On the outputs of linear control systems

• Published in: Journal of mathematical analysis and applications Vol. 340; no. 1; pp. 116 - 125
• Main Author: Shapiro, Joel H.
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 01.04.2008; Elsevier
• Subjects: Algebra; Algebraic geometry; Exact sciences and technology; Functional analysis; Linear control system; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Sciences and techniques of general use; Several complex variables and analytic spaces; Sobolev space; Volterra operator; Sobolev space; Linear control system; Volterra operator; Complex function; Approximation; Linear system; Mathematical analysis; Control system; Bounded operator; Time interval; Banach space; Application
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2007.08.032

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.