*3.3. Iterative Pseudo-State Enclosures for Box-Type and Exponential Representations of Mittag-Leffler Functions*

Both box-type and exponential enclosures are used in this subsection to evaluate the iteration Formula (19) of Corollary 1. In this subsection, an interval subdivision scheme with respect to the time interval [*t*] is employed to reduce the effect of overestimation.

For that purpose, we assume that [*t*] = [0 ; *T*] is subdivided into Ξ not necessarily equally wide subintervals

$$\mathbb{E}\left[t\right] = \bigcup\_{\xi=1}^{\Xi} \left[t\_{\xi}^{\varepsilon} - 1 \; ; \; t\_{\xi}^{\varepsilon}\right] = \bigcup\_{\xi=1}^{\Xi} \left[t\right]\_{\xi} \; \; \prime \tag{30}$$

where *t*<sup>0</sup> := 0, *t*<sup>Ξ</sup> := *T*, and *t*<sup>0</sup> < *t*<sup>1</sup> < ... < *t*Ξ.

Then, a subinterval-based evaluation of the iteration Formula (19) of Corollary 1 is given by

$$\begin{split} \mathbb{E}\left[\boldsymbol{\lambda}\_{i}\right]^{(\kappa+1)} &:= \bigsqcup\_{\boldsymbol{\xi}=1}^{\mathbb{E}} \left( a\_{\boldsymbol{\eta}} \Big( [\mathbf{x}\_{\varepsilon}]^{(\kappa)} \big( [t]\_{\boldsymbol{\xi}} \big) \right) + \\\\ & \sum\_{\begin{subarray}{c} \boldsymbol{\eta} = 1 \\ \boldsymbol{\eta} \neq i \end{subarray}}^{n} \left\{ a\_{\boldsymbol{\eta}} \Big( [\mathbf{x}\_{\varepsilon}]^{(\kappa)} \big( [t]\_{\boldsymbol{\xi}} \big) \right) \cdot \frac{E\_{\nu,1} \left( \left[ \boldsymbol{\lambda}\_{j} \right]^{(\kappa)} \cdot [t]\_{\boldsymbol{\xi}}^{\nu} \right)}{E\_{\nu,1} \left( [\boldsymbol{\lambda}\_{i}]^{(\kappa)} \cdot [t]\_{\boldsymbol{\xi}}^{\nu} \right)} \cdot \frac{\left[ \mathbf{x}\_{\varepsilon,j} \right](0)}{\left[ \mathbf{x}\_{\varepsilon,i} \right](0)} \right\} \end{split} \tag{31}$$

where the symbol 6 denotes the convex interval hull around all arguments of this operator. Moreover, the expression

$$\mathbf{E}\left[\mathbf{x}\_{\mathfrak{c}}\right]^{\langle\mathbf{x}\rangle}\left(\left[t\right]\_{\mathfrak{f}}\right) = \mathbf{E}\_{\nu,\mathbf{l}}\left(\left[\mathbf{A}\right]^{\langle\mathbf{x}\rangle}\cdot \left[t\right]\_{\mathfrak{f}}^{\nu}\right) \cdot \left[\mathbf{x}\_{\mathfrak{c}}\right](0) \,, \left[\mathbf{x}\_{\mathfrak{c}}\right](0) = \left[\mathbf{x}\_{0}\right] \tag{32}$$

in (31) represents the evaluation of the Mittag-Leffler-type pseudo-state enclosure for each temporal subinterval [*t*]*<sup>ξ</sup>* .

To replace the evaluation of the iteration Formula (31) by a counterpart that exploits the novel exponential enclosures of the Mittag-Leffler function for each temporal subinterval, define the enclosure

$$E\_{\nu,1}\left(\lambda \cdot t^{\nu}\right) \in \mathfrak{e}\left[\mathbb{Z}\_{i,\mathbb{S}}^{\langle\kappa\rangle} : \overline{\eta}\_{i,\mathbb{S}}^{\langle\kappa\rangle}\right] \cdot \left[t\_{\mathbb{S}-1} : t\_{\mathbb{S}}\right]^{\nu} \tag{3.3}$$

for *λ* ∈ [*λ*] *κ <sup>i</sup>* and *<sup>t</sup>* <sup>∈</sup> [*t*]*<sup>ξ</sup>* . The interval bounds *<sup>η</sup>κ <sup>i</sup>*,*<sup>ξ</sup>* and *<sup>η</sup>κ <sup>i</sup>*,*<sup>ξ</sup>* on the right-hand side of (33) are obtained by replacing *λ* with *λκ <sup>i</sup>* , *<sup>λ</sup>* with *<sup>λ</sup>κ <sup>i</sup>* , *t* with *tξ*−1, and *t* with *t<sup>ξ</sup>* in the Equations (25) and (26) that are defined in Theorem 3.

