Structured preconditioning of conjugate gradients for path-graph network optimal control problems

A structured preconditioned conjugate gradient (PCG) solver is developed for the Newton steps in second-order methods for a class of constrained network optimal control problems. Of specific interest are problems with discrete-time dynamics arising from the path-graph interconnection of $N$ heterogeneous sub-systems. The computational complexity of each PGC step is shown to be $O(NT)$, where $T$ is the length of the time horizon. The proposed preconditioning involves a fixed number of block Jacobi iterations per PCG step. A decreasing analytic bound on the effective conditioning is given in terms of this number. The computations are decomposable across the spatial and temporal dimensions of the optimal control problem, into sub-problems of size independent of $N$ and $T$. Numerical results are provided for a mass-spring-damper chain.