hal-03153733
https://ensta-bretagne.hal.science/hal-03153733
https://ensta-bretagne.hal.science/hal-03153733/document
https://ensta-bretagne.hal.science/hal-03153733/file/fractalfract-05-00017.pdf
doi:10.3390/fractalfract5010017
[UNIV-BREST] Université de Bretagne occidentale - Brest (UBO)
[INSTITUT-TELECOM] Institut Mines Télécom
[ENSTA-BRETAGNE] ENSTA Bretagne
[CNRS] CNRS - Centre national de la recherche scientifique
[UNIV-UBS] Université de Bretagne Sud
[ENSTA-BRETAGNE-STIC] Département STIC
[INSMI] CNRS-INSMI - INstitut des Sciences Mathématiques et de leurs Interactions
[ENIB] Ecole Nationale d'Ingénieurs de Brest
[LAB-STICC] Laboratoire des sciences et techniques de l'information, de la communication et de la connaissance
[TDS-MACS] Réseau de recherche en Théorie des Systèmes Distribués, Modélisation, Analyse et Contrôle des Systèmes
[INSTITUTS-TELECOM] composantes instituts telecom
[LAB-STICC_ROBEX] Equipe ROBotics for EXploration
Novel Techniques for a Verified Simulation of Fractional-Order Differential Equations
Rauh, Andreas
Jaulin, Luc
[MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS]
ART
fractional-order differential equations
interval methods
guaranteed enclosures
bounded uncertainty
Mittag-Leffler functions
nonlinearities
Verified simulation techniques have been investigated intensively by researchers who are dealing with ordinary and partial differential equations. Tasks that have been considered in this context are the solution to initial value problems and boundary value problems, parameter identification, as well as the solution of optimal control problems in cases in which bounded uncertainty in parameters and initial conditions are present. In contrast to system models with integer-order derivatives, fractional-order models have not yet gained the same attention if verified solution techniques are desired. In general, verified simulation techniques rely on interval methods, zonotopes, or Taylor model arithmetic and allow for computing guaranteed outer enclosures of the sets of solutions. As such, not only the influence of uncertain but bounded parameters can be accounted for in a guaranteed way. In addition, also round-off and (temporal) truncation errors that inevitably occur in numerical software implementations can be considered in a rigorous manner. This paper presents novel iterative and series-based solution approaches for the case of initial value problems to fractional-order system models, which will form the basic building block for implementing state estimation schemes in continuous-discrete settings, where the system dynamics is assumed as being continuous but measurements are only available at specific discrete sampling instants.
2021
2021-02-26
en
Fractal and Fractional
MDPI