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This monograph explores applications of Carleman estimates in the
study of stabilization and controllability properties of partial
differential equations, including quantified unique continuation,
logarithmic stabilization of the wave equation, and
null-controllability of the heat equation. Where the first
volume derived these estimates in regular open sets in Euclidean
space and Dirichlet boundary conditions, here they are extended to
Riemannian manifolds and more general boundary conditions. The book
begins with the study of Lopatinskii-Sapiro boundary conditions for
the Laplace-Beltrami operator, followed by derivation of Carleman
estimates for this operator on Riemannian manifolds.Â
Applications of Carleman estimates are explored next: quantified
unique continuation issues, a proof of the logarithmic
stabilization of the boundary-damped wave equation, and a spectral
inequality with general boundary conditions to derive the
null-controllability result for the heat equation. Two additional
chapters consider some more advanced results on Carleman
estimates. The final part of the book is devoted to
exposition of some necessary background material: elements of
differential and Riemannian geometry, and Sobolev spaces and
Laplace problems on Riemannian manifolds.
This monograph explores applications of Carleman estimates in the
study of stabilization and controllability properties of partial
differential equations, including the stabilization property of the
damped wave equation and the null-controllability of the heat
equation. All analysis is performed in the case of open sets
in the Euclidean space; a second volume will extend this treatment
to Riemannian manifolds. The first three chapters illustrate the
derivation of Carleman estimates using pseudo-differential calculus
with a large parameter. Continuation issues are then
addressed, followed by a proof of the logarithmic stabilization of
the damped wave equation by means of two alternative proofs of the
resolvent estimate for the generator of a damped wave
semigroup. The authors then discuss null-controllability of
the heat equation, its equivalence with observability, and how the
spectral inequality allows one to either construct a control
function or prove the observability inequality. The final
part of the book is devoted to the exposition of some necessary
background material: the theory of distributions, invariance under
change of variables, elliptic operators with Dirichlet data and
associated semigroup, and some elements from functional analysis
and semigroup theory.
This brief provides unified methods for the stabilization of some
fractional evolution systems, nicely complementing existing
literature on fractional calculus. The volume is divided into three
chapters, the first of which considers the stabilization for some
abstract evolution equations with a fractional damping, the second
of which validates the abstract results of chapter 1 on concrete
examples, and the third of which studies the stabilization of
fractional evolution systems with memory.
This monograph explores applications of Carleman estimates in the
study of stabilization and controllability properties of partial
differential equations, including quantified unique continuation,
logarithmic stabilization of the wave equation, and
null-controllability of the heat equation. Where the first volume
derived these estimates in regular open sets in Euclidean space and
Dirichlet boundary conditions, here they are extended to Riemannian
manifolds and more general boundary conditions. The book begins
with the study of Lopatinskii-Sapiro boundary conditions for the
Laplace-Beltrami operator, followed by derivation of Carleman
estimates for this operator on Riemannian manifolds. Applications
of Carleman estimates are explored next: quantified unique
continuation issues, a proof of the logarithmic stabilization of
the boundary-damped wave equation, and a spectral inequality with
general boundary conditions to derive the null-controllability
result for the heat equation. Two additional chapters consider some
more advanced results on Carleman estimates. The final part of the
book is devoted to exposition of some necessary background
material: elements of differential and Riemannian geometry, and
Sobolev spaces and Laplace problems on Riemannian manifolds.
This monograph explores applications of Carleman estimates in the
study of stabilization and controllability properties of partial
differential equations, including the stabilization property of the
damped wave equation and the null-controllability of the heat
equation. All analysis is performed in the case of open sets in the
Euclidean space; a second volume will extend this treatment to
Riemannian manifolds. The first three chapters illustrate the
derivation of Carleman estimates using pseudo-differential calculus
with a large parameter. Continuation issues are then addressed,
followed by a proof of the logarithmic stabilization of the damped
wave equation by means of two alternative proofs of the resolvent
estimate for the generator of a damped wave semigroup. The authors
then discuss null-controllability of the heat equation, its
equivalence with observability, and how the spectral inequality
allows one to either construct a control function or prove the
observability inequality. The final part of the book is devoted to
the exposition of some necessary background material: the theory of
distributions, invariance under change of variables, elliptic
operators with Dirichlet data and associated semigroup, and some
elements from functional analysis and semigroup theory.
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