Toward the Development of Iteration Procedures for the Interval-Based Simulation of Fractional-Order Systems

Abstract

In recent years, numerous interval-based simulation techniques have been developed which allow for a verified computation of outer interval enclosures for the sets of reachable states of dynamic systems represented by finitedimensional sets of ordinary differential equations (ODEs). Here, especially the evaluation of IVPs is of interest, when both the systems’ initial conditions and parameters can only be defined by finitely large domains, often represented by interval boxes. Suitable simulation techniques make use of series expansions of the solutions of IVPs with respect to time and (possibly) the uncertain initial conditions as well as of verified Runge-Kutta techniques. Solution sets are then typically represented by means of multi-dimensional intervals, zonotopes, ellipsoids, or Taylor models, cf. [5]. In most of these approaches, variants of the Picard iteration [1] are involved, which either determine the sets of possible solutions or at least worst-case outer enclosures with which time discretization errors are quantified. An example for a solution routine based entirely on this iteration is the exponential enclosure technique published in [9] and the references therein. It is applicable to systems with nonoscillatory and oscillatory behavior if the solution of the IVP of interest shows an asymptotically stable behavior. For non-oscillatory