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)
L -norm minimal control of the wave equation: on the weakness of the bang-bang principle. (English) Zbl 1133.49006
Summary: For optimal control problems with ordinary differential equations where the L -norm of the control is minimized, often bang-bang principles hold. For systems that are governed by a hyperbolic partial differential equation, the situation is different: even if a weak form of the bang-bang principle still holds for the wave equation, it implies no restriction on the form of the optimal control. To illustrate that for the Dirichlet boundary control of the wave equation in general not even feasible controls of bang-bang type exist, we examine the states that can be reached by bang-bang-off controls, that is controls that are allowed to attain only three values: Their maximum and minimum values and the value zero. We show that for certain control times, the difference between the initial and the terminal state can only attain a finite number of values. For the problems of optimal exact and approximate boundary control of the wave equation where the L -norm of the control is minimized, we introduce dual problems and present the weak form of a bang-bang principle, that states that the values of L -norm minimal controls are constrained by the sign of the dual solutions. Since these dual solutions are in general given as measures, this is no restriction on the form of the control function: the dual solution may have a finite support, and when the dual solution vanishes, the control is allowed to attain all values from the interval between the two extremal control values.
49J30Optimal solutions belonging to restricted classes (existence)
49K20Optimal control problems with PDE (optimality conditions)
35L05Wave equation (hyperbolic PDE)
35B37PDE in connection with control problems (MSC2000)