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)
Explicit observability estimate for the wave equation with potential and its application. (English) Zbl 0976.93038

The author obtains an observability estimate of the type

|w 0 | L 2 (Ω) 2 +|w 1 | H -1 (Ω) 2 K 0 T ω |w| 2 dxdt,

where w denotes the weak solution of the problem w '' -w=q(t,x)w in Q=(0,T)×Ω, w=0 on Σ=(0,T)×Ω, w(0)=w 0 ,w ' (0)=w 1 in Ω. The potential q is assumed essentially bounded in Q and ω denotes a subdomain of the bounded domain Ω in n with C 1,1 boundary Ω. The main novelty of the paper is the explicit estimate of the constant K with respect to =|q| L (Q) : in fact, it is proved that K=O(exp(exp(exp()))) as . As usual, this type of estimates can be applied to get the exact internal controllability in H 0 1 (Ω)×L 2 (Ω) at (sufficiently large) time T of the semilinear wave equation y '' -y=f(y)+χ ω (x)u(t,x) in Q, where fC 1 () with f ' L (), by choosing the control u in L 2 (Q).

MSC:
93C20Control systems governed by PDE
93B07Observability
93B05Controllability
35L05Wave equation (hyperbolic PDE)