Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T22:54:47.395Z Has data issue: false hasContentIssue false
  • English
  • Français

Null-control and measurable sets∗

Published online by Cambridge University Press:  02 May 2012

Jone Apraiz  and
Luis Escauriaza
Jone Apraiz
Affiliation:
Universidad del País Vasco/Euskal Herriko Unibertsitatea, Departamento de Matemática Aplicada, Escuela Universitaria Politécnica de Donostia-San Sebastián, Plaza de Europa 1, 20018 Donostia-San Sebastián, Spain. [email protected]
Luis Escauriaza
Affiliation:
Universidad del País Vasco/Euskal Herriko Unibertsitatean, Dpto. de Matemáticas, Apto. 644, 48080 Bilbao, Spain; [email protected]
Get access

Abstract

We prove the interior and boundary null-controllability of some parabolic evolutions with controls acting over measurable sets.

Keywords

Null-controllability
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 1 , January 2013 , pp. 239 - 254
Copyright
© EDP Sciences, SMAI, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ahlfors, L. and Bers, L., Riemann’s mapping theorem for variable metrics. Ann. Math. 72 (1960) 265–296. Google Scholar
Alessandrini, G. and Escauriaza, L., Null-controllability of one-dimensional parabolic equations. ESAIM : COCV 14 (2008) 284–293. Google Scholar
Astala, K., Area distortion under quasiconformal mappings. Acta Math. 173 (1994) 37–60. Google Scholar
Benabdallah, A. and Naso, M.G., Null controllability of a thermoelastic plate. Abstr. Appl. Anal. 7 (2002) 585–599. Google Scholar
Benabdallah, A., Dermenjian, Y. and Le Rousseau, J., On the controllability of linear parabolic equations with an arbitrary control location for stratified media. C. R. Acad. Sci. Paris, Sér. 1 344 (2007) 357–362. Google Scholar
L. Bers and L. Nirenberg, On a representation theorem for linear elliptic systems with discontinuous coefficients and applications, in Convegno Internazionale sulle Equazioni alle Derivate Parziali. Cremonese, Roma (1955) 111–138.
L. Bers, F. John and M. Schechter, Partial Differential Equations. Interscience. New York (1964).
Cho, S., Dong, H. and Kim, S., Global estimates for Green’s matrix of second order parabolic systems with application to elliptic systems in two dimensional domains. Potential Anal. 36 (2012) 339–372. Google Scholar
L.C. Evans, Partial differential equations. American Mathematical Society, Providence, RI (1998).
A. Fursikov and O.Yu. Imanuvilov, Controllability of Evolution Equations. Seoul National University, Korea. Lect. Notes Ser. 34 (1996).
M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems. Princeton University Press (1983).
D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition. Springer-Verlag (1983).
F. John, Plane Waves and Spherical Means Applied to Partial Differential Equations. Interscience Publishers, Inc., New York (1955).
F. John, Partial Differential Equations. Springer-Verlag, New York (1982).
Léautaud, M., Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems. J. Funct. Anal. 258 (2010) 2739–2778. Google Scholar
Lebeau, G. and Robbiano, L., Contrôle exact de l’équation de la chaleur. Commun. Partial Differ. Equ. 20 (1995) 335–356. Google Scholar
Lebeau, G. and Zuazua, E., Null controllability of a system of linear thermoelasticity. Arch. Rational Mech. Anal. 141 (1998) 297–329. Google Scholar
Malinnikova, E., Propagation of smallness for solutions of generalized Cauchy-Riemann systems. Proc. Edinb. Math. Soc. 47 (2004) 191–204. Google Scholar
A.I. Markushevich, Theory of Functions of a Complex Variable. Prentice Hall, Englewood Cliffs, NJ (1965).
Miller, L., On the controllability of anomalous diffusions generated by the fractional laplacian. Math. Control Signals Syst. 3 (2006) 260–271. Google Scholar
C.B. Morrey, Multiple Integrals in the Calculus of Variations. Springer (1966).
Morrey, C.B. and Nirenberg, L., On the analyticity of the solutions of linear elliptic systems of partial differential equations. Commun. Pure Appl. Math. X (1957) 271–290. Google Scholar
Nadirashvili, N.S., A generalization of Hadamard’s three circles theorem. Mosc. Univ. Math. Bull. 31 (1976) 30–32. Google Scholar
Nadirashvili, N.S., Estimation of the solutions of elliptic equations with analytic coefficients which are bounded on some set. Mosc. Univ. Math. Bull. 34 (1979) 44–48. Google Scholar
J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations. ESAIM : COCV, doi:10.1051/cocv/2011168. CrossRef
Le Rousseau, J. and Robbiano, L., Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces. Invent. Math. 183 (2011) 245–336. Google Scholar
Russel, D.L., A unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Stud. Appl. Math. 52 (1973) 189–221. Google Scholar
Vessella, S., A continuous dependence result in the analytic continuation problem. Forum Math. 11 (1999) 695–703. Google Scholar
H.F. Weinberger, A first course in partial differential equations with complex variables and transform methods. Dover Publications, New York (1995).