**2. Fundamentals of Verified Mittag-Leffler-Type Pseudo-State Enclosures for Fractional Differential Equations**

*2.1. System Models under Consideration*

Throughout this article, we analyze and derive set-valued simulation approaches for commensurate fractional-order differential equations of the form

$$\mathbf{x}^{(v)}(t) = \mathbf{f}(\mathbf{x}(t)) \; , \; \mathbf{f} : \mathbb{R}^n \mapsto \mathbb{R}^n \; , \tag{1}$$

with the order 0 < *ν* ≤ 1, where the right-hand side **f x**(*t*) is assumed to be given by a continuous function. Moreover, we assume that the system model (1) is initialized by the pseudo state **x**(0) at the time instant *t* = 0, where **x**(*t*) = **x**(0) holds for all *t* < 0. This case corresponds to the Caputo definition of fractional derivatives as described, for example, in [1,2].

To account for uncertainty in the pseudo-state initialization, we use the interval representation

$$\mathbf{x}(0) \in [\mathbf{x}](0) := \begin{bmatrix} \underline{\mathbf{x}}(0) \ \vdots \ \overline{\mathbf{x}}(0) \end{bmatrix} \,, \tag{2}$$

for which *xi*(0) ≤ *xi*(0) holds for each vector component *i* ∈ {1, ... , *n*}. Note that the property of temporally constant initializations for *t* < 0 is still assumed to hold.
