Implicit Third‐Order Peer Triplets with Variable Stepsizes for Gradient‐Based Solutions in Large‐Scale ODE‐Constrained Optimal Control

ABSTRACT This paper is concerned with the theory, construction and application of variable‐stepsize implicit Peer two‐step methods that are super‐convergent for variable stepsizes, that is, preserve their classical order achieved for uniform stepsizes when applied in a gradient‐based solution algorithm to solve ODE‐constrained optimal control problems in a first‐discretize‐then‐optimize setting. Gradients of the objective function can be computed most efficiently using approximate adjoint variables. High accuracy with moderate computational effort can be achieved through time integration methods that satisfy a sufficiently large number of adjoint order conditions for variable stepsizes and provide gradients with higher‐order consistency. In this paper, we enhance our previously developed variable implicit two‐step Peer triplets constructed in [J. Comput. Appl. Math. 460, 2025] to get ready for large‐scale dynamical systems with varying time scales without losing efficiency. A key advantage of Peer methods is their use of multiple stages with the same high stage order, which prevents order reduction—an issue commonly encountered in semi‐discretized PDE problems with boundary control. Two third‐order methods with four stages, good stability properties, small error constants, and a grid adaptation by equi‐distributing global errors are constructed and tested for a 1D boundary heat control problem and an optimal control of cytotoxic therapies in the treatment of prostate cancer.