On the Stability and Null-Controllability of an Infinite System of Linear Differential Equations

Abstract

In this work, the null controllability problem for a linear system in ℓ2 is considered, where the matrix of a linear operator describing the system is an infinite matrix with $\lambda \in \mathbb {R}$ λ ∈ ℝ on the main diagonal and 1s above it. We show that the system is asymptotically stable if and only if λ ≤− 1, which shows the fine difference between the finite and the infinite-dimensional systems. When λ ≤− 1 we also show that the system is null controllable in large. Further we show a dependence of the stability on the norm, i.e. the same system considered $\ell ^{\infty }$ ℓ ∞ is not asymptotically stable if λ = − 1.

On a Linear Differential Game of Pursuit with Integral Constraints in ℓ2

“…We are going to consider the cases λ < −1 and λ = −1 separately to prove global null controllability. The proof is similar to that of [14], here we bring it for completeness.…”

“…We will show that ∥W −1 (τ)x 0 ∥ 2 → 0 as τ → +∞. In order to do so, we need to improve the inequality in Equation (14).…”

“…These schemes converge, but the approximating controls do not stabilize the original system, and the costs do not converge. A different example was considered in [14], where the stability and controllability of a system in the form of an infinite Jordan block, with λ ∈ R on the main diagonal is considered.…”

On a linear differential game in the Hilbert space $\ell^2$

“…A motivation for the setup comes from control problems for evolutionary PDEs, where using suitable decomposition of the control problem (see for example [4,5,[8][9][10]) x i would be a Fourier coefficients of an unknown function, while u i and v i would be that of control parameters. Also, the setup is of independent interest as a controlled system in a Banach space (for works in this spirit see for example [6,13]). Differential games for infinite dimensional systems are also well studied, for example, when the evolution of the system is governed by parabolic equations pursuit evasion problems are considered in [21][22][23], where the problem for the partial differential equations is reduced to an infinite system of ordinary differential equations.…”

On a Linear Differential Game in the Hilbert Space ℓ2

“…Motivation for the setup comes from control problems for evolutionary PDEs, where using suitable decomposition of the control problem (see, for example, [1][2][3][4][5]) x i would be a Fourier coefficient of an unknown function, while u i and v i would be that of control parameters. Also, the setup is of independent interest as a controlled system in a Banach space (for works in this spirit see for example [6][7][8][9][10][11][12][13]). Differential games for infinite dimensional systems are also well studied, for example, when the evolution of the system is governed by parabolic equations pursuit-evasion problems are considered in [14][15][16], where the problem for the partial differential equations is reduced to an infinite system of ordinary differential equations.…”