# Implicit A-Stable Peer Triplets for ODE Constrained Optimal Control Problems
## 2 Materials and Methods
### Error Analysis
#### 2.2.2 Bounds for the Global Error  
In this section, the errors \(\tilde{Y}_{nj}:=y(t_{nj})-Y_{nj}\), \(\tilde{P}_{nj}:=p(t_{nj})-P_{nj}\), \(n=0,\ldots,N\), \(j=1,\ldots,s\), are analyzed. According to [13], the equation for the errors \(\tilde{Y}^{\mathsf{T}}=(\tilde{Y}_{0}^{\mathsf{T}},\ldots,\tilde{Y}_{N}^{ \mathsf{T}})\) and \(\tilde{P}^{\mathsf{T}}=(\tilde{P}_{0}^{\mathsf{T}},\ldots,\tilde{P}_{N}^{ \mathsf{T}})\) is a linear system of the form  
\[\mathbb{M}_{h}\tilde{Z}=\tau,\ \tilde{Z}= \begin{pmatrix}\tilde{Y}\\ \tilde{P}\end{pmatrix} ,\ \tau= \begin{pmatrix}\tau^{Y}\\ \tau^{P}\end{pmatrix} , \tag{35}\]  
where the matrix \(\mathbb{M}_{h}\) has a \(2\times 2\)-block structure and \((\tau^{Y},\tau^{P})\) denote the corresponding truncation errors. Deleting all \(h\)-depending terms from \(\mathbb{M}_{h}\), the block structure of the remaining matrix \(\mathbb{M}_{0}\) is given by  
\[\mathbb{M}_{0}= \begin{pmatrix}M_{11}\otimes I_{m}&0\\ M_{21}\otimes\
abla_{yy}C_{N}&M_{22}\otimes I_{m}\end{pmatrix}  \tag{36}\]  
with \(M_{11},M_{21},M_{22}\in\mathbb{R}^{s(N+1)\times s(N+1)}\) and a mean value \(\
abla_{yy}C_{N}\in\mathbb{R}^{m\times m}\) of the symmetric Hessian matrix of \(C\). The index ranges of all three matrices are copied from the numbering of the grid, \(0,\ldots,N\), for convenience. In fact, \(M_{21}=(e_{N}\otimes\mathbf{I})(e_{N}\otimes w)^{\mathsf{T}}\) has rank one only with \(e_{N}=(\delta_{Nj})_{j=0}^{N}\). The diagonal blocks of \(\mathbb{M}_{0}\) are nonsingular and its inverse has the form  
\[\mathbb{M}_{0}^{-1}= \begin{pmatrix}M_{11}^{-1}\otimes I_{m}&0\\ -(M_{22}^{-1}M_{21}M_{11}^{-1})\otimes\
abla_{yy}C_{N}&M_{22}^{-1}\otimes I_{m} \end{pmatrix} . \tag{37}\]The diagonal blocks \(M_{11}\), \(M_{22}\) have a bi-diagonal block structure with identity matrices \(I_{s}\) in the diagonal. The individual \(s\times s\)-blocks of their inverses are easily computed with \(M_{11}^{-1}\) having lower triangular block form and \(M_{22}^{-1}\) upper triangular block form, with blocks  
\[(M_{11}^{-1})_{nk}=\bar{B}_{n}\cdots\bar{B}_{k+1},\,k\leq n,\quad(M_{22}^{-1}) _{nk}=\bar{B}_{n+1}^{\mathsf{T}}\cdots\bar{B}_{k}^{\mathsf{T}},\,k\geq n, \tag{38}\]  
with the abbreviations \(\bar{B}_{n}:=A_{n}^{-1}B_{n}\), \(1\leq n\leq N\) and \(\bar{B}_{n+1}^{\mathsf{T}}:=(B_{n+1}A_{n}^{-1})^{\mathsf{T}}\), \(0\leq n<N\). Empty products for \(k=n\) mean the identity \(I_{s}\). Defining \(\mathbb{U}:=h^{-1}(\mathbb{M}_{n}-\mathbb{M}_{0})\) and rewriting (35) in fixed-point form  
\[\tilde{Z}=h\mathbb{M}_{0}^{-1}\mathbb{U}\tilde{Z}+\mathbb{M}_{0}^{-1}\tau\,, \tag{39}\]  
it has been shown in the proof of Theorem 4.1 of [13] for smooth right hand sides \(f\) and \(h\leq h_{0}\) that  
\[\|\tilde{Z}\|\leq 2\max\{\|M_{11}^{-1}\tau^{Y}\|,\|M_{22}^{-1}\tau^{P}\|\} \tag{40}\]  
in suitable norms, where these norms are discussed in more detail in Section 2.2.3 below. Moreover, due to the lower triangular block structure of \(\mathbb{M}_{0}\) the estimate for the error in the state variable may be refined (Lemma 4.2 in [13]) to  
\[\|\tilde{Y}\|\leq\|M_{11}^{-1}\tau^{Y}\|+hL\|\tilde{Z}\| \tag{41}\]  
with some constant \(L\). Without additional conditions, estimates of the terms on the right-hand side of (40) in the form \(\|\tilde{Z}\|=O(h^{-1}\|\tau\|)\) lead to the loss of one order in the global error. However, this loss may be avoided by one additional superconvergence condition on the forward and the adjoint method each, which will be considered next. In our convergence result Theorem 1 below, we will assume the existence of the discrete solution satisfying (22)-(27), for simplicity. However, at least for quadratic objective functions \(C\), this existence follows quite simply along the same lines used in this paragraph here. Lemma 1: _Let the right-hand side \(f\) of (2) be smooth with bounded second derivatives and let the function \(C\) be a polynomial of degree two, at most.