# Combinatorial Integral Approximation Decompositions for Mixed-Integer Optimal Control
## 4 A Priori Bounds for CIA Decompositions
### Combinatorial Integral Approximation  
We recapitulate a variant of [PERSON]'s Lemma from [41], which is needed for the main theorem. **Lemma 1** (A variant of [PERSON]'s Lemma, see [41], Lemma 1). : _Let \(z_{1},z_{2}:\ \mathcal{T}\to\mathbb{R}\) be real-valued integrable functions and let \(z_{2}\) also belong to \(L^{\infty}(\mathcal{T},\mathbb{R})\). If, for a constant \(L\geq 0\), the following holds:_  
\[z_{1}(t)\leq z_{2}(t)+L\int_{t_{0}}^{t}z_{1}(\tau)\,d\tau\qquad\text{for a.a. }t\in\mathcal{T},\]  
_then we have:_  
\[z_{1}(t)\leq\|z_{2}\|_{\infty}\,\mathrm{e}^{L(t-t_{0})}\qquad\text{for a.a. }t\in\mathcal{T}. \tag{16}\]  
Proof. : See [41], proof of Lemma 1. In the following results, we analyze the evolution of two trajectories, \(\mathbf{x}\) and \(\mathbf{y}\), based on the same ODE system (2b) but driven by two different controls, \(\mathbf{\alpha}\) and \(\mathbf{\omega}\). The following theorem gives a statement on the distance between the two trajectories depending on the distance of the controls. **Theorem 1**. : _Consider \(\mathbf{\alpha}\) and \(\mathbf{\omega}\in\Omega\). We reuse the model functions \(\mathbf{f}_{0},\mathbf{f}_{i}:\mathcal{T}\times\mathbb{R}^{n_{x}}\to\mathbb{R}^{n_{x}}\) from Definition 2 for \(i\in[n_{w}]\). Let \(\mathbf{x}(\cdot)\) and \(\mathbf{y}(\cdot)\) be the unique solutions of the IVP:_  
\[\dot{\mathbf{x}}(t) =\mathbf{f}_{0}(t,\mathbf{x}(t))+\sum_{i=1}^{n_{w}}\alpha_{i}(t)\mathbf{ f}_{i}(t,\mathbf{x}(t)),\qquad\mathbf{x}(t_{0})=\mathbf{x}_{0}, \tag{17a}\] \[\dot{\mathbf{y}}(t) =\mathbf{f}_{0}(t,\mathbf{y}(t))+\sum_{i=1}^{n_{w}}\omega_{i}(t)\mathbf{ f}_{i}(t,\mathbf{y}(t)),\qquad\mathbf{y}(t_{0})=\mathbf{y}_{0}, \tag{17b}\]  
_where \(\mathbf{x}_{0},\mathbf{y}_{0}\in\mathbb{R}^{n_{x}}\). Assume that there are positive constants \(L,C\in\mathbb{R}^{+}\), together with a vector norm \(||\cdot||\), such that, for a.a., \(t\in\mathcal{T}\) holds:_  
\[||\mathbf{f}_{i}(t,\mathbf{x}(t))-\mathbf{f}_{i}(t,\mathbf{y}(t))|| \leq L||\mathbf{x}(t)-\mathbf{y}(t)||,\qquad\text{for }i\in[n_{w}]_{0}, \tag{17c}\] \[\left\|\frac{\mathrm{d}}{\mathrm{d}t}\mathbf{f}_{i}(t,\mathbf{x}(t))\right\| \leq C,\qquad\text{for }i\in[n_{w}]. \tag{17d}\]_Furthermore, let \(f_{i}(\cdot,\mathbf{x}(\cdot))\), \(i\in[n_{\omega}]\) be essentially bounded by \(B\in\mathbb{R}^{+}\) on \(\mathcal{T}\), and assume that for all \(t\in\mathcal{T}\). It holds that:_  
\[\left\|\int_{t_{0}}^{t}\alpha(\tau)-\omega(\tau)\ \mathrm{d}\tau\right\|\leq\theta, \tag{17e}\]  
_with the constant \(\theta\geq 0\). Then, for a.a. \(t\in\mathcal{T}\) we also have:_  
\[\left\|\mathbf{x}(t)-\mathbf{y}(t)\right\|\leq(\left\|\mathbf{x}_{0}-\mathbf{ y}_{0}\right\|+\theta n_{\omega}(B+C(t-t_{0})))\mathrm{e}^{L(t-t_{0})}. \tag{17f}\]  
Proof. : This Theorem is an (equivalent) reformulation of Theorem 2 from [41]. The only differences are modified notations and the usage of \(f_{i},\ i\in[n_{\omega}]_{0}\) as differentiable mapping instead of \(A\) in [41]. We first recognize that Theorem 1 is applicable for the CIA problem with any vector norm, which can be guessed from the equivalence of norms. **Corollary 1** (Approximation bounds via (CIA), cf. Corollary 5.2 in [50]). : _Consider the setting of Theorem 1; in particular, let the regularity assumptions on \(f_{i},i\in[n_{\omega}]_{0}\), hold. 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 relaxed control \(\mathbf{a}\) and \(\mathbf{y}_{\mathrm{CIA}}\) is based on \(\mathbf{w}^{\ast}\), which is the optimal solution of (CIAno), \(no\in\{\max,1\}\), with objective value \(\theta^{\ast}_{\mathrm{CIA}}\) from Definition 6. Then, the state approximation error is bounded for a.a. \(t\in\mathcal{T}\) by:_  
\[\left\|\mathbf{x}(t)-\mathbf{y}_{\mathrm{CIA}}(t)\right\|\leq\theta^{\ast}_{ \mathrm{CIA}}(B+C(t-t_{0}))\mathrm{e}^{L(t-t_{0})}. \tag{18}\]  
Proof. : This corollary is a direct result from Theorem 1 with \(\mathbf{x}_{0}=\mathbf{y}_{0}\).