zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Null and approximate controllability for weakly blowing up semilinear heat equations. (English) Zbl 0970.93023
Summary: We consider the semilinear heat equation in a bounded domain of $\bbfR^d$, with control on a subdomain and homogeneous Dirichlet boundary conditions. We prove that the system is null-controllable at any time provided a globally defined and bounded trajectory exists and the nonlinear term $f(y)$ is such that $|f(s)|$ grows slower than $|s|\log^{3/2}(1+|s|)$ as $|s|\to\infty$. For instance, this condition is fulfilled by any function $f$ growing at infinity like $|s|\log^p(1+|s|)$ with $1< p<3/2$ (in this case, in the absence of control, blow-up occurs). We also prove that, for some functions $f$ that behave at infinity like $|s|\log^p(1+|s|)$ with $p> 2$, null controllability does not hold. The problem remains open when $f$ behaves at infinity like $|s|\log^p(1+|s|)$, with $3/2\le p\le 2$. Results of the same kind are proved in the case of approximate controllability.

MSC:
93C20Control systems governed by PDE
93B05Controllability
93C10Nonlinear control systems
35B37PDE in connection with control problems (MSC2000)
WorldCat.org
Full Text: DOI Numdam EuDML
References:
[1] Anita, S.; Barbu, V.: Internal null controllability of nonlinear heat equation, preprint. (1999)
[2] Barbu, V.: Exact controllability of the superlinear heat equation, preprint. (1998)
[3] Brezis, H.; Cazenave, Th; Martel, Y.; Ramiandrisoa, A.: Blow up for ut-${\Delta}u=g(u)$ revisited. Adv. differential equations 1, No. 1, 73-90 (1996) · Zbl 0855.35063
[4] Cannarsa P.M., Komornik V., Loreti P., Well posedness and control of the semilinear wave equation with iterated logarithms, ESAIM: COCV (to appear) · Zbl 0959.93026
[5] Cazenave, T.; Haraux, A.: Equations d’évolution avec nonlinéarité logarithmique. Ann. fac. Sci. Toulouse 2, 21-51 (1980) · Zbl 0411.35051
[6] Cazenave, T.; Haraux, A.; Barbu, V.: Introduction aux problèmes d’évolution semilinéaires. Mathématiques & applications (1989)
[7] Fabre, C.; Puel, J. P.; Zuazua, E.: Approximate controllability of the semilinear heat equation. Proc. royal soc. Edinburgh 125A, 31-61 (1995) · Zbl 0818.93032
[8] Fernández-Cara, E.: Null controllability of the semilinear heat equation. Esaim: cocv 2, 87-107 (1997) · Zbl 0897.93011
[9] Fernández-Cara E., Zuazua E., The cost of approximate controllability for heat equations: The linear case, Adv. Differential Equations (to appear) · Zbl 1007.93034
[10] Fursikov, A.; Imanuvilov, O. Yu: Controllability of evolution equations. Lecture notes 34 (1996) · Zbl 0862.49004
[11] Galaktionov, V.: On blow-up and degeneracy for the semilinear heat equation with source. Proc. royal soc. Edinburgh 115A, 19-24 (1990) · Zbl 0715.35006
[12] Galaktionov, V.; Vázquez, J. L.: Regional blow-up in a semilinear heat equation with convergence to a Hamilton--Jacobi equation. SIAM J. Math. anal 24, No. 5, 1125-1276 (1993) · Zbl 0813.35033
[13] Glass, O.: Sur la contrôlabilité des fluides parfaits incompressibles, ph.d. Thesis. (January 2000)
[14] Henry, J.: Contrôle d’un réacteur enzymatique à l’aide de modèles à paramètres distribués: quelques problèmes de contrôlabilité de systèmes paraboliques, ph.d. Thesis. (1978)
[15] Imanuvilov, O. Yu: Exact boundary controllability of the parabolic equation. Russian math. Surveys 48, 211-212 (1993)
[16] Khapalov, A.: Some aspects of the asymptotic behavior of the solutions of the semilinear heat equation and approximate controllability. J. math. Anal. appl. 194, 858-882 (1995) · Zbl 0846.35059
[17] Zuazua, E.: Exact boundary controllability for the semilinear wave equation. Nonlinear partial differential equations and their applications, vol. X, 357-391 (1991)
[18] Zuazua, E.: Finite dimensional null-controllability of the semilinear heat equation. J. math. Pures et appl. 76, 237-264 (1997) · Zbl 0872.93014
[19] Zuazua, E.: Exact controllability for the semilinear wave equation in one space dimension. Ann. IHP, analyse non linéaire 10, 109-129 (1996)