# Combinatorial Integral Approximation Decompositions for Mixed-Integer Optimal Control
## 4 A Priori Bounds for CIA Decompositions
### \(\lambda\)-Combinatorial Integral Approximation  
The \(\lambda\)-approximation stems from the aforementioned theorem of approximating differential states, but with assessing the difference of the cost-to-go function. We define it in a more general MIOCP setting with a Bolza objective:  
\[\Phi(\mathbf{x}(t_{\text{f}}))+\int\limits_{t\in\mathcal{T}}L(\mathbf{x}(\tau ),\omega(\tau))\ \ \mathrm{d}\tau, \tag{20}\]  
where \(L\in C^{1}(\mathbb{R}^{n_{\text{x}}+1}\times\mathbb{R}^{n_{\omega}},\mathbb{ R})\). **Definition 9** (**Cost-to-go function by [PERSON])**. : _Let the cost-to-go function \(\int L^{\infty}(\mathbb{R}^{n_{\text{x}}}\times\mathcal{T},\mathbb{R})\) with Bolza objective (20) be implicitly defined as:_  
\[ \begin{split} J(\mathbf{x}(t_{\text{f}}),t_{\text{f}})& =\Phi(\mathbf{x}(t_{\text{f}})),\\ -\frac{\partial I}{\partial t}(\mathbf{x}(t),t)&= \min_{\omega\in\Omega}L(\mathbf{x}(t),\omega(t))+\frac{\partial I}{\partial x }(\mathbf{x}(t),t)\Bigg{(}f_{0}(t,\mathbf{x}(t))+\sum_{i=1}^{n_{\omega}} \omega_{i}(t)\mathbf{f}_{i}(t,\mathbf{x}(t))\Bigg{)}.\end{split} \]  
We recognize that \(\frac{\partial I}{\partial x}\) can be interpreted as Lagrange multiplier or dual variables of the ODE constraints in (MIOCP). Therefore, we write, in short, \(\lambda(t)\) instead of \(\frac{\partial I}{\partial x}\). With the groundwork above, we are ready to deduce the corresponding bound. **Corollary 3** (**Approximation bounds via (\(\lambda\)CIA), cf. Corollary 5.4 in [50]). : _Consider the setting of Theorem 1. In particular, let the regularity assumptions (17d), (17c), and the essential boundedness of \(\mathbf{f}\) be true. Assume that \(\mathbf{x}\) and \(\mathbf{y}_{\lambda\text{CIA}}\) are the solutions of the IVP (2b) and (2c), where \(\mathbf{x}\) is based on a given \(\mathbf{a}\), and \(\mathbf{y}_{\lambda\text{CIA}}\) is based on \(\mathbf{w}^{*}\), which is the optimal solution of (\(\lambda\)CIA1).__Let \(J\) be the cost-to-go function as defined in Definition 9 for (MIOCP) and \(\lambda(t)\) be the adjoint vector at \(t\in\mathcal{T}\) for the ODE system (2b). For a.a. \(t\in\mathcal{T}\), it follows that:_  
\[|J(\mathbf{x}(t),t)-J(\mathbf{y}_{\lambda\mathrm{CIA}}(t),t)|\leq\theta_{ \lambda\mathrm{CIA}}^{*}\mathrm{e}^{L(t-t_{0})}+o\Big{(}\|\mathbf{x}(t)- \mathbf{y}_{\lambda\mathrm{CIA}}(t)\|^{2}\Big{)},\]  
_where \(o\) refers to [PERSON]'s little-\(o\) notation._  
Proof. : Let us consider the difference of the cost-to-go functions by approximation with a partial first-order Taylor expansion around \(J(\mathbf{x}(t),t)\). We perform the approximation with respect to the trajectories \(\mathbf{x},\ \mathbf{y}_{\lambda\mathrm{CIA}}\). Thus, we apply [PERSON]'s theorem for \(t\in\mathcal{T}\):  
\[J(\mathbf{y}_{\lambda\mathrm{CIA}}(t),t)-J(\mathbf{x}(t),t)=\frac{\partial I} {\partial\mathbf{x}}(\mathbf{x}(t),t)(\mathbf{y}_{\lambda\mathrm{CIA}}(t)- \mathbf{x}(t))+o\Big{(}\|\mathbf{y}_{\lambda\mathrm{CIA}}(t)-\mathbf{x}(t) \|^{2}\Big{)}. \tag{21}\]  
As pointed out above, the dual variables of the ODE constraint (2b) are equal to \(\frac{\mathrm{d}J}{\mathrm{d}\mathbf{x}}(\mathbf{x}(t),t)\). We use the notation \(\|\mathbf{x}(t)\|_{\lambda(t)}:=\left|\sum_{k\in[n_{x}]}\lambda_{k}(t)x_{k}(t)\right|\) for \(t\in\mathcal{T}\), which defines a semi-norm.