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Let the matrix operator L = D∂xx + q (x) A0, with D = diag (1, ν), ν ≠ 1, q ∈ L∞ (0, π), and A0 is a Jordan block of order 1. We analyze the boundary null controllability for the system yt − Ly = 0. When ν ∉ Q + * and q is constant, q = 1 for instance, there exists a family of root vectors of ( L * , D ( L * )) forming a Riesz basis of L2(0,π;ℝ2). Moreover F. Ammar Khodja et al. [J. Funct. Anal.267 (2014) 2077–2151] shows the existence of a minimal time of control depending on condensation of eigenvalues of ( L * , D ( L * )) , that is to say the existence of T0 (ν) such that the system is null controllable at time T > T0 (ν) and not null controllable at time T < T0 (ν). In the same paper, the authors prove that for all τ ∈ [0, + ∞], there exists ν ∈] 0, + ∞ [ such that T0 (ν) = τ. When q depends on x, the property of Riesz basis is no more guaranteed. This leads to a new phenomena: simultaneous condensation of eigenvalues and eigenfunctions. This condensation affects the time of null controllability. [ABSTRACT FROM AUTHOR]

Abstract The paper determines the minimum number of general or dedicated controllers which are required to guarantee the controllability of a class of dynamical systems. For this purpose, we begin by reformulating Kalman controllability condition for linear systems (A, B) in the context of the similarity equivalence class associated with A , and then we provide new alternative characterizations of such controllability condition based on the cases when matrix A is in Jordan or Frobenius form, so that theoretical aspects and computational requirements are comparatively commented. In the second part of the paper, given a system state matrix A , we solve the optimal design problem of determining the minimum number of controllers (columns of B) required for making (A, B) controllable, resorting also to the Jordan and Frobenius canonical forms in the similarity equivalence class associated with A. These canonical forms are proven to be also fundamental to provide bounds on the minimum number of dedicated controllers (corresponding to columns of B with a single non-zero element) required to make (A, B) controllable. Some examples serve to illustrate the fundamental results. [ABSTRACT FROM AUTHOR]