In this paper, a novel way to handle the analysis and design of control systems with bounded asymmetrical signals is presented. Instead of the standard peak-to-peak norm, normally used in l(1)-optimal control, this paper proposes to use an asymmetric objective functional, which simplifies the consideration of asymmetrically bounded signals. Necessary and sufficient conditions are derived to check if some performance specifications, expressed using this objective functional, are fulfilled. The analysis and the synthesis problems are also formulated with respect to this objective functional: as in the standard l(1) control, this approach leads to a linear programming problem which can be solved by available optimization tools. A numerical example is also studied to clarify this proposed approach.