We study the problem of null controllability for viscous Hamilton–Jacobi equations in bounded domains of the Euclidean space in any space dimension and with controls localized in an arbitrary open nonempty subset of the domain where the equation holds. We prove the null controllability of the system in the sense that, every bounded (and in some cases uniformly continuous) initial datum can be driven to the null state in a sufficiently large time. The proof combines decay properties of the solutions of the uncontrolled system and local null controllability results for small data obtained by means of Carleman inequalities. We also show that there exists a waiting time so that the time of control needs to be large enough, as a function of the norm of the initial data, for the controllability property to hold. We give sharp asymptotic lower and upper bounds on this waiting time both as the size of the data tends to zero and infinity. These results also establish a limit on the growth of nonlinearities that can be controlled uniformly on a time independent of the initial data.

Keywords: Null controllability, Viscous Hamilton–Jacobi equations, Decay estimates

@article{AIHPC_2012__29_3_301_0, author = {Porretta, Alessio and Zuazua, Enrique}, title = {Null controllability of viscous {Hamilton{\textendash}Jacobi} equations}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {301--333}, publisher = {Elsevier}, volume = {29}, number = {3}, year = {2012}, doi = {10.1016/j.anihpc.2011.11.002}, zbl = {1244.93027}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2011.11.002/} }

TY - JOUR AU - Porretta, Alessio AU - Zuazua, Enrique TI - Null controllability of viscous Hamilton–Jacobi equations JO - Annales de l'I.H.P. Analyse non linéaire PY - 2012 SP - 301 EP - 333 VL - 29 IS - 3 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2011.11.002/ DO - 10.1016/j.anihpc.2011.11.002 LA - en ID - AIHPC_2012__29_3_301_0 ER -

%0 Journal Article %A Porretta, Alessio %A Zuazua, Enrique %T Null controllability of viscous Hamilton–Jacobi equations %J Annales de l'I.H.P. Analyse non linéaire %D 2012 %P 301-333 %V 29 %N 3 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2011.11.002/ %R 10.1016/j.anihpc.2011.11.002 %G en %F AIHPC_2012__29_3_301_0

Porretta, Alessio; Zuazua, Enrique. Null controllability of viscous Hamilton–Jacobi equations. Annales de l'I.H.P. Analyse non linéaire, Volume 29 (2012) no. 3, pp. 301-333. doi : 10.1016/j.anihpc.2011.11.002. http://www.numdam.org/articles/10.1016/j.anihpc.2011.11.002/

[1] Null controllability for the dissipative semilinear heat equation, Appl. Math. Optim. 46 no. 2–3 (2002), 97-105 | Zbl

, ,[2] Exact controllability of the superlinear heat equation, Appl. Math. Optim. 42 (2000), 73-89 | Zbl

,[3] On the generalized Dirichlet problem for viscous Hamilton–Jacobi equations, J. Math. Pures Appl. (9) 83 no. 1 (2004), 53-75 | Zbl

, ,[4] Uniqueness for unbounded solutions to stationary viscous Hamilton–Jacobi equations, Ann. Scuola Norm. Sup. Pisa 5 (2006), 107-136 | EuDML | Numdam | Zbl

, ,[5] 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 | Zbl

, , ,[6] Decay estimates for a viscous Hamilton–Jacobi equation with homogeneous Dirichlet boundary conditions, Asymptot. Anal. 51 no. 3–4 (2007), 209-229 | Zbl

, , ,[7] Existence results for some quasilinear parabolic equations, Nonlinear Anal. 13 (1989), 373-392 | Zbl

, , ,[8] Userʼs guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. 27 no. 1 (1992)

, , ,[9] Approximate controllability for some nonlinear parabolic problems, System Modelling and Optimization, Compiègne, 1993, Lecture Notes in Control and Inform. Sci. vol. 197, Springer, London (1994), 128-143 | Zbl

,[10] On the controllability of parabolic systems with a nonlinear term involving the state and the gradient, SIAM J. Control Optim. 41 no. 3 (2002), 798-819 | Zbl

, , , ,[11] On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials, Ann. Inst. H. Poincaré Analyse Nonlin. 25 (2008), 1-41 | EuDML | Numdam | Zbl

, , ,[12] Null controllability of the Burgers system with distributed controls, Systems Control Lett. 56 (2007), 366-372 | Zbl

, ,[13] Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. Inst. H. Poincaré Analyse Nonlin. 17 no. 5 (2000), 583-616 | EuDML | Numdam | Zbl

, ,[14] Controllability of Evolution Equations, Lecture Notes vol. 34, Seoul National University, Korea (1996) | Zbl

, ,[15] Linear and Quasilinear Equations of Parabolic Type, Trans. Math. Monogr. vol. 23 (1967)

, , ,[16] Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints. I. The model problem, Math. Ann. 283 no. 4 (1989), 583-630 | EuDML | Zbl

, ,[17] Contrôle exact de lʼéquation de la chaleur, Comm. Partial Differential Equations 20 (1995), 335-356 | Zbl

, ,[18] Large solutions for a class of nonlinear elliptic equations with gradient terms, Adv. Nonlinear Stud. 7 no. 2 (2007), 237-269 | Zbl

,[19] Regularizing effects for first-order Hamilton Jacobi equations, Appl. Anal. 20 no. 3–4 (1985), 283-307 | Zbl

,[20] ${L}^{\infty}\left(Q\right)$-estimate and existence of solutions for some nonlinear parabolic equations, Boll. Un. Mat. Ital. B 6 (1992), 631-647 | Zbl

, ,[21] Global Solutions of Reaction–Diffusion Systems, Springer-Verlag (1984) | Zbl

,[22] Compact sets in ${L}^{p}(0,T;B)$, Ann. Mat. Pura Appl. 146 no. 4 (1987), 65-96 | Zbl

,[23] Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions, Differential Integral Equations 15 no. 2 (2002), 237-256 | Zbl

,[24] Global solutions of inhomogeneous Hamilton–Jacobi equations, J. Anal. Math. 99 (2006), 355-396 | Zbl

, ,[25] Large time behavior of solutions of viscous Hamilton–Jacobi equations with superquadratic Hamiltonian, Asymptot. Anal. 66 no. 3–4 (2010), 161-186 | Zbl

,[26] Exact controllability for the semilinear wave equation in one space dimension, Ann. Inst. H. Poincaré Analyse Nonlin. 10 (1993), 109-129 | EuDML | Numdam | Zbl

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