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

• Volume: 20, Issue: 1, page 222-235
• ISSN: 1292-8119

top

## Abstract

top
The liner parabolic equation $\frac{y}{t}-\frac{1}{2}\phantom{\rule{0.166667em}{0ex}}𝔻y+F·y={\stackrel{\to }{1}}_{{}_{0}}u$ ∂y ∂t − 1 2   Δy + F · ∇ y = 1 x1d4aa; 0 u with Neumann boundary condition on a convex open domain x1d4aa; ⊂ ℝd with smooth boundary is exactly null controllable on each finite interval if &#x1d4aa;0is an open subset of x1d4aa; which contains a suitable neighbourhood of the recession cone of  x1d4aa; . Here,F : ℝd → ℝd is a bounded, C1-continuous function, and F = ∇g, where g is convex and coercive.

## How to cite

top

Barbu, Viorel. "Exact null internal controllability for the heat equation on unbounded convex domains." ESAIM: Control, Optimisation and Calculus of Variations 20.1 (2014): 222-235. <http://eudml.org/doc/272944>.

@article{Barbu2014,
abstract = {The liner parabolic equation $\frac\{y\}\{t\}-\frac\{1\}\{2\}\,\mathbb \{D\}y+F\cdot y=\{\vec\{1\}\}_\{_0\}u$ ∂y ∂t − 1 2   Δy + F · ∇ y = 1 x1d4aa; 0 u with Neumann boundary condition on a convex open domain x1d4aa; ⊂ ℝd with smooth boundary is exactly null controllable on each finite interval if &#x1d4aa;0is an open subset of x1d4aa; which contains a suitable neighbourhood of the recession cone of $$x1d4aa; . Here,F : ℝd → ℝd is a bounded, C1-continuous function, and F = ∇g, where g is convex and coercive.}, author = {Barbu, Viorel}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, keywords = {parabolic equation; null controllability; convex set; Carleman inequality}, language = {eng}, number = {1}, pages = {222-235}, publisher = {EDP-Sciences}, title = {Exact null internal controllability for the heat equation on unbounded convex domains}, url = {http://eudml.org/doc/272944}, volume = {20}, year = {2014}, } TY - JOUR AU - Barbu, Viorel TI - Exact null internal controllability for the heat equation on unbounded convex domains JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 PB - EDP-Sciences VL - 20 IS - 1 SP - 222 EP - 235 AB - The liner parabolic equation \frac{y}{t}-\frac{1}{2}\,\mathbb {D}y+F\cdot y={\vec{1}}_{_0}u ∂y ∂t − 1 2 Δy + F · ∇ y = 1 x1d4aa; 0 u with Neumann boundary condition on a convex open domain x1d4aa; ⊂ ℝd with smooth boundary is exactly null controllable on each finite interval if &#x1d4aa;0is an open subset of x1d4aa; which contains a suitable neighbourhood of the recession cone of$$ x1d4aa; . Here,F : ℝd → ℝd is a bounded, C1-continuous function, and F = ∇g, where g is convex and coercive.
LA - eng
KW - parabolic equation; null controllability; convex set; Carleman inequality
UR - http://eudml.org/doc/272944
ER -

## References

top
1. [1] S. Aniţa and V. Barbu, Null controllability of nonlinear convective heat equation. ESAIM: COCV 5 (2000) 157–173. Zbl0938.93008MR1744610
2. [2] V. Barbu, Exact controllability of the superlinear heat equations. Appl. Math. Optim.42 (2000) 73–89. Zbl0964.93046MR1751309
3. [3] V. Barbu, Controllability of parabolic and Navier-Stokes equations. Scientiae Mathematicae Japonicae56 (2002) 143–211. Zbl1010.93054MR1911840
4. [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. Zbl1130.35137MR2180301
5. [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. Zbl1155.60034MR2450160
6. [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. Zbl0997.35012MR1876411
7. [7] E. Cepá, Multivalued stochastic differential equations. C.R. Acad. Sci. Paris, Ser. 1, Math. 319 (1994) 1075–1078. Zbl0809.60071MR1305679
8. [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. Zbl1038.93041MR1939871
9. [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. Zbl1092.93006MR1932966
10. [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. Zbl1121.35017MR2257228
11. [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. Zbl0970.93023MR1791879
12. [12] A. Fursikov, Imanuvilov and O. Yu, Controllability of Evolution Equations, Lecture Notes #34. Seoul National University Korea (1996). Zbl0862.49004MR1406566
13. [13] G. Lebeau and L. Robbiano, Contrôle exact de l’équation de la chaleur. Commun. Partial Differ. Eq.30 (1995) 335–357. Zbl0819.35071
14. [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. Zbl1262.35206MR3041662
15. [15] J. Le Rousseau and L. Robbiano, Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaqces. Inventiones Mathematicae183 (2011) 245–336. Zbl1218.35054MR2772083
16. [16] S. Micu and E. Zuazua, On the lack of null controllability of the heat equation on the half-line. Trans. AMS353 (2000) 1635–1659. Zbl0969.35022MR1806726
17. [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. Zbl0991.35010MR1820835
18. [18] L. Miller, Unique continuation estimates for the Laplacian and the heat equation on non-compact manifolds, Math. Res. Lett.12 (2005) 37–47. Zbl1065.58019MR2122728
19. [19] R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton, N.Y. (1970). Zbl0193.18401MR274683
20. [20] C. Zalinescu, Convex Analysis in General Vector Spaces. World Scientific Publishing, River Edge, N.Y. (2002). Zbl1023.46003MR1921556
21. [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). Zbl1226.93031MR2828003
22. [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. Zbl1134.74030MR2549154

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.