Structural stability, asymptotic stability and exponential stability for linear multidimensional systems: the good, the bad and the ugly

ABSTRACT

In this paper, we investigate three concepts of stability for linear two-dimensional systems: the ‘good’ structural stability (an algebraic property linked to the location of the roots of a certain characteristic polynomial), the ‘bad’ asymptotic stability (roughly the trajectory converges to the equilibrium point) and the ‘ugly’ exponential stability (the rate of convergence is at least exponential). More precisely, we show that for a usual set of boundary conditions taken along the positive semi-axes, structural stability and exponential stability are equivalent notions. For this particular set of boundary conditions, we further prove that structural stability implies asymptotic stability but a counterexample shows that asymptotic stability does not imply structural stability which is a major difference compared to the one-dimensional case. This also highlights the importance of the boundary conditions when one works with multidimensional systems.

KEYWORDS:

• Multidimensional systems
• linear systems
• discrete systems
• structural stability
• asymptotic stability
• exponential stability

Acknowledgments

We would like to thank the anonymous reviewers for their valuable comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. In the 2D case (i.e. m = 2), $x\in c_0({\mathbb{N}}^2,{\mathbb{R}}^d)$ means: $\forall ϵ>0,\exists N_{ϵ}\in \mathbb{N}$ such that $\forall (i,j)\in {\mathbb{N}}^2,i+j\ge N_{ϵ}\Rightarrow \parallel x(i,j)\parallel \le ϵ$.

2. Note that the same remark applies to the sequence of matrices ${\left(U(i,j)\right)}_{(i,j)\in {\mathbb{N}}^2}$ defined by Equation (3(3) $$\begin{array}{c}\hfill \left\{\begin{array}{c}U(i+1,j+1)=AU(i,j+1)+BU(i+1,j),\hfill \\ \forall i,j\in \mathbb{N},U(i,0)=A^i,U(0,j)=B^j.\hfill \end{array}\right.\end{array}$$(3) ).

3. Recall that the induced matrix norm on ${\mathbb{R}}^{d\times d}$ is defined by: $\parallel A\parallel :=sup\left\{∥Ax∥/x\in {\mathbb{R}}^d,∥x∥=1\right\}$.

4. It can be obtained by differentiating n times the geometric power series ∑ _k x ^k and evaluating the result at x = 1/2.

5. For the readers familiar with time-delay systems, this is similar to the tradeoff between the delay independent or delay dependent stability conditions.

Additional information

Funding

Agence Nationale de la Recherche [grant number MSDOS ANR-13-BS03-0005].