# Combinatorial Integral Approximation Decompositions for Mixed-Integer Optimal Control
## 4 A Priori Bounds for CIA Decompositions
### Scaled Combinatorial Integral Approximation  
The proof of Theorem 1 contains a motivation for the (SCIA) problems as derived in the following result.  
**Corollary 2** (**Approximation bounds via (Scia), cf. Corollary 5.3 in [50]).: _Consider the setting of Theorem 1, and let \(\|\cdot\|_{n_{0}}\) refer to the maximum or 1-norm, i.e., \(no\in\{\max,1\}\). Assume that \(\mathbf{x}\) and \(\mathbf{y}_{\mathrm{CIA}}\) are the solutions of the IVP (2b) and (2c), where \(\mathbf{x}\) is based on a given \(\mathbf{a}\), and \(\mathbf{y}_{\mathrm{CIA}}\) is driven by \(\mathbf{w}^{\ast}\), which is the optimal solution of (SCIAno). Then, for a.a. \(t\in\mathcal{T}\), the state approximation error is bounded by:_  
\[\left\|\mathbf{x}(t)-\mathbf{y}_{\mathrm{SCIA}}(t)\right\|\leq\theta^{\ast}_{ \mathrm{CIA}}\mathrm{e}^{L(t-t_{0})}\leq\theta^{\ast}_{\mathrm{CIA}}(B+C(t-t_{ 0}))\mathrm{e}^{L(t-t_{0})}, \tag{19}\]  
_where \(\theta^{\ast}_{\mathrm{SCIA}}\) is the optimal objective value of (SCIAno), \(no\in\{\max,1\}\)._  
Proof.: From the second and last inequalities in the proof of Theorem 2 from [41] and Corollary 1, it follows that, for a.a. \(t\in\mathcal{T}\) and any \(\boldsymbol{\omega}\in\Omega\):  
\[\left\|\int_{t_{0}}^{t}\sum_{i=1}^{n_{\omega}}(\alpha_{i}(\tau)-\omega_{i}( \tau))f_{i}(\mathbf{x}(\tau))\ \mathrm{d}\tau\right\|\leq\left\|\int_{t_{0}}^{t}\boldsymbol{\alpha}(\tau)- \boldsymbol{\omega}(\tau)\ \mathrm{d}\tau\right\|(B+C(t-t_{0})).\]Let \(\mathbf{\omega}^{\text{CIA}}\) denote the control based on the optimal solution \(\mathbf{w}^{*}\) of (CIA_no_), \(no\in\{\max,1\}\). We take the minimum in the above inequality and obtain:  
\[\theta^{*}_{\text{SCIA}}\leqslant\left\|\int_{t_{0}}^{t}\sum_{i=1}^{n_{\omega}}( \alpha_{i}(\tau)-\omega_{i}^{\text{CIA}}(\tau))f_{i}(\mathbf{x}(\tau))\ \mathrm{d}\tau\right\|\leqslant\theta^{*}_{\text{CIA}}(B+C(t-t_{0})).\]  
\(\Box\)  
We conclude from Corollary 1 that the approximation and convergence results of the decomposition still hold if (SCIA_no_), \(no\in\{\max,1\}\), is used to construct the binary control. The approximation bound based on (SCIA_no_) is tighter than the existing (CIA_no_)-related bound. Hence, it is an obvious choice to consult these alternative binary controls for an approximation study. Ideally, the binary control constructed in this way will result in an improved state approximation and objective value for (MIOCP). However, we oppose this hope next.  
**Remark 1** (**Construction by (**Scia**) does not guarantee superior quality).: _The binary control based on the optimal solution of (SCIA_no_), \(no\in\{\max,1\}\) and used in the decomposition does not necessarily result in a state approximation or objective value that is superior to that obtained using (CIA_no_). It may hold that:_  
\[\|\mathbf{x}(t)-\mathbf{y}_{\text{CIA}}(t)\|<\|\mathbf{x}(t)-\mathbf{y}_{ \text{SCIA}}(t)\|<\theta^{*}_{\text{SCIA}},\quad\text{ for some }t\in\mathcal{T},\]  
_where we use the notation from Corollaries 1 and 2. Because of a possible non-convex objective, the computed trajectories may lead to a superior objective value for the solution based on (CIA_no_), compared with that based on (SCIA_no_), even if the above inequality does not hold._