# Implicit A-Stable Peer Triplets for ODE Constrained Optimal Control Problems
## 2 Materials and Methods
### Existence of Boundary Methods Imposes Restrictions on the Standard Method
#### 2.3.1 Combined Conditions for the End Method  
We remind that we now are looking for methods having local order \(q_{1}=s\) and \(q_{2}=s-1\) everywhere, which we abbreviate from now on as \((q_{1},q_{2})=(s,q)\). In particular this means that \(A^{\mathsf{T}}V_{q}+K^{\mathsf{T}}V_{q}\bar{E}_{q}=B^{\mathsf{T}}V_{q}\mathcal{ P}_{q}\) for the standard method. Looking for bottlenecks in the design of these methods, we try to identify crucial necessary conditions and consider the three order conditions for the end method \((A_{N},B_{N},K_{N})\) in combination  
\[ \begin{array}{rl}A_{N}V_{s}-B_{N}V_{s}\mathcal{P}_{s}^{-1}-K_{N}V_{s}\bar{E}_ {s}&=0,\\ B_{N}^{\mathsf{T}}V_{q}\mathcal{P}_{q}&=A^{\mathsf{T}}V_{q}+K^{\mathsf{T}}V_{q }\bar{E}_{q}=B^{\mathsf{T}}V_{q}\mathcal{P}_{q},\\ A_{N}^{\mathsf{T}}V_{q}+K_{N}^{\mathsf{T}}V_{q}\bar{E}_{q}&=w\mathbbm{1}_{q}^{ \mathsf{T}}.\end{array}  \tag{63}\]  
From these conditions the matrices \(A_{N},B_{N}\) may be eliminated, revealing the first restrictions on \(B\). Here, the singular matrix map  
\[\mathcal{L}_{q,s}:\,\mathbb{R}^{q\times s}\to\mathbb{R}^{q\times s},\quad X \mapsto\bar{E}_{q}^{\mathsf{T}}X+X\bar{E}_{s} \tag{64}\]  
plays a crucial role. **Lemma 3**. : _A necessary condition for a boundary method \((A_{N},B_{N},K_{N})\) to satisfy (63) is_  
\[\mathcal{L}_{q,s}(V_{q}^{\mathsf{T}}K_{N}V_{s})= \mathbbm{1}_{q}\mathbbm{1}_{s}^{\mathsf{T}}-V_{q}^{\mathsf{T}}BV _{s}\mathcal{P}_{s}^{-1}. \tag{65}\]  
Proof. : The second condition, \(V_{q}^{\mathsf{T}}B_{N}=V_{q}^{\mathsf{T}}B\), in (63) leads to the necessary equation  
\[V_{q}^{\mathsf{T}}A_{N}V_{s}-V_{q}^{\mathsf{T}}K_{N}V_{s}\bar{E}_{s}=V_{q}^{ \mathsf{T}}BV_{s}\mathcal{P}_{s}^{-1}\]  
due to the first condition. The transposed third condition  
\[V_{q}^{\mathsf{T}}A_{N}V_{s}+\bar{E}_{q}^{\mathsf{T}}V_{q}^{\mathsf{T}}K_{N}V_{ s}=\mathbbm{1}_{q}w^{\mathsf{T}}V_{s}=\mathbbm{1}_{q}\mathbbm{1}_{s}^{ \mathsf{T}}\]  
multiplied by the nonsingular matrix \(V_{s}\) may be used to eliminate \(A_{N}\) and leads to  
\[\bar{E}_{q}^{\mathsf{T}}V_{q}^{\mathsf{T}}K_{N}V_{s}+V_{q}^{\mathsf{T}}K_{N}V_{ s}\bar{E}_{s}=\mathbbm{1}_{q}\mathbbm{1}_{s}^{\mathsf{T}}-V_{q}^{\mathsf{T}} BV_{s}\mathcal{P}_{s}^{-1}\]  
which is the equation (65) from the statement. Unfortunately, this lemma leads to several restrictions on the design of the methods due to the properties of the map \(\mathcal{L}_{q,s}\). Firstly, for diagonal matrices \(K_{N}\) the image \(\mathcal{L}_{q,s}\big{(}V_{q}^{\mathsf{T}}K_{N}V_{s}\big{)}\) has a very restricted shape. **Lemma 4**. : _If \(K\in\mathbb{R}^{s\times s}\) is a diagonal matrix, then \(V_{q}^{\mathsf{T}}KV_{s}\) and \(\mathcal{L}_{4,s}(V_{q}^{\mathsf{T}}KV_{s})\) are Hankel matrices with constant entries along anti-diagonals._  
Proof. : With \(K=\operatorname{diag}(\kappa_{i})\), we have \(x_{ij}:=e_{i}^{\mathsf{T}}(V_{q}^{\mathsf{T}}KV_{s})e_{j}=\sum_{k=1}^{s} \kappa_{k}c_{k}^{i+j-2}\), showing Hankel form of \(X=(x_{ij})=:(\xi_{i+j-1})\) for \(i=1,\ldots,s-1\), \(j=1,\ldots,s\). Now, \(e_{i}^{\mathsf{T}}(\tilde{E}_{q}^{\mathsf{T}}X+X\tilde{E}_{s})e_{j}=(i-1)x_{i- 1,j}+x_{i,j-1}(j-1)=(i+j-2)\xi_{i+j-2}\), which shows again Hankel form. This lemma means that an end method with diagonal \(K_{N}\) only exists if also \(V_{q}^{\mathsf{T}}BV_{s}\mathcal{P}_{s}^{-1}\) on the right-hand side of (65) has Hankel structure. Unfortunately it was observed that for standard methods with definite \(K\) this is the case for \(q_{2}\leq 2\) only (there exist methods with an explicit stage \(\kappa_{33}=0\)). **Remark 5**. : _Trying to overcome this bottleneck with diagonal matrices \(K_{N}\), one might consider adding additional stages of the end method. However, using general end nodes \((\xi_{1},\ldots,\xi_{s})\) with \(\hat{s}\geq s\) does not remove this obstacle.