Sufficient conditions for function space controllability and feedback stabilizability of linear retarded systems

Abstract

New sufficient conditions for function space controllability and hence feedback stabilizability of linear retarded systems are presented. These conditions were obtained by treating the retarded systems as a special case of an abstract equation in Hilbert space <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R^{n}\times L_{2}([- h, 0], R^{n})</tex> (denoted as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M_{2</tex> }). For systems of type <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\cdot{x}(t)=A_{0}x(t)+A_{1}x(t-h)+Bu(t)</tex> , it is shown that most of controllability properties are described by a certain polynomial matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P(\lambda)</tex> , whose columns can be generated by an algorithm comparing <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A_{0}^{i}B,A_{0}^{i} B</tex> and mixed powers of A <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> and A <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> multiplied by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B.</tex> It is shown that the M <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> -approximate controllability of the system is guaranteed by certain triangularity properties of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P(\lambda)</tex> . By using the Luenberger canonical form, it is shown that the system is M <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> -approximately controllable if the pair <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(A_{1},B)</tex> is controllable and if each of the spaces spanned by columns of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[B,A_{1}B,... ,A_{1}^{j}B], j=O...n-1</tex> , is invariant under transformation A <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> . Other conditions of this type are also given. Since the M <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> -approximate controllability implies controllability of all the eigenmodes of the system, the feedback stabilizability with an arbitrary exponential decay rate is guaranteed under hypotheses leading to M <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> -approximate controllability. Some examples are given.