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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.
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