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)
Optimal Neumann control for the two-dimensional steady-state Navier-Stokes equations. (English) Zbl 1200.35214
Fursikov, Andrei V. (ed.) et al., New directions in mathematical fluid mechanics. The Alexander V. Kazhikhov memorial volume. Boston, MA: Birkhäuser (ISBN 978-3-0346-0151-1/hbk). Advances in Mathematical Fluid Mechanics, 193-221 (2010).
The paper focuses on an optimal control problem for a 2D stationary Navier-Stokes system. The authors consider a domain Ω of 2 consisting of a rectangle without a subset bounded by a smooth curve S. The Navier-Stokes system -Δv+v·v+p=0, ·v=0 is considered in Ω. Dirichlet or Neumann boundary conditions are imposed on the different pieces of the boundary of this domain. The optimal control problem consists to minimize the functional J= S n·σ·e 1 dx under the action of controls u 1 and u 2 imposed on subintervals of the horizontal parts of the boundary of the domain Ω through Neumann boundary conditions. Here e 1 is the first unit vector of 2 and n·σ=-pn+2𝒟(v)n, where n is the unit outer normal and 2𝒟(v)=( j v i + i v j ) i,j=1,2 . The applied controls are supposed to satisfy u 1 2 +u 2 2 γ 2 in the L 2 -norm of their respective domains, for some positive constant γ. The first main result of the paper proves the existence of a generalized solution of this optimal control problem. The authors start proving further properties of the generalized solution of the associated Stokes problem. They also prove the existence of a generalized solution of the Navier-Stokes problem assuming the existence of admissible collections (v,p,u 1 ,u 2 ) for this Navier-Stokes problem and that the boundary data are small enough. This is done building a contraction operator associated to this Navier-Stokes problem. The proof of the existence of an optimal solution is obtained rewriting the original problem as a minimization problem for a continuous functional on a compact set. The second main result of the paper establishes the optimality system for the optimal solution. The authors here use the abstract Lagrange principle framework. The paper ends with the presentation of some briefly described numerical simulations.
MSC:
35Q30Stokes and Navier-Stokes equations
76D55Flow control and optimization
76D05Navier-Stokes equations (fluid dynamics)
49J20Optimal control problems with PDE (existence)
49K20Optimal control problems with PDE (optimality conditions)
93C20Control systems governed by PDE