# 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)
Approximate controllability of the semilinear heat equation. (English) Zbl 0818.93032
Summary: This article is concerned with the study of approximate controllability for the semilinear heat equation in a bounded domain ${\Omega }$ when the control acts on any open and nonempty subset of ${\Omega }$ or on a part of the boundary. In the case of both an internal and a boundary control, the approximate controllability in ${L}^{p}\left({\Omega }\right)$ for $1\le p<+\infty$ is proved when the nonlinearity is gobally Lipschitz with a control in ${L}^{\infty }$. In the case of the interior control, we also prove approximate controllability in ${C}_{0}\left({\Omega }\right)$. The proof combines a variational approach to the controllability problem for linear equations and a fixed point method. We also prove that the control can be taken to be of “quasi bang-bang” form.

##### MSC:
 93C20 Control systems governed by PDE 35K60 Nonlinear initial value problems for linear parabolic equations 93C10 Nonlinear control systems 93B05 Controllability