Exact solutions to a two-block l1 optimal control problem

In this paper we illustrate some new ideas in the theory of l1–norm minimisation. A simple looking mathematical programming problem, namely the minimisation of the sum of the one norms of two signals connected by convolution constraints, is investigated. This describes an l1 model matching problem for which there are no zero interpolation conditions, and just one rank interpolation condition. Despite its apparent simplicity, finding exact solutions for this problem is a challenging task. We extend the class of problems for which exact optimal solutions can be found by combining a primal/dual formulation with dynamic programming ideas. These solutions have the desirable feature of yielding a control law in feedback form.