Exact null internal controllability for the heat equation on unbounded convex domains

Barbu, Viorel

ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 1, pp. 222-235

The linear parabolic equation $\frac{\partial y}{\partial t}-\frac{1}{2}\Delta y+F·\nabla y={\mathbb{1}}_{{𝒪}_0}u$ with Neumann boundary condition on a convex open domain $𝒪\subset {ℝ}^d$ with smooth boundary is exactly null controllable on each finite interval if ${𝒪}_0$ is an open subset of $𝒪$ which contains a suitable neighbourhood of the recession cone of $\overline{𝒪}$. Here, $F:{ℝ}^d\to {ℝ}^d$ is a bounded, $C^1$-continuous function, and $F=\nabla g$ where $g$ is convex and coercive.

Classification : 93B07, 35K50, 47D07

[1] S. Aniţa and V. Barbu, Null controllability of nonlinear convective heat equation. ESAIM: COCV 5 (2000) 157-173. | Zbl | MR | Numdam

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

[3] V. Barbu, Controllability of parabolic and Navier-Stokes equations. Scientiae Mathematicae Japonicae 56 (2002) 143-211. | Zbl | MR

[4] V. Barbu and G. Da Prato, The Neumann problem on unbounded domains of Rd and stochastic variational inequalities. Commun. Partial Differ. Eq. 11 (2005) 1217-1248. | Zbl | MR

[5] V. Barbu and G. Da Prato, The generator of the transition semigroup corresponding to a stochastic variational inequality. Commun. Partial Differ. Eq. 33 (2008) 1318-1338. | Zbl | MR

[6] V.I. Bogachev, N.V. Krylov and M. Röckner, On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions. Commun. Partial Differ. Eq. 26 (2001) 11-12. | Zbl | MR

[7] E. Cepá, Multivalued stochastic differential equations. C.R. Acad. Sci. Paris, Ser. 1, Math. 319 (1994) 1075-1078. | Zbl | MR

[8] A. Dubova, E. Fernandez Cara and M. Burges, On the controllability of parabolic systems with a nonlinear term involving state and gradient. SIAM J. Control Optim. 41 (2002) 718-819. | Zbl | MR

[9] A. Dubova, A. Osses and J.P. Puel, Exact controllability to trajectories for semilinear heat equations with discontinuous coefficients. ESAIM: COCV 8 (2002) 621-667. | Zbl | MR | Numdam

[10] E. Fernandez Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability. SIAM J. Control Optim. 45 (2006) 1395-1446. | Zbl | MR

[11] E. Fernandez Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, vol. 17 of Annales de l'Institut Henri Poincaré (C) Nonlinear Analysis (2000) 583-616. | Numdam | Zbl | MR | EuDML

[12] A. Fursikov, Imanuvilov and O. Yu, Controllability of Evolution Equations, Lecture Notes #34. Seoul National University Korea (1996). | Zbl | MR

[13] G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur. Commun. Partial Differ. Eq. 30 (1995) 335-357. | Zbl | MR

[14] J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applicatiosn to unique continuation and control of parabolic equations. ESAIM: COCV 18 (2012) 712-747. | Zbl | MR | Numdam

[15] J. Le Rousseau and L. Robbiano, Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaqces. Inventiones Mathematicae 183 (2011) 245-336. | Zbl | MR

[16] S. Micu and E. Zuazua, On the lack of null controllability of the heat equation on the half-line. Trans. AMS 353 (2000) 1635-1659. | Zbl | MR

[17] S. Micu and E. Zuazua, On the lack of null controllability of the heat equation on the half-space. Part. Math. 58 (2001) 1-24. | Zbl | MR

[18] L. Miller, Unique continuation estimates for the Laplacian and the heat equation on non-compact manifolds, Math. Res. Lett. 12 (2005) 37-47. | Zbl | MR

[19] R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton, N.Y. (1970). | Zbl | MR

[20] C. Zalinescu, Convex Analysis in General Vector Spaces. World Scientific Publishing, River Edge, N.Y. (2002). | Zbl | MR

[21] Zhang, Xu, A unified controllability/observability theory for some stochastic and deterministic partial differential equations, Proc. of the International Congress of Mathematicians, vol. IV, 3008-3034. Hindustan Book Agency, New Delhi (2010). | Zbl | MR

[22] X. Zhang and E. Zuazua, On the optimality of the observability inequalities for Kirchoff plate systems with potentials in unbounded domains, in Hyperbolic Prloblems: Theory, Numerics and Applications, edited by S. Benzoni-Gavage and D. Serre. Springer (2008) 233-243. | Zbl | MR