User gilles gnacadja - MathOverflow

Matrix Operations Preserving Hurwitz Stability

2012-05-16T03:03:35Z

I begin with terminology I use in the question. A real square matrix $A$ is

negative-stable if for every eigenvalue $\lambda$ of $A$, ${\mathrm{Re}}(\lambda) < 0$;
$\ast$-negative-stable if for every eigenvalue $\lambda$ of $A$, either $\lambda = 0$ or ${\mathrm{Re}}(\lambda) < 0$;
nonpositive-stable if for every eigenvalue $\lambda$ of $A$, ${\mathrm{Re}}(\lambda) \leqslant 0$.

I made up the term '$\ast$-negative-stable' and I would welcome better and/or established terminology. For example, the Laplacian matrix of a nonnegatively weighted (directed or undirected) graph is $\ast$-negative-stable.

To put it broadly, I am looking for what is known about matrix operations that preserve the above stability properties.

Let $A$ be a real $n{\times}n$ matrix and let $u$, $v$, $w$ be real $n{\times}1$ vectors. Consider the real $n{\times}n$ matrices $D = \mathrm{diag}(u)$ and $B = vw^{\mathrm{T}}$, and the real number $\alpha = w^{\mathrm{T}}v$. I am particularly interested in what additional conditions on the matrix $A$ would make the following implications true. (I do not mean simultaneously true.) They concern preserving stability from $A$ to $AD$ for the first three and from $A$ to $A+B$ for the last three.

( $A$ is negative-stable and $u$ is positive ) $\Rightarrow$ ( $AD$ is negative-stable )
( $A$ is $\ast$-negative-stable and $u$ is nonnegative ) $\Rightarrow$ ( $AD$ is $\ast$-negative-stable )
( $A$ is nonpositive-stable and $u$ is nonnegative ) $\Rightarrow$ ( $AD$ is nonpositive-stable )
( $A$ is negative-stable and $\alpha < 0$ ) $\Rightarrow$ ( $A + B$ is negative-stable )
( $A$ is $\ast$-negative-stable and $\alpha \leqslant 0$ ) $\Rightarrow$ ( $A + B$ is $\ast$-negative-stable )
( $A$ is nonpositive-stable and $\alpha \leqslant 0$ ) $\Rightarrow$ ( $A + B$ is nonpositive-stable )

In implications 2 and 5 about $\ast$-negative-stability, it would be acceptable to assume that $A$ is similar to a Laplacian matrix (but Laplacian matrices should not be assumed to be symmetric). Would that be sufficient?

Addendum 1

Here is a way $\ast$-negative stability can be useful in studying negative/Hurwitz stability. Suppose I know that a matrix $A$ is similar to $C = \begin{pmatrix} O_{p{\times}p} & O_{p{\times}q} \\ S & T \end{pmatrix}$, where $T$ is nonsingular. Then $T$ is negative-stable if and only if $A$ is $\ast$-negative-stable, a potentially useful observation if $A$ looks easier to work with than $T$. The notion of nonpositive stability can become useful in similar (but not identical) circumstances.

Preserving the algebraic multiplicity of the zero eigenvalue

2011-09-20T17:20:22Z

Let $M$ and $N$ be two real square matrices of size $p+q$. The matrix $M$ is nonsingular. The matrix $N$ has the following block structure, where $A$ is a $q{\times}p$ matrix. $N = \left(\begin{array}{cc} \text{O}_{p,p} & \text{O}_{p,q} \\ A & \text{Id}_{q} \end{array}\right)$ . I would like to conclude that the algebraic multiplicity of zero as an eigenvalue of the product matrix $MN$ is $p$. (The geometric multiplicity of the zero eigenvalue is preserved as $p$ from $N$ to $MN$.) I am looking for arguments to prove (or disprove) that conclusion.

Equilibria Exist in Compact Convex Forward-Invariant Sets

2011-06-19T00:12:11Z

Theorem. Consider a continuous map $f : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}$ and suppose that the autonomous dynamical system $\dot{x} = f(x)$ has a semiflow $\varphi : {\mathbb{R}}_{\geq{0}} \times {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}$. Let $K \subseteq {\mathbb{R}}^{n}$. If $K$ is nonempty, compact, convex and forward-invariant, then $K$ contains an equilibrium of the dynamical system, i.e. a zero of the map $f$.

According to a reliable source, the above theorem is a standard result everyone uses in dynamical systems without proof. I propose a proof in "Equilibria Exist in Compact Convex Forward-Invariant Sets" at http://math.GillesGnacadja.info/files/EquilExists.html. I am interested in comments on this proof, in references to this or other proofs in the literature, and in new/better proofs.

Map Transformation to Force Convergence to Unique Fixed Point

2010-12-13T02:29:02Z

Is there a transformation $\mathcal{T}$ of maps $\mathbb{R}_{{\geq}0}^{n} \rightarrow \mathbb{R}_{{\geq}0}^{n}$ with the following property?

If a map $F : \mathbb{R}_{{\geq}0}^{n} \rightarrow \mathbb{R}_{{\geq}0}^{n}$ is smooth and order-reversing with respect to the product order and possesses a unique fixed point $\omega$, then

The point $\omega$ is the unique fixed point of ${\mathcal{T}}\!F$; and

For some (known) point $a \in \mathbb{R}_{{\geq}0}^{n}$, the sequence of iterates of ${\mathcal{T}}\!F$ starting at $a$ converges to the (unknown) point $\omega$.

For more on this question, please refer to the 3-page PDF document at this address: http://math.gillesgnacadja.info/files/FixedPointAlgo_OPEN.html. I would have liked to post everything here but I could not find a way to save and preview the question before posting it. The content of the document is as follows.

The Question
Why this Question?
The Trivial Case $n = 1$
Where Does This Question Come From?
Satisfying the Hypotheses of the Question
An Analogous Question

Comment by Gilles Gnacadja

2012-08-11T22:48:59Z

A colleague showed me an article that essentially has the result: "The Brouwer Fixed Point Theorem Applied to Rumour Transmission", dx.doi.org/10.1016/j.aml.2006.02.007. The article is dated 2005/2006. There have to be earlier references.

Comment by Gilles Gnacadja

2012-05-16T17:32:56Z

Thanks, Federico. I recognize that "$\ast$-negative-stable" and "nonpositive-stable" can seem strange without explanation of where they come from. Also, note that $\ast$-negative-stable matrices do not have purely imaginary eigenvalues, except possibly zero. I augmented my question with Addendum 1 to provide some motivation.

Comment by Gilles Gnacadja

2011-09-20T18:53:23Z

Thank you for this method of generating counterexamples.

Comment by Gilles Gnacadja

2011-09-20T18:39:34Z

Thank you for the obvious counterexample.

Comment by Gilles Gnacadja

2011-06-22T01:37:42Z

Thanks Jaap, for catching and illustrating this insufficiency. I changed the statement. Now I explicitly require the existence of the semiflow. In my intended application, the map $f$ is a polynomial describing the kinetics of a chemical reaction network and time runs from zero to infinity. So I believe it would be too strong to require (global) Lipschitz continuity and too weak to require local Lipschitz continuity. Thanks again.

Comment by Gilles Gnacadja

2010-12-13T16:58:45Z

Thanks to fedja for the question and to Ricky Demer for the answer. The order is indeed the product order. I added the precision in the question.