Then, the iteration Formula (19) of Corollary 1 is replaced with the expression

$$\begin{split} [\lambda\_i]^{(\kappa+1)} &:= \bigsqcup\_{\xi=1}^{\Sigma} \left( a\_{\overline{u}} \left( [\mathbf{y}\_c]^{(\kappa)} \left( [t]\_{\xi} \right) \right) + \\ & \quad \left( \sum\_{\substack{j=1 \\ j \neq i}}^n a\_{ij} \left( [\mathbf{y}\_c]^{(\kappa)} \left( [t]\_{\xi} \right) \right) \cdot e \left( \begin{bmatrix} \underline{u}\_{i,\xi}^{(\kappa)} \ \overline{\boldsymbol{\eta}}\_{i,\overline{\boldsymbol{\eta}}}^{(\kappa)} \end{bmatrix} - \begin{bmatrix} \underline{u}\_{i,\overline{\boldsymbol{\xi}}}^{(\kappa)} \ \overline{\boldsymbol{\eta}}\_{i,\overline{\boldsymbol{\xi}}}^{(\kappa)} \end{bmatrix} \right) \cdot [t]\_{\overline{\boldsymbol{\xi}}}^{\overline{\boldsymbol{\eta}}} \cdot \begin{bmatrix} \underline{x}\_{\varepsilon,j} \end{bmatrix} \right) \right) \;, \end{split} \tag{34}$$

where

$$\left[\left[\mathbf{x}\_{\varepsilon}\right]^{\left(\mathbf{x}\right)}\right]\left(\left[t\right]\_{\xi}\right) = \exp\left(\text{diag}\left\{\left[\underline{\eta}\_{1,\xi}^{\left(\mathbf{x}\right)};\,\overline{\eta}\_{1,\xi}^{\left(\mathbf{x}\right)}\right] \quad \dots \quad \left[\underline{\eta}\_{1,\xi}^{\left(\mathbf{x}\right)};\,\overline{\eta}\_{1,\xi}^{\left(\mathbf{x}\right)}\right]\right\}\right) \cdot \left[\left.t\right]\_{\xi}^{\left(\mathbf{y}\right)}\right) \tag{35}$$

is a direct substitute for (32) that was included before in (31).

As a preparation for the evaluation of Formulas (31) and (34) in the following section, the true range as well as both the box-type and exponential enclosures of the quotient

$$\frac{E\_{\nu,1}(\lambda\_1 t^{\nu})}{E\_{\nu,1}(\lambda\_2 t^{\nu})} \tag{36}$$

are illustrated in Figure 3 for the mutually independent interval parameters *λ*<sup>1</sup> ∈ [−2 ; −1] and *λ*<sup>2</sup> ∈ [−1.9 ; −1]. It becomes obvious that the box-type enclosure (as already discussed in Figure 1) is much more pessimistic for small points in time than for larger ones. Therefore, it can be expected that an intersection of both enclosure approaches leads to less pessimism during the evaluation of the iteration Formula (34). Note that this comes with practically no additional computational effort because the box-type range bounds form the basis for the application of Theorem 3.

**Figure 3.** Illustration of the two considered guaranteed enclosure methods for the quotient of two Mittag-Leffler functions with uncertain parameters *λ*<sup>1</sup> ∈ [*λ*1] and *λ*<sup>2</sup> ∈ [*λ*2] for *ν* = 0.5: (**a**) Box-type enclosures vs. exact range for the quotient (36); (**b**) Exponential enclosures vs. exact range for the quotient (36).

In the following section, both subdivision-based Formulas (31) and (34) are compared for the computation of guaranteed pseudo-state enclosures for a quasi-linear model of the charging/discharging dynamics of a Lithium-ion battery. The comparison is based on a quantification of the tightness of the obtained solution enclosures and a count of the numbers of iterations *κ* for identical numbers Ξ of temporal subintervals.
