7 research outputs found
Local feedback stabilisation to a non-stationary solution for a damped non-linear wave equation
We study a damped semi-linear wave equation in a bounded domain with smooth
boundary. It is proved that any sufficiently smooth solution can be stabilised
locally by a finite-dimensional feedback control supported by a given open
subset satisfying a geometric condition. The proof is based on an investigation
of the linearised equation, for which we construct a stabilising control
satisfying the required properties. We next prove that the same control
stabilises locally the non-linear problem.Comment: 29 page
Exact controllability for multidimensional semilinear hyperbolic equations
In this paper, we obtain a global exact controllability result for a class of multidimensional semilinear hyperbolic equations with a superlinear nonlinearity and variable coefficients. For this purpose, we establish an observability estimate for the linear hyperbolic equation with an unbounded potential, in which the crucial observability constant is estimated explicitly by a function of the norm of the potential. Such an estimate is obtained by a combination of a pointwise estimate and a global Carleman estimate for the hyperbolic differential operators and analysis on the regularity of the optimal solution to an auxiliary optimal control problem
Exact Controllability and Stabilization of the Wave Equation
These Notes originated from a course I delivered at the Institute of
Mathematics of the Universidade Federal do Rio de Janeiro, Brazil (UFRJ) in
July-September 1989, were initially published in 1989 in Spanish under the
title "Controlabilidad Exacta y Estabilizaci\'on de la Ecuaci\'on de Ondas" in
the Lecture Notes Series of the Institute.
Despite the significant evolution of the topic over the last three decades, I
believe that the text, with its synthetic presentation of fundamental tools in
the field, remains valuable for researchers in the area, especially for younger
generations. It is written from the perspective of the young mathematician I
was when I authored the Notes, needing to learn many things in the process and,
therefore, taking care to develop details often left to the reader or not
readily available elsewhere.
These Notes were written one year after completing my PhD at the Universit\'e
Pierre et Marie Curie in Paris and drafting the lectures of Professor
Jacques-Louis Lions at Coll\`ege de France in the academic year 1986-1987,
later published as a book in 1988. Parts of these Notes offer a concise
presentation of content developed in more detail in that book, supplemented by
work on the decay of dissipative wave equations during my PhD under the
supervision of Professor Alain Haraux in Paris