On Uncontrollable Discrete-Time Bilinear Systems Which are “Nearly” Controllable

Study of a Class of Almost‐controllable Discrete‐time Bilinear Systems

“…One illustrates that a discrete bilinear system is almost everywhere on R n and is nearly controllable, although the system itself is not controllable. The other indicates that the result in [20] is a particular case of our results and that the controllability counterexample in [21] can be extended to n-dimensional bilinear systems.…”

Approximately nearly controllability of discrete-time bilinear control systems with control input characteristic

“…Some recent notable sequence works on bilinear control systems have been carried out by Tie and his co-authors [10,11,8,12]. In [10,12] they consider the following discrete-time bilinear systems with a single input:…”

“…In [10,12], Tie, et al proposed the concept nearly controllable which means there exists an exactly controllable subset with full Lebesgue measure in the state space R n . They have shown that: The system (1) is nearly controllable if B is a real diagonal matrix with the elements in diagonal being non-zero and pair-wisely distinct in [10].…”

“…They have shown that: The system (1) is nearly controllable if B is a real diagonal matrix with the elements in diagonal being non-zero and pair-wisely distinct in [10]. More recently, Tie in [12] generalized it to the case that B has real non-zero eigenvalues (not necessary to be diagonalizable) and obtained that the system (1) is nearly controllable if and only if each eigenvalue of B has only one Jordan block and the order of each block is at most two by different approach.…”

On small-controllability of a class of Hammerstein-nonlinear systems – an implicit function approach

“…, u(l − 1) such that ξ can be transferred to η at k = l. Controllability of discretetime nonlinear systems has been studied (Fliess and Normand-Cyrot 1981;Jakubczyk and Sontag 1990;Albertini and Sontag 1994;Wirth 1998;Klamka 2002) challenges remain. Even for the special class of nonlinear systems, say, discrete-time bilinear systems, most controllability problems are unsolved (Tarn, Elliott, and Goka 1973;Cheng, Tarn, and Elliott 1975;Elliott 2009;Tie, Cai, and Lin 2010;Tie and Cai 2012). The main difficulty in investigating the controllability problems of discretetime nonlinear systems is that semigroups tend to appear so that less algebraic structure of the systems is available.…”