Repetitive control tools for an original approach to convex optimization problems under affine periodic constraints

This paper provides a solution to the online convex optimization problem under a class of affine constraints, periodic with a known period. Functions whose minimizer vector exhibits a constant component within the kernel space of the constraint horizontal matrix are considered. By resorting to the latest developments in the repetitive control (RC) theory, two algorithms are originally presented: the first one resorting to the point-wise use of the delay as a universal periodic signal generator, the second one relying on the PDE (Partial Differential Equation) transport-equation-based theory. Both of them naturally extend the standard primal–dual algorithm acting in the constant constraint scenario, while guaranteeing global asymptotic convergence properties. Indeed, the two main different RC approaches in the literature are applied to the same optimization problem, while drawing original conclusions under the adoption of a common view. The derivation of an internal-model-based finite-dimensional (spectral) approximation for the latter introduces a further interpretation of the renowned adaptive learning control.