*2.2. Linear Scalar System Models*

For the special case of scalar fractional differential equations

$$\mathbf{x}^{(v)}(t) = \boldsymbol{\lambda} \cdot \mathbf{x}(t) \quad \text{with} \quad \mathbf{x}(0) = \mathbf{x}\_0 \tag{3}$$

which are linear in the pseudo state *<sup>x</sup>*(*t*) with the parameter *<sup>λ</sup>* <sup>∈</sup> <sup>R</sup> and which are initialized according to the previous subsection in the Caputo-type sense, it is well-known according to [20,21] that the exact solution is given in the form

$$\mathbf{x}(t) = E\_{\nu,1}(\lambda t^{\nu}) \cdot \mathbf{x}(0) \ . \tag{4}$$

In (4), *Eν*,*β*(*ζ*) is the two-parameter Mittag-Leffler function. It is defined by the infinite series

$$E\_{\nu,\beta}(\zeta) = \sum\_{i=0}^{\infty} \frac{\zeta^i}{\Gamma(\nu i + \beta)}\tag{5}$$

for the general argument *<sup>ζ</sup>* <sup>∈</sup> <sup>C</sup>. In (4), <sup>Γ</sup> *νi* + *β* denotes the gamma function of the respective argument and *ν* is the derivative order as introduced in (1). To obtain the solution in (4), the parameter *β* is set to the value *β* = 1.

**Remark 1.** *The classical exponential function e<sup>ζ</sup> is obtained as a special case of the Mittag-Leffler function (5) when setting ν* = 1 *and β* = 1*.*

**Remark 2.** *For an interval extension of the Mittag-Leffler function on the basis of the accurate floating-point* MATLAB *implementation by R. Garrappa [18,24], see [13].*

*2.3. Mittag-Leffler Functions as Pseudo-State Enclosures for Fractional-Order Differential Equations*

For nonlinear scalar and vector-valued system models (1), the Mittag-Leffler functions introduced in the previous subsection can be used to define guaranteed pseudo-state enclosures according to Definition 1.

**Definition 1** (Mittag-Leffler-type pseudo-state enclosure)**.** *A verified Mittag-Leffler-type pseudo-state enclosure for the system model (1) with (2) is defined by the time-dependent enclosure function*

$$\mathbf{x}^\*(t) \in [\mathbf{x}\_\ell](t) = \mathbb{E}\_{\nu, 1} \left( [\mathbf{A}] \cdot t^\nu \right) \cdot [\mathbf{x}\_\ell](0) \,, \ [\mathbf{x}\_\ell](0) = [\mathbf{x}\_0] \tag{6}$$

*with the diagonal parameter matrix* [**Λ**] := diag [*λi*] *, i* ∈ {1, ... , *n*}*, if it is determined according to Theorem 1. In (6), the generalization of the scalar Mittag-Leffler function Eν*,1 *to the matrix case* **E***ν*,1 *is given by the following diagonal matrix*

$$\mathbb{E}\_{\boldsymbol{\nu},1}\left(\left[\boldsymbol{\Lambda}\right]\cdot\boldsymbol{t}^{\boldsymbol{\nu}}\right) = \text{diag}\left\{ \left[E\_{\boldsymbol{\nu},1}\left(\left[\boldsymbol{\lambda}\_{1}\right]\cdot\boldsymbol{t}^{\boldsymbol{\nu}}\right) \quad \dots \quad E\_{\boldsymbol{\nu},1}\left(\left[\boldsymbol{\lambda}\_{n}\right]\cdot\boldsymbol{t}^{\boldsymbol{\nu}}\right) \right] \right\}.\tag{7}$$

**Theorem 1** ([6,13,25] Iteration for Mittag-Leffler-type enclosures)**.** *All reachable pseudo states* **x**∗(*T*) *are enclosed in accordance with Theorem 1 by the Mittag-Leffler-type pseudo-state enclosure*

$$\mathbf{x}^\*(T) \in [\mathbf{x}\_\ell](T) = \mathbf{E}\_{\nu, 1}([\mathbf{A}] \cdot T^\vee) \cdot [\mathbf{x}\_\ell](0) \tag{8}$$

