Computing Guaranteed Bounds for Uncertain Cooperative and Monotone Nonlinear Systems

Abstract

Models of biological or technical applications are represented by nonlinear systems, which are defined by ordinary differential equations. These systems generally contain multiple uncertain or unknown parameters. These uncertainties result from measurement errors or from modeling, e.g. numerical modeling. For several applications, the guaranteed enclosure of all possible solutions of an initial value problem (IVP) of a given uncertain system is demanded. In general the calculation of guaranteed bounds of the given uncertain nonlinear system cannot be done directly, because the solution set of an IVP can be solved algebraically only in certain cases. Furthermore, most numerical methods which compute the solution of IVPs cannot handle systems with uncertain parameters. But for the class of cooperative systems tight guaranteed bounds for all solutions of the IVP can be computed. This class satisfies certain monotony conditions. Moreover the computation of guaranteed lower and upper bounds can be applied to a larger class of ordinary differential equations, which does not satisfy all conditions for uncertain cooperative systems. For this class of monotone systems the guaranteed enclosure can show some overestimation. Some examples illustrate the methods described in this contribution.