Topological classification and structural stability of linear systems

Abstract In this paper it is shown that the linear systems Σi defined by Σ i : x i = A i x i + B i u i , i = 1, 2 , are topologically equivalent if and only if they have the same Kronecker indices and the flows defined by considering trajectories modulo their controllable subspace are topologically equivalent. From some recent work of N. H. Kuiper (in “Manifolds—Tokyo 1973,” Univ. of Tokyo Press, Tokyo 1975) it is known exactly what this last condition amounts to. With these results at hand it is then not difficult to investigate the structural stability of ∑: x = Ax + Bu and, in fact, structural stability turns out to be generic.