# Implicit A-Stable Peer Triplets for ODE Constrained Optimal Control Problems
## 1 Introduction  
The design of efficient time integrators for the numerical solution of optimal control problems constrained by systems of ordinary differential equations (ODEs) is still an active research field. Such systems typically arise from semi-discretized partial differential equations describing, e.g., the dynamics of heat and mass transfer or fluid flow in complex physical systems. Symplectic one-step Runge-Kutta methods [1, 2] exploit the Hamiltonian structure of the first-order optimality system--the necessary conditions to find an optimizer--and automatically yield a consistent approximation of the adjoint equations, which can be used to compute the gradient of the objective function. The first-order symplectic Euler, second-order Stormer-Verlet and higher-order Gauss methods are prominent representatives of this class, which are all implicit for general Hamiltonian systems, see the monograph [3]. Moreover, compositions of basic integrators with different steps sizes and splitting methods have been investigated. Generalized partitioned Runge-Kutta methods which allow one to compute exact gradients with respect to the initial condition are studied in [4]. To avoid the solution of large systems of nonlinear equations, semi-explicit W-methods [5] and stabilized explicit Runge-Kutta-Chebyshev methods [6] have been proposed, too. However, as all one-step methods, also symplectic Runge-Kutta schemes join the structural suffering of order reductions, which may lead to poor results in their application, e.g., to boundary control problems such as external cooling and heating in a manufacturing process; see [7, 8] for a detailed study of this behaviour. In contrast, multistep methods including Peer two-step methods avoid order reductions and allow a simple implementation [9; 10]. However, the discrete adjoint schemes of linear multistep methods are in general not consistent or show a significant decrease of their approximation order [11; 12]. Recently, we have developed implicit Peer two-step methods [13] with three stages to solve ODE constrained optimal control problems of the form  
\[\text{minimize}\ C\big{(}y(T)\big{)} \tag{1}\] \[\text{subject to}\ y^{\prime}(t)= f\big{(}y(t),u(t)\big{)},\quad u(t)\in U_{ad},\ t\in(0,T],\] (2) \[y(0)= y_{0}, \tag{3}\]  
with the state \(y(t)\in\mathbb{R}^{m}\), the control \(u(t)\in\mathbb{R}^{d}\), \(f:\mathbb{R}^{m}\times\mathbb{R}^{d}\mapsto\mathbb{R}^{m}\), the objective function \(C:\mathbb{R}^{m}\mapsto\mathbb{R}\), where the set of admissible controls \(U_{ad}\subset\mathbb{R}^{d}\) is closed and convex. Introducing for any \(u\in U_{ad}\) the normal cone mapping  
\[N_{U}(u)=\{w\in\mathbb{R}^{d}:w^{T}(v-u)\leq 0\ \text{for all}\ v\in U_{ad}\}, \tag{4}\]  
the first-order Karush-Kuhn-Tucker (KKT) optimality system [14; 15] reads  
\[y^{\prime}(t)= f\big{(}y(t),u(t)\big{)},\quad t\in(0,T],\quad y(0)=y_{0}, \tag{5}\] \[p^{\prime}(t)= -\
abla_{y}f\big{(}y(t),u(t)\big{)}^{\mathsf{T}}p(t),\quad t\in[ 0,T),\quad p(T)=\
abla_{y}C\big{(}y(T)\big{)}^{\mathsf{T}},\] (6) \[-\
abla_{u}f\big{(}y(t),u(t)\big{)}^{\mathsf{T}}p(t)\in N_{U} \big{(}u(t)\big{)},\quad t\in[0,T]. \tag{7}\]  
In this paper, we assume the existence of a unique local solution \((y^{\star},p^{\star},u^{\star})\) of the KKT system with sufficient regularity properties to justify the use of higher order Peer triplets, see, e.g., the smoothness assumption in Section 2 of [14]. The objective function \(C(y(T))\) in (1) is specified in the so-called Mayer form using terminal solution values only. Terms given in the Lagrange form  
\[C_{l}(y,u):=\int_{0}^{T}l(y(t),u(t))\ dt \tag{8}\]  
can be equivalently reduced to the Mayer form by introducing an additional state \(y_{m+1}\), the new state vector \(\tilde{y}:=(y_{1},\ldots,y_{m},y_{m+1})^{\mathsf{T}}\), and an additional differential equation \(y^{\prime}_{m+1}(t)=l(y(t),u(t))\) with initial values \(y_{m+1}(0)=0\). Then, the Lagrange term (8) simply reduces to \(y_{m+1}(T)\).