Stability of linear systems via eigenvalues

Builds on Homogeneous linear system with constant coefficients Matrix exponential as the solution of a linear system Lyapunov stability Asymptotic stability

Statement

Let $A$ be a real $n \times n$ matrix and consider the linear system with constant coefficients $x' = A x$ on $\mathbb{R}^n$ , with the equilibrium point $0$ . (1) The point $0$ is asymptotically stable if and only if each eigenvalue $\lambda$ of $A$ has $\operatorname{Re} \lambda < 0$ . (2) If some eigenvalue $\lambda$ of $A$ has $\operatorname{Re} \lambda > 0$ , then $0$ is unstable. (3) The point $0$ is Lyapunov stable if and only if two conditions hold. Each eigenvalue has $\operatorname{Re} \lambda \le 0$ . Each eigenvalue with $\operatorname{Re} \lambda = 0$ has equal algebraic and geometric multiplicities.

Remarks

The proof uses the matrix exponential. The entries of $e^{tA}$ are sums of terms $t^k e^{\lambda t}$ . Here $\lambda$ is an eigenvalue and $k$ is less than the size of its largest Jordan block. In case (1), there are constants $C, \alpha > 0$ with $\|e^{tA}\| \le C e^{-\alpha t}$ for $t \ge 0$ . So the origin is globally and exponentially stable. For $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ , both eigenvalues are $0$ but the solution $(t, 1)$ is unbounded. So $0$ is unstable, in accordance with (3). The Routh–Hurwitz criterion tests the condition of (1) from the coefficients of the characteristic polynomial, without computing the roots.

Reading path

To understand this page from the ground up, read these 21 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Differential equation concept
Function concept
Interval concept
Ordinary differential equation definition
Lipschitz condition definition
Grönwall's inequality statement
Order of a differential equation definition
Solution of an ordinary differential equation definition
Autonomous system definition
Initial value problem definition
Linear differential equation of order n definition
Equilibrium point definition
System of linear differential equations definition
Uniqueness of solutions for Lipschitz vector fields statement
Homogeneous linear system with constant coefficients definition
Maximal solution definition
Matrix exponential as the solution of a linear system statement
Flow of a vector field definition
Lyapunov stability definition
Asymptotic stability definition
Stability of linear systems via eigenvalues statement

Where this leads

12 pages build on this one.

Tag	`0SKVEMXJWF`
URI	`m2://0SKVEMXJWF`
Kind	statement, theorem
Writing	W0 stub
MSC	34D20, 34A30
Status	This copy of the site has no build data. Run `m2 status 0SKVEMXJWF`.
Uses	Homogeneous linear system with constant coefficients, Matrix exponential as the solution of a linear system, Lyapunov stability, Asymptotic stability

Markdown · JSON · Edit on GitHub

What this page needs for W1

an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.