Computing Enclosures for the Matrix Mittag–Leffler Function

Quantification of Time-Domain Truncation Errors for the Reinitialization of Fractional Integrators

“…In fact, the Grünwald-Letnikov discretization can be interpreted as a generalization of the wellknown Euler discretization scheme for integer-order models so that the true state evolution can be approximated accurately for sufficiently small values of ∆T k . For methods that allow a rigorous quantification of time discretization errors, the reader is referred to [27][28][29], where an exponential state enclosure technique is generalized to fractional models by using an iteration scheme exploiting an interval extension of Mittag-Leffler functions [10,19], or to [1,18,26] where series expansion approaches and Picard iteration schemes were generalized to the fractional case.…”

“…Using this modification, the simulation is continued after the point t = t k + T for the differential inclusion model defined by the expression f x(t) and the pseudo state values x(t k ) as initial condition, while the entire past for t < t k is no longer required for a further system simulation. Under the assumption of cooperativity of the state equations, see [3,5,25,33] for further details, independent lower and upper bounding trajectories can be extracted from the modified system model ( 19) so that set-based integration routines such as the one based on interval extensions of the Mittag-Leffler function from [27][28][29] can be avoided when solving the corresponding initial value problem for the differential inclusion problem (19) after the inflation of the right-hand side f x(t) of the original system. Remark 2.…”

“…Remark 2. To limit the pessimism introduced by the additive error bounds in (19), the following two aspects should be accounted for:…”