Controlabilidad de ecuaciones escalares parabólicas: análisis de diferentes técnicas

Abstract

The controllability of parabolic partial differential equations has been a topic of increasing interest in the last decades that it has attracted to good part of the mathematical community dedicated to the control. This type of equations and systems arise, for example, in the study of diverse phenomena related to the Physics, the Biology or the Chemistry. In this work we try to analyze different technologies that have allowed to give positive results of controllability of parabolic equations. The first positive results of controllability of parabolic equations were obtained in 1971 (Arch. Rational Mech. Anal. 43, 1971) y 1974 (Quart. Appl. Math. 32, 1974/75) por H.O. Fattorini y D.L. Russe. These proved the boundary controllability of the unidimensional heat equation by means of the method of the moments. The general results of controllability to zero of N-dimensional parabolic equations were obtained independently for G. Lebeau and L. Robbiano and for A. Fursikov in O. Imanuvilov in 1995 - 1996. The first work is valid for autonomous parabolic equations and the result obtains as consequence of a spectral inequality obtained as consequence of local inequalites Carleman’s. The second work is applicable to not autonomous parabolic equations and the positive result of controllability is obtained from inequalites of Carleman for the attached operator. In this work we analyze the different technologies developed by the authors mentioned previously to prove the controllability of parabolic equations. The analysis will begin for the method of the moments, will happen for the study of global inequality of Carleman for uniformly parabolic operators and will finish with Lebeau-Robbiano’s method.