Online parameter identification in time-dependent differential equations as a non-linear inverse problem

Online parameter identification in time-dependent differential equations from time course observations related to the physical state can be understood as a non-linear inverse and ill-posed problem and appears in a variety of applications in science and engineering. The feature as well as the challenge of online identification is that sensor data have to be continuously processed during the operation of the real dynamic process in order to support simulation-based decision making. In this paper we present an online parameter identification method that is based on a non-linear parameter-to-output operator and, as opposed to methods available so far, works both for finite- and infinite-dimensional dynamical systems, e.g., both for ordinary differential equations and time-dependent partial differential equations. A further advantage of the method suggested is that it renders typical restrictive assumptions such as full state observability, linear parametrisation of the underlying model and data differentiation or filtering unnecessary. Assuming existence of a solution for exact data, a convergence analysis based on Lyapunov theory is presented. Numerical illustrations given are by means of online identification both of aerodynamic coefficients in a 3DoF-longitudinal aircraft model and of a (distributed) conductivity coefficient in a heat equation.