533 research outputs found
A global Nullstellensatz for ideals of Denjoy-Carleman functions
We prove a Nullstellensatz result for global ideals of Denjoy-Carleman
functions in both finitely generated and infinitely generated cases.Comment: 5 page
Semilinear nonautonomous parabolic equations with unbounded coefficients in the linear part
We study the Cauchy problem for the semilinear nonautonomous parabolic
equation in ,
, in the spaces and in . Here is a Borel measure defined via a
tight evolution system of measures for the evolution operator
associated to the family of time depending second order uniformly elliptic
operators . Sufficient conditions for existence in the large
and stability of the null solution are also given in both and
contexts. The novelty with respect to the literature is that the coefficients
of the operators are allowed to be unbounded
A theory of the infinite horizon LQ-problem for composite systems of PDEs with boundary control
We study the infinite horizon Linear-Quadratic problem and the associated
algebraic Riccati equations for systems with unbounded control actions. The
operator-theoretic context is motivated by composite systems of Partial
Differential Equations (PDE) with boundary or point control. Specific focus is
placed on systems of coupled hyperbolic/parabolic PDE with an overall
`predominant' hyperbolic character, such as, e.g., some models for
thermoelastic or fluid-structure interactions. While unbounded control actions
lead to Riccati equations with unbounded (operator) coefficients, unlike the
parabolic case solvability of these equations becomes a major issue, owing to
the lack of sufficient regularity of the solutions to the composite dynamics.
In the present case, even the more general theory appealing to estimates of the
singularity displayed by the kernel which occurs in the integral representation
of the solution to the control system fails. A novel framework which embodies
possible hyperbolic components of the dynamics has been introduced by the
authors in 2005, and a full theory of the LQ-problem on a finite time horizon
has been developed. The present paper provides the infinite time horizon
theory, culminating in well-posedness of the corresponding (algebraic) Riccati
equations. New technical challenges are encountered and new tools are needed,
especially in order to pinpoint the differentiability of the optimal solution.
The theory is illustrated by means of a boundary control problem arising in
thermoelasticity.Comment: 50 pages, submitte
On globally defined semianalytic sets
In this work we present the concept of -semianalytic subset of a real
analytic manifold and more generally of a real analytic space. -semianalytic
sets can be understood as the natural generalization to the semianalytic
setting of global analytic sets introduced by Cartan (-analytic sets for
short). More precisely is a -semianalytic subset of a real analytic
space if each point of has a neighborhood such
that is a finite boolean combinations of global analytic equalities
and strict inequalities on . By means of paracompactness -semianalytic
sets are the locally finite unions of finite boolean combinations of global
analytic equalities and strict inequalities on .
The family of -semianalytic sets is closed under the same operations as
the family of semianalytic sets: locally finite unions and intersections,
complement, closure, interior, connected components, inverse images under
analytic maps, sets of points of dimension , etc. although they are defined
involving only global analytic functions. In addition, we characterize
subanalytic sets as the images under proper analytic maps of -semianalytic
sets.
We prove also that the image of a -semianalytic set under a proper
holomorphic map between Stein spaces is again a -semianalytic set. The
previous result allows us to understand better the structure of the set
of points of non-coherence of a -analytic subset of a real analytic
manifold . We provide a global geometric-topological description of
inspired by the corresponding local one for analytic sets due to
Tancredi-Tognoli (1980), which requires complex analytic normalization. As a
consequence it holds that is a -semianalytic set of dimension
.Comment: 32 pages, 3 figure
- …