Bifurcation and periodic points in the $l_{1}$-norm minimization problem

We explore an optimization problem which arises naturally in the design of feedback controllers to achieve optimal robustness. Stated mathematically, the problem imposes an $l_{1}$-norm objective on the input and output signals of a linear discrete-time dynamic system. Recently I presented an algorithm which systematically determines initial conditions for which exact solutions can be found. The contribution of this article is twofold. Firstly, we illustrate the usefulness of the algorithm in understanding optimal dynamic response for a specific example. Secondly, we investigate the apparent disappearance of an attracting periodic point as an input data parameter is varied. I conjecture that the dynamic evolution of optimal solutions may exhibit chaos.