*at the point of time t* = *T* > 0 *if* [**Λ**] *is set to the outcome of the converging iteration*

$$\left[\boldsymbol{\lambda}\_{i}\right]^{\langle\mathbf{x}+1\rangle} := \frac{f\_{i}\left(\mathbb{E}\_{\nu,1}\left([\mathbf{A}]^{\langle\mathbf{x}\rangle} \cdot [t]^{\nu}\right) \cdot [\mathbf{x}\_{\varepsilon}](0)\right)}{E\_{\nu,1}\left([\boldsymbol{\lambda}\_{i}]^{\langle\mathbf{x}\rangle} \cdot [t]^{\nu}\right) \cdot [\mathbf{x}\_{\varepsilon,i}](0)}\,,\tag{9}$$

*i* ∈ {1, ... , *n*}*, with the prediction horizon* [*t*] = [0 ; *T*]*. To ensure convergence, the value x*<sup>∗</sup> *<sup>i</sup>* = 0 *must not belong to the solution for any vector component i* ∈ {1, . . . , *n*}*.*

**Remark 3.** *Typically, the iteration according to Theorem 1 is initialized with intervals centered around the eigenvalues of the Jacobian of the right-hand side of (1), evaluated for the midpoint of the interval vector (2).*

As a preparation for the derivation of the exponential enclosure approach presented for the first time in this paper, the following proof of Theorem 1, according to [6,13,25], is summarized.

**Proof.** Formulate a Picard iteration (iteration index *κ*) for computing pseudo-state enclosures in the differential form

$$\left(\mathbf{x}^{(\nu)}\right)^{(\mathbf{x})}\left([0\ ;\ T]\right)\supset \left[\mathbf{x}^{(\nu)}\right]^{(\mathbf{x}+1)}\left([0\ ;\ T]\right) = \mathbf{f}\left([\mathbf{x}^{(\nu)}]^{(\mathbf{x})}\left([0\ ;\ T]\right)\right)\,,\tag{10}$$

where [**x**(*ν*)] *κ* [0 ; *T*] is an interval extension of the temporal derivative of order *ν* of the inclusion function [**x**] *κ* (*t*) over the time interval *t* ∈ [*t*] = [0 ; *T*]. Substituting the pseudo-state enclosure given in Definition 1 into (10) yields the expression

$$\begin{split} \mathbf{x}^{\left(\nu\right)}(t) &\in \left( [\mathbf{A}]^{\left(\kappa+1\right)} \right) \cdot \mathbb{E}\_{\nu,1} \left( [\mathbf{A}]^{\left(\kappa+1\right)} \cdot t^{\nu} \right) \cdot [\mathbf{x}\_{\epsilon}](0) \\ &= \mathbf{f} \left( \mathbb{E}\_{\nu,1} \left( [\mathbf{A}]^{\left(\kappa\right)} \cdot t^{\nu} \right) \cdot [\mathbf{x}\_{\epsilon}](0) \right) \quad \text{for all} \quad t \in \left[t\right] \ . \end{split} \tag{11}$$

A converging iteration implies the set-valued relation

$$\left[\mathbf{x}\_{\mathfrak{e}}\right]^{\langle\mathbf{x}+1\rangle}\left(\left[t\right]\right) \subset \left[\mathbf{x}\_{\mathfrak{e}}\right]^{\langle\mathbf{x}\rangle}\left(\left[t\right]\right) \quad , \tag{12}$$

which corresponds to the relation

$$[\lambda\_i]^{\langle \kappa+1 \rangle} \subset [\lambda\_i]^{\langle \kappa \rangle} \tag{13}$$

for the unknown intervals of the solution parameters *λi*.

