Optimality of linear ℓ1 compensators in nonlinear feedback class

A simple scalar ℓ¹ optimal compensator design problem is stated for 1-block scalar system but in the class of all nonlinear causal feedbacks. It is well known that nonlinear state-space compensators are not worse than linear dynamic ones in the standard ℓ¹ optimal control problem. The inverse assertion is proved now for special cases only [1]. The question is, under which conditions the optimal linear compensator can not be improved by nonlinear admissible compensators? This minimax control design problem is reduced for the general case to a linear maximization on the binary graph [2]. For the special case of a strong nonminimum phased plant the problem was solved using elementary technique.