On the outputs of linear control systems

Abstract

This paper studies autonomous, single-input, single-output linear control systems on finite time intervals. The object of interest is the output operator O , which associates to each input function and initial state vector the corresponding system output. Main result: If the system has relative degree r < ∞ , then for any “admissible” Banach space U of inputs, O is a bounded operator taking U × C n onto the “Sobolev space” of complex functions f ∈ C ( r − 1 ) ( [ 0 , T ] ) for which the ( r − 1 ) -order derivative f ( r − 1 ) is absolutely continuous, with f ( r ) ∈ U . This completes recent results of Jönsson and Martin [Ulf Jönsson, Clyde Martin, Approximation with the output of linear control systems, J. Math. Anal. Appl. 329 (2007) 798–821] who showed that if the system is minimal and U is either L 2 ( [ 0 , T ] ) or C ( [ 0 , T ] ) , then O : U × C n → U has dense range.