On the outputs of linear control systems

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×Cn 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 L2([0,T]) or C([0,T]), then has dense range.