Constrained finite-horizon linear-quadratic optimal control problems are studied within the context of discrete-time dynamics that arise from the series interconnec- tion of subsystems. A structured algorithm is devised for computing the Newton-like steps of primal-dual interior-point methods for solving a particular re-formulation of the problem as a quadratic program. This algorithm has the following properties: (i) the computation cost scales linearly in the number of subsystems along the cascade; and (ii) the computations can be distributed across a linear proces- sor network, with localized problem data dependencies between the processor nodes and low communication overhead. The computation cost of the approach, which is based on a fixed permutation of the primal and dual variables, scales cubically in the time horizon of the original optimal control problem. Limitations in these terms are explored as part of a numerical example. This example involves application of the main results to model data for the cascade dynamics of an automated irrigation channel in particular.