zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Controllability of the heat equation with memory. (English) Zbl 0990.93008
The present paper is concerned with the controllability of the equation $$\cases y_t(x,t)- \gamma\Delta y(x,t)- \int^t_0 a(t-s) \Delta y(x,s) ds= m(x) u(x,y)\text{ in }Q,\\ y(x,0)= y_0(x)\quad\text{in }\Omega,\quad y(x,t)= 0\quad\text{in }\Sigma,\endcases\tag 1.1$$ where $\gamma> 0$, $\Omega$ is an open bounded subset of $\bbfR^n$ with the boundary $\partial\Omega$ of class $C^2$, $Q=\Omega\times (0,T)$, $\Sigma= \partial\Omega\times (0, T)$, $m(\cdot)$ is the characteristic function of an open subset $\omega$ of $\Omega$, and $a\in C^\infty(0,+\infty)$ is a locally integrable completely monotone kernel. The main result tells us that under some assumptions related to the kernel $a$, the problem (1.1) is approximately controllable. In particular, the approximate boundary controllability of the problem follows: $$\cases y_t(x,t)- \gamma\Delta y(x,t)- \int^t_0 a(t-s)\Delta y(x,s) ds= 0\text{ in }Q,\\ y(x,0)= y_0(x)\quad\text{in }\Omega,\quad y(x,t)= u(x,t)\quad\text{in }\Sigma.\endcases$$ In the last section the controllability of the one-dimensional linear viscoelasticity equation is studied: $$\cases y_t(x,t)- \int^t_0 a(t-s) y_{xx}(x, s) ds= m(x) u(x,t),\ (x,t)\in Q,\\ y(0,t)= y(\ell,t)= 0,\quad t\in (0,T),\quad y(x,0)= y_0(x),\quad x\in (0,1),\endcases$$ where $Q= (0,\ell)\times (0,T)$, $\omega= (a_1,a_2)\subset (0,\ell)$ and $a(t)\in C[0,+\infty)\cap C^\infty (0,+\infty)$, $(-1)^j a^{(j)}(t)\ge 0$, $t> 0$, $j= 0,1,\dots$ .

35B37PDE in connection with control problems (MSC2000)
35K05Heat equation
35R10Partial functional-differential equations
93C20Control systems governed by PDE