Relative Controllability of Linear Difference Equations

Abstract

In this paper, we study the relative controllability of linear difference equations with multiple delays in the state by using a suitable formula for the solutions of such systems in terms of their initial conditions, their control inputs, and some matrix-valued coefficients obtained recursively from the matrices defining the system. Thanks to such formula, we characterize relative controllability in time T in terms of an algebraic property of the matrix-valued coefficients, which reduces to the usual Kalman controllability criterion in the case of a single delay. Relative controllability is studied for solutions in the set of all functions and in the function spaces L p and C k . We also compare the relative controllability of the system for different delays in terms of their rational dependence structure, proving that relative controllability for some delays implies relative controllability for all delays that are "less rationally dependent" than the original ones, in a sense that we make precise. Finally, we provide an upper bound on the minimal controllability time for a system depending only on its dimension and on its largest delay.Notations In this paper, we denote by N and N * the sets of nonnegative and positive integers, respectively. For a, b ∈ R, we write the set of all integers between a and b as a,The cardinality of a set N is denoted by #N. For ξ ∈ R N , we use ξ min and ξ max to denote the smallest and the largest components of ξ , respectively. For ξ ∈ R, the symbol ⌊ξ ⌋ is used to the denote the integer part of ξ , i.e., the unique integer such that ξ − 1 < ⌊ξ ⌋ ≤ ξ .The set of d × m matrices with coefficients in

Approximate and exact controllability of linear difference equations

“…Throughout the paper, we rely on a basic tool for the controllability analysis of (1.1), namely a suitable representation formula, describing a solution at time t in terms of its initial condition, the control input, and some matrix-valued coefficients computed recursively (see Proposition 2.4). Such a formula, already proved in [24], generalizes the ones obtained in [4,Th. 3…”

“…This notion of solution, already used in [24] and similar to the one used in [3], requires no regularity on x 0 , u, or x. Nonetheless, such a weak framework is enough to guarantee existence and uniqueness of solutions.…”

“…Proposition 2.2 can be easily proved from (1.1), which is already an explicit representation formula for the solution in terms of the initial condition and the control when t < Λ min . Its proof can be found in [24,Prop. 2.2] and is very similar to that of [3,Prop.…”

“…Due to the infinite-dimensional nature of the dynamics of difference equations and neutral functional differential equations, several different notions of controllability can be used, such as approximate, exact, spectral, or relative controllability [34,5,12,31,24]. Relative controllability was originally introduced in the study of control systems with delays in the control input [5] and consists in controlling the value of x(T ) ∈ C d at some prescribed time T .…”

“…Relative controllability was originally introduced in the study of control systems with delays in the control input [5] and consists in controlling the value of x(T ) ∈ C d at some prescribed time T . In the context of difference equations of the form (1.1), it was characterized in some particular situations with integer delays in [12,31], with a complete characterization on the general case provided in [24].…”