Bifurcation and Stable Fixed Points in the l 1 -norm Minimization Problem

The topic of this paper is an optimization problem which arises naturally in the design of feedback controllers to achieve optimal robustness. Stated mathematically, the problem imposes an l1-norm objective on the input and output signals of a linear discrete-time dynamic system. In a recent paper an algorithm has been presented which systematically determines initial conditions for which exact solutions can be found. The contribution of this paper is twofold. Firstly, we illustrate the usefulness of the algorithm in understanding optimal dynamic response for a speci…c example. Secondly, we investigate the creation and apparent disappearance of stable …xed points as an input data parameter is varied. For the …rst time to our knowledge the conjecture is made that the dynamic evolution of optimal solutions may exhibit chaos.