*4.2. Simulation with the Help of Box-Type Enclosures*

In Figure 5, the iteration Formula (31) has been used to compute guaranteed enclosures for the pseudo states of the controlled Lithium-ion battery for the two differently wide enclosures of initial values.

Due to the fact that the solution parameters *λ<sup>i</sup>* are determined in such a way that they are valid for the complete time interval [0 ; *T*], longer simulation horizons lead to inflating interval bounds as long as the integrator reset approach derived in [25] and its extended version published in [7] are not employed. The simulations show clearly that the overestimation of the range enclosures for the pseudo states is larger for those variables that change faster. In this scenario, the faster changing variable is the voltage *v*1(*t*) as compared to the state of charge *σ*(*t*). However, it can also be seen that the blow-up of the solution bounds can be reduced by tighter bounds for the pseudo-state initialization which gives rise to the option to repeat the simulations after paving the possible initial pseudo-state domain.

**Figure 5.** Use of box-type enclosures for the evaluation of the iteration Formula (31) for the computation of guaranteed pseudo-state enclosures: (**a**) State of charge *σ*(*t*) for **x**(0) ∈ [**x**]1(0); (**b**) State of charge *σ*(*t*) for **x**(0) ∈ [**x**]2(0); (**c**) Voltage *v*1(*t*) for **x**(0) ∈ [**x**]1(0); (**d**) Voltage *v*1(*t*) for **x**(0) ∈ [**x**]2(0); (**e**) Interval diameter for the enclosure of *σ*(*t*); (**f**) Interval diameter for the enclosure of *v*1(*t*).