Overapproximating the interval evaluation of the Mittag-Leffler-type enclosure

$$\mathbf{E}\left[\mathbf{x}\right]^{\langle\kappa+1\rangle}(t) = \mathbf{E}\_{\nu,1}\left(\left[\mathbf{A}\right]^{\langle\kappa+1\rangle} \cdot t^{\nu}\right) \cdot \left[\mathbf{x}\_{\mathfrak{e}}\right](0) \tag{14}$$

in the iteration step *κ* + 1 on the first line of (11) by the enclosure [**x***e*] *κ* ([*t*]) in the case of convergence, i.e., using the relation

$$\mathbb{E}\left( [\mathbf{A}]^{\langle \mathbf{x}+1\rangle} \right) \cdot \mathbb{E}\_{\mathbf{v}, \mathbf{1}} \Big( [\mathbf{A}]^{\langle \mathbf{x}+1\rangle} \cdot t^{\nu} \Big) \cdot [\mathbf{x}\_{\varepsilon}](0) \subset \left( [\mathbf{A}]^{\langle \mathbf{x}+1\rangle} \right) \cdot \mathbb{E}\_{\mathbf{v}, \mathbf{1}} \Big( [\mathbf{A}]^{\langle \mathbf{x}\rangle} \cdot t^{\nu} \Big) \cdot [\mathbf{x}\_{\varepsilon}](0) \,, \tag{15}$$

leads to the new iteration formula

$$\text{diag}\left\{ \left[ \tilde{\lambda}\_{i} \right]^{\langle \kappa+1 \rangle} \right\} \cdot \left[ \mathbf{x}\_{\mathfrak{c}} \right]^{\langle \kappa \rangle} (\left[ t \right]) = \mathbf{f} \left( \left[ \mathbf{x}\_{\mathfrak{c}} \right]^{\langle \kappa \rangle} (\left[ t \right]) \right) \; , \tag{16}$$

where

$$\text{diag}\left\{ \left[ \tilde{\lambda}\_{i} \right]^{(\kappa+1)} \right\} \supseteq \text{diag}\left\{ \left[ \lambda\_{i} \right]^{(\kappa+1)} \right\} \;. \tag{17}$$

Solving expression (16) for [*λ*˜ *<sup>i</sup>*] *κ*+1 with subsequent renaming of this parameter into [*λi*] *κ*+1 completes the proof of Theorem 1. For further details, the reader is referred to the references [6,13,25].

**Corollary 1.** *In the case that the fractional-order differential equations given in Equation (1) with the initial conditions (2) can be rewritten into the quasi-linear form*

$$\mathbf{x}^{(\upsilon)}(t) = \mathfrak{A}\left(\mathbf{x}(t)\right) \cdot \mathbf{x}(t) \quad \text{with} \quad 0 < \upsilon \le 1 \; \; \; \; \tag{18}$$

*with an equilibrium at* **x** = **0** *and the state-dependent matrix* **A x**(*t*) *, Theorem 1 simplifies to the iteration scheme*

$$\left[\left[\lambda\_{i}\right]^{\left(\mathbf{x}+\mathbf{1}\right)} := a\_{i\mathbf{i}}\left(\left[\mathbf{x}\_{\mathbf{c}}\right]^{\left(\mathbf{x}\right)}\left(\left[\mathbf{t}\right]\right)\right) + \sum\_{j=1 \atop j\neq i}^{n} \left\{ a\_{ij}\left(\left[\mathbf{x}\_{\mathbf{c}}\right]^{\left(\mathbf{x}\right)}\left(\left[\mathbf{t}\right]\right)\right) \cdot \frac{E\_{\mathbf{v},\mathbf{l}}\left(\left[\lambda\_{j}\right]^{\left(\mathbf{x}\right)} \cdot \left[\mathbf{t}\right]^{\mathbf{v}}\right)}{E\_{\mathbf{v},\mathbf{l}}\left(\left[\lambda\_{i}\right]^{\left(\mathbf{x}\right)} \cdot \left[\mathbf{t}\right]^{\mathbf{v}}\right)} \cdot \frac{\left[\mathbf{x}\_{\mathbf{c},j}\right]\left(\mathbf{0}\right)}{\left[\mathbf{x}\_{\mathbf{c},j}\right]\left(\mathbf{0}\right)}\right\}\right.\tag{19}$$

**Remark 4.** *In (19), the quotient of two Mittag-Leffler functions can usually only be simplified further for ν* = 1*. In all other cases, where ν* = 1*, overestimation due to the so-called dependency effect [23] (which arises due to multiple dependencies on common interval parameters) can be reduced by exploiting the monotonicity properties of the Mittag-Leffler function that were analyzed in detail in [13].*
