On integral inequalities and applications to the stability of some systems distributed non dissipative

First we prove some new integral inequalities to obtain a precise estimate on behavior at infinity of a positive and not necessarily decreasing functon. This extends in many directions and improves in certain cases some integral inequalities due to A. Haraux, V. Komornik, P. Martinez, M. Eller et al. and F. Alabau-Boussouira concerning decreasing functions. Then we give applications to (internal or boundary, linear or nonlinear) stabilization of certain nondissipative distributed systems, which improve and generalize many stabilization results known in the dissipative case. The variety of systems considered proves that the method developed in this paper is direct and very flexible; it can be applied to various nondissipative problems and it allows one to obtain general estimates (exponential, polynomial, logarithmic or other) of stability. The main results, of this paper constituted the first part of my HDR thesis presented in 2006 at Paul Verlaine Metz University, some of them had been announced without proof in C. R. Acad. Sci. Paris and under less restrictive conditions in SIAM J. Control Optim., and are investigated in this paper in more detail.