Implicit A-Stable Peer Triplets for ODE Constrained Optimal Control Problems

TL;DR

This paper develops new implicit Peer triplet methods with high order and A-stability for solving nonlinear ODE-constrained optimal control problems, demonstrating their effectiveness through theoretical analysis and numerical tests.

Contribution

It introduces novel implicit Peer triplet methods with superconvergence, A-stability, and higher order accuracy, extending existing methods like BDF4 for optimal control applications.

Findings

Constructed four practical Peer triplets with order four and three for state and adjoint variables.

Extended BDF4 to a Peer triplet with higher order and A-stability.

Numerical tests confirm the theoretical stability and accuracy of the proposed methods.

Abstract

This paper is concerned with the construction and convergence analysis of novel implicit Peer triplets of two-step nature with four stages for nonlinear ODE constrained optimal control problems. We combine the property of superconvergence of some standard Peer method for inner grid points with carefully designed starting and end methods to achieve order four for the state variables and order three for the adjoint variables in a first-discretize-then-optimize approach together with A-stability. The notion triplets emphasizes that these three different Peer methods have to satisfy additional matching conditions. Four such Peer triplets of practical interest are constructed. Also as a benchmark method, the well-known backward differentiation formula BDF4, which is only $A(73.35^o)$-stable, is extended to a special Peer triplet to supply an adjoint consistent method of higher order and BDF type with equidistant nodes. Within the class of Peer triplets, we found a diagonally implicit $A(84^o)$-stable method with nodes symmetric in $[0,1]$ to a common center that performs equally well. Numerical tests with three well established optimal control problems confirm the theoretical findings also concerning A-stability.