Controllability of Linear Discrete Systems with Constant Coefficients and Pure Delay

On relative controllability of delayed difference equations with multiple control functions

“…for some v ∈ ker K, where u * = (u * 1 (0),..., u * N (0), u * 1 (1),...,u * N (k 1 − 1)) T ∈ R Nk 1 , ξ and K are given by (6) and (10), respectively. Remark 1.…”

“…However, only recently the method was used to obtain representations of solutions of delayed differential [8,9,2,3,14] and difference equations [5,10]. The representations have various applications (see papers [2,3,4,6,11,12,13,14,15] and the references therein) in theory of stability, controllability, observability, boundary value problems, etc.…”

Fredholm’s boundary-value problems for differential systems with a single delay

“…Due to the delayed matrix exponential (11), this matrix can be constructed explicitly. In the space C 1 [0, T ] of functions continuously differentiable on [0, T ], the periodic boundary-value problem (7), (17) does not always have a solution even if det Q = 0 because the transformation of system (7) to system (12) has been performed ignoring the condition of solution the z and its initial function ψ being continuous at t = 0.…”

Representation of solutions of discrete linear delay systems with non permutable matrices

“…Advantage of discrete delayed exponential matrix is to help transferring the classical idea to represent the solution of linear ordinary differential equations into linear delay discrete equations. Although there are many continued contributions in a discrete linear system with pure delay with permutable matrices, to stability theory [5], [6], [7], [9], [11], controllability theory [12], [13], [10], delay oscillating systems [14], discrete linear system with two delays [3], [4], Fredholm integral equations [15], no results were obtained for such systems with non permutable matrices. It should be mentioned that recently non permutable case for the contiunous delay linear systems was considered in [8].…”