# Combinatorial Integral Approximation Decompositions for Mixed-Integer Optimal Control
## 2 Problem Class, Definitions, and Main Algorithm  
We denote the considered time horizon by \(\mathcal{T}:=[t_{0},t_{t}]\subset\mathbb{R}\), and write \"for a.a. \(t\in\mathcal{T}\)\" for all \(t\in\mathcal{T}\), except on a set of measure zero. Let \([n]:=\{1,\ldots,n\}\), \([n]_{0}:=0\cup[n]\), for \(n\in\mathbb{N}\). The null vector is written as \(\mathbf{o}\). We are interested in the following class of mixed-integer optimal control problems. **Definition 1**. : _(MIOCP) We refer to the following control problem (2) as (MIOCP)._  
\[\inf_{\mathbf{x},\boldsymbol{\omega}} \Phi(\mathbf{x}(t_{\!f\!})) \tag{2a}\] \[s.t. \dot{\mathbf{x}}(t)=\boldsymbol{f}_{0}(t,\mathbf{x}(t))+\sum_{i= 1}^{n_{\boldsymbol{\omega}}}\omega_{i}(t)\boldsymbol{f}_{i}(t,\mathbf{x}(t)), \text{for a.a. }t\in\mathcal{T},\] (2b) \[\mathbf{x}(t_{0})=\mathbf{x}_{0},\] (2c) \[1=\sum_{i=1}^{n_{\boldsymbol{\omega}}}\omega_{i}(t) \text{for a.a. }t\in\mathcal{T},\] (2d) \[\boldsymbol{\omega}(t)\in\{0,1\}^{n_{\boldsymbol{\omega}}} \text{for a.a. }t\in\mathcal{T},\] (2e) \[\boldsymbol{o}\leq\mathbf{c}(t,\mathbf{x}(t)) \text{for a.a. }t\in\mathcal{T}. \tag{2f}\]  
We minimize a Mayer term \(\Phi\in C^{1}(\mathbb{R}^{n_{x}},\mathbb{R})\) over differential states \(\mathbf{x}\in\mathcal{W}^{1,\infty}(\mathcal{T},\mathbb{R}^{n_{x}})\) for binary control functions \(\boldsymbol{\omega}\in L^{\infty}(\mathcal{T},\{0,1\}^{n_{\boldsymbol{\omega }}})\). The system of ordinary differential equations (ODE), (2b), is written using partial outer convexification to model the switched system, using a one-hot encoding (Ihot) constraint (2d), a drift term \(\boldsymbol{f}_{0}\), and \(n_{\boldsymbol{\omega}}\geq 2\) functions \(\boldsymbol{f}_{i}\), ref. [27], both out of \(C^{0}(\mathbb{R}^{n_{x}+1},\mathbb{R}^{n_{x}})\). We assume fixed initial values \(\mathbf{x}_{0}\in\mathbb{R}^{n_{x}}\) for the differential states. The functions \(\mathbf{c}\in C^{1}(\mathbb{R}^{n_{x}},\mathbb{R}^{n_{c}})\) model nonlinear state inequalities. In the following, we assume that (MIOCP) has an optimal solution, so that we can write \"min\" instead of \"inf\". We refer to [19; 29; 49] for a discussion of the generality of (MIOCP) and extensions to cope with, e.g., Lagrange functionals, boundary and multi-point constraints, vanishing constraints, free final time, control values, and the like. Particularly interesting are continuous control functions \(\mathbf{u}\in L^{\infty}(\mathcal{T},\mathcal{U})\) that often enter (2b) and (2f) in practical applications. From a theoretical point of view, in the interest of comparability, and for computational speed, it is convenient to consider the continuous controls \(u(\cdot)\) as fixed to the solution that was obtained by solving the continuous relaxation of (MIOCP) in our approach. It is also possible, though, and often makes sense in practice, to improve the objective function value by reoptimizing \(\mathbf{u}\) when (MIOCP) is evaluated for fixed \(\boldsymbol{\omega}\). For notational convenience and without loss of generality, we omit the continuous controls \(\mathbf{u}\) in the following. An _evaluation_ of (MIOCP) for fixed \(\boldsymbol{\omega}\) is then the solution of an initial value problem. We stress that there are only \(n_{\omega}\) possible solutions for the integer part of the problem due to the constraint (2d). Our algorithm solves continuous relaxations of (MIOCP), defined as follows. **Definition 2**. : _(OCP) We define (OCP) as the canonical relaxation of (MIOCP) with respect to (2e), where we substitute \(\boldsymbol{\omega}\in L^{\infty}(\mathcal{T},\{0,1\}^{n_{\omega}})\) for \(\boldsymbol{\alpha}\in L^{\infty}(\mathcal{T},[0,1]^{n_{\omega}})\)._  
The problem (OCP) in function space can be solved by different approaches, as mentioned above.