate Bellman equations and applications to stochastic control. ESAIM: Control, Optimisation and Calculus of Variations, Volume 22 (2016) no. 3, pp. 842-861. doi : 10.1051/cocv/2015033. http://archive.numdam.org/articles/10.1051/cocv/2015033/

Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations. Mem. Amer. Math. Soc. 204 (2010) 960. | MR | Zbl

and ,A. Arapostathis, V.S. Borkar and M.K. Ghosh, Ergodic control of diffusion processes. Cambridge University Press, Cambridge (2012). | MR | Zbl

On ergodic stochastic control. Commun. Partial Differ. Equ. 23 (1998) 2187–2217. | DOI | MR | Zbl

and ,M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton−Jacobi−Bellman equations. Birkhäuser Boston, Boston, MA (1997). | MR | Zbl

M. Bardi and P. Goatin, Invariant sets for controlled degenerate diffusions: a viscosity solutions approach, in “Stochastic Analysis, Control, Optimization and Applications: A Volume in Honor of W.H. Fleming”, edited by W.M. McEneaney, G.G. Yin and Q. Zhang. Birkhäuser, Boston (1999) 191–208. | MR | Zbl

Propagation of maxima and strong maximum principle for fully nonlinear degenerate elliptic equations. Part I: convex operators. Nonlinear Anal. 44 (2001) 991–1006. | DOI | MR | Zbl

and ,A geometric characterization of viable sets for controlled degenerate diffusions. Set-Valued Anal. 10 (2002) 129–141. | DOI | MR | Zbl

and ,Propagation of maxima and strong maximum principle for fully nonlinear degenerate elliptic equations. Part II: concave operators. Indiana Univ. Math. J. 52 (2003) 607–627. | DOI | MR | Zbl

and ,Fully nonlinear Neumann type boundary conditions for second-order elliptic and parabolic equations. J. Differ. Equ. 106 (1993) 90–106. | DOI | MR | Zbl

,The Dirichlet problem for semilinear second-order degenerate elliptic equations and applications to stochastic exit time control problems. Commun. Partial Differ. Eq. 20 (1995) 129–178. | DOI | MR | Zbl

and ,A strong comparison result for the Bellman equation arising in stochastic exit time control problems and its applications. Commun. Partial Differ. Equ. 23 (1998) 11–12, 1995–2033. | DOI | MR | Zbl

and ,On the boundary ergodic problem for fully nonlinear equations in bounded domains with general nonlinear Neumann boundary conditions. Ann. Inst. Henri Poincaré Anal. Non Linéaire 22 (2005) 521–541. | DOI | Numdam | MR | Zbl

and ,On the large time behavior of solutions of the Dirichlet problem for subquadratic viscous Hamilton-Jacobi equations. J. Math. Pures Appl. 94 (2010) 497–519. | DOI | MR | Zbl

, and ,A. Bensoussan and J. Frehse, Ergodic control Bellman equation with Neumann boundary conditions. Vol. 280 of Lect. Notes Control Inform. Sci. Springer, Berlin (2002) 59–71. | MR | Zbl

Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators. J. Math. Pures Appl. 103 (2015) 1276–1293. | DOI | MR | Zbl

, , , ,Ergodic control for constrained diffusions: characterization using HJB equations. SIAM J. Control Optim. 43 (2004/05) 1467–1492. | DOI | MR | Zbl

and ,Invariant measures associated to degenerate elliptic operators. Indiana Univ. Math. J. 59 (2010) 53–78. | DOI | MR | Zbl

, and ,Hadamard and Liouville type results for fully nonlinear partial differential inequalities. Commun. Contemp. Math. 5 (2003) 435–448. | DOI | MR | Zbl

, ,Lyapunov Stabilizability of Controlled Diffusions via a Superoptimality Principle for Viscosity Solutions. Appl. Math. Optim. 53 (2006) 1–29. | DOI | MR | Zbl

,On Liouville type theorems for fully nonlinear elliptic equations with gradient term. J. Differ. Eq. 255 (2013) 2167–2195. | DOI | MR | Zbl

and ,User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992) 1–67. | DOI | MR | Zbl

, and ,Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine. Atti Accad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I. 5 (1956) 1–30. | MR | Zbl

,W.H. Fleming and H.M. Soner, Controlled Markov processes and viscosity solutions. 2nd edition. Springer, New York (2006). | MR

D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1988). | MR | Zbl

On the existence of optimal controls. SIAM J. Control Optim. 28 (1990) 851–902. | DOI | MR | Zbl

and ,Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints. I. The model problem. Math. Ann. 283 (1989) 583–630. | DOI | MR | Zbl

and ,Gradient bounds for elliptic problems singular at the boundary. Arch. Ration. Mech. Anal. 202 (2011) 663–705. | DOI | MR | Zbl

and ,Quelques remarques sur les problèmes elliptiques quasilinéaires du second ordre. J. Anal. Math. 45 (1985) 234–254. | DOI | MR | Zbl

,Régularité optimale de racines carrées. C. R. Acad. Sci. Paris Sér. I Math. 321 (1995) 1537–1541. | MR | Zbl

and ,The “ergodic limit” for a viscous Hamilton-Jacobi equation with Dirichlet conditions. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 21 (2010) 59–78. | DOI | MR | Zbl

,On the classical solution of Bellman’s elliptic equation. Sov. Math. Dokl. 30 (1984) 482–485. | Zbl

,N.S. Trudinger, On regularity and existence of viscosity solutions of nonlinear second order, elliptic equations. Partial differential equations and the calculus of variations. Vol. II of Progr. Nonlin. Differ. Equ. Appl. Birkhäuser Boston, Boston, MA (1989) 939–957. | MR | Zbl

*Cited by Sources: *