J Glob Optim (2008) 40:169–174 DOI 10.1007/s10898-007-9186-5 On the perturbation of the observability equation in linear control systems Mohamed Ait Mansour Received: 29 May 2007 / Accepted: 29 May 2007 / Published online: 31 July 2007 © Springer Science+Business Media, LLC 2007 Abstract The aim of this paper is to investigate stability and sensitivity of the observability variable in linear control systems, (LCS) for short. We first present two results of Hölder con- tinuity in the abstract framework of the ordinary differential equation initial-value problem x ′ (t ) = f (t , x (t )), x (t 0 ) = x 0 . Afterwards, we apply our results to automatic systems, providing henceforth the sharpest bounds for the parametric input-output relation in LCS. Keywords Linear control systems · Ordinary differential equations · Perturbation · Observability · Quantitative stability 1 Introduction The classical Cauchy-Lipschitz problem of ordinary differential equation (ODE) initial-value problem x ′ (t ) = f (t , x (t )), x (t 0 ) = x 0 is considered. One of the fashion topics in this prob- lem concerns sensitivity of the solutions. In regard to perturbation on the initial value, results are now available. In this respect we refer the reader to the recent paper by Jong-Shi Pang and David Stewart [4] and references therein. In some systems, some external parameters may perturb their states. A typical instance in this context we can give is the input-output relation in linear control where all of the data in both evolution and observability equations are permanently subject to non-negligible change. In this case, one can possibly convert the question to one of change of initial conditions, by augmenting the given state variables of the ODE, which may work pretty well and fit many sensitivity problems to a nice abstract unifying framework. However, we believe that, for heterogeneity considerations, the appli- cability of these eventual results-that can be obtained via change state variable approach-may cause some hard time for some engineers and people working in automatic at the practical level. They actually need to separate parameters and compute the sharpest error bound for each parameter to obtain a better performance of their systems. The present note poses new M. A. Mansour (B ) Département de Mathématiques, Université de Perpignan, 52, Av Paul Alduy, 66860 Perpignan, France e-mail: webmaster@aitmansour.com 123