# 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)
${L}^{\infty }$-null controllability for the heat equation and its consequences for the time optimal control problem. (English) Zbl 1165.93016
Summary: We establish a certain ${L}^{\infty }$-null controllability for the internally controlled heat equation in ${\Omega }×\left[0,T\right]$, with the control restricted to a product set of an open nonempty subset in ${\Omega }$ and a subset of positive measure in the interval $\left[0,T\right]$. Based on this, we obtain a bang-bang principle for the time optimal control of the heat equation with controls taken from the set ${𝒰}_{\text{ad}}=\left\{u\left(·,t\right):\left[0,\infty \right)\to {L}^{2}\left({\Omega }\right)$ measurable; $u\left(·,t\right)\in U,$ a.e. in $t\right\}$, where $U$ is a closed and bounded subset of ${L}^{2}\left({\Omega }\right)$. Namely, each optimal control ${u}^{*}\left(·,t\right)$ of the problem satisfies necessarily the bang-bang property: ${u}^{*}\left(·,t\right)\in \partial U$ for almost all $t\in \left[0,{T}^{*}\right]$, where $\partial U$ denotes the boundary of the set $U$ and ${T}^{*}$ is the optimal time. We also get the uniqueness of the optimal control when the target set $S$ is convex and the control set $U$ is a closed ball.
##### MSC:
 93B05 Controllability 93C35 Multivariable systems, multidimensional control systems 93C05 Linear control systems 35K05 Heat equation 49J30 Optimal solutions belonging to restricted classes (existence)