Generalization of the concept of diagonal dominance with applications to matrix D-stability

Abstract

In this paper, we introduce the class of diagonally dominant with respect to a given LMI region $\mathfrak{D}\subset \mathbb{C}$ matrices. They are shown to possess the analogues of well-known properties of (classical) diagonally dominant matrices, e.g. their spectra are localized inside the region $\mathfrak{D}$. Moreover, we show that in some cases, diagonal $\mathfrak{D}$-dominance implies $(\mathfrak{D},\mathcal{D})$-stability (i.e. the preservation of matrix spectra localization under multiplication by a positive diagonal matrix).

Introduction

Here, we introduce the following notations:

• ${\mathcal{M}}^{n\times n}$ for the set of real $n\times n$ matrices;

• ${\mathcal{M}}^{n\times n}(\mathbb{C})$ for the set of complex $n\times n$ matrices;

• $\sigma (\mathbf{A})$ for the spectrum of a matrix A $\in {\mathcal{M}}^{n\times n}$ (i.e. the set of all eigenvalues of A defined as zeroes of its characteristic polynomial $f_{\mathbf{A}}(\lambda ):=\det (\lambda \mathbf{I}-\mathbf{A})$);

• ${\mathbb{C}}^{-}$ for the open left-hand side of the complex plane $\mathbb{C}$, i.e.

  $${\mathbb{C}}^{-}=\{z\in \mathbb{C}:\mathrm{Re}(z)<0\};$$

• $\overline{{\mathbb{C}}^{+}}$ for the closed right-hand side of the complex plane $\mathbb{C}$, i.e.

  $$\overline{{\mathbb{C}}^{+}}=\{z\in \mathbb{C}:\mathrm{Re}(z)\ge 0\};$$

• ${\mathcal{D}}^{+}\subset {\mathcal{M}}^{n\times n}$ for the set of all positive diagonal matrices (i.e. matrices with positive entries on the principal diagonal while the entries outside the principal diagonal are zeroes);

• ${\mathcal{D}}_{(0,1]}^{+}\subset {\mathcal{M}}^{n\times n}$ for the subclass of ${\mathcal{D}}^{+}$, defined as follows:

  $${\mathcal{D}}_{(0,1]}^{+}=\{\mathbf{D}=\mathrm{diag}\{d_{11},\dots ,d_{nn}\}:0<d_{ii}\le 1,i=1,\dots ,n\};$$

• ${\mathcal{D}}_{\ge 1}^{+}\subset {\mathcal{M}}^{n\times n}$ for the subclass of ${\mathcal{D}}^{+}$, defined as follows:

  $${\mathcal{D}}_{\ge 1}^{+}=\{\mathbf{D}=\mathrm{diag}\{d_{11},\dots ,d_{nn}\}:1\le d_{ii}<+\infty ,i=1,\dots ,n\}.$$

In this paper, we generalize the concept of diagonal dominance. First, let us recall the classical definition of diagonally dominant matrices (see, for example, [7], also [15], [16]).

Definition 1

A matrix $\mathbf{A}\in {\mathcal{M}}^{n\times n}(\mathbb{C})$ is called strictly row diagonally dominant if the following inequalities hold:

$$|a_{ii}|>\sum _{i\ne j}|a_{ij}|i=1,\dots ,n.$$

A matrix A is called strictly column diagonally dominant if ${\mathbf{A}}^T$ is strictly row diagonally dominant.

Definition 1'. A matrix $\mathbf{A}\in {\mathcal{M}}^{n\times n}(\mathbb{C})$ is called generalized diagonally dominant if there exist positive scalars (weights) $m_i$, $i=1,\dots ,n$ such that

$$m_i|a_{ii}|>\sum _{i\ne j}{m}_j|a_{ij}|,i=1,\dots ,n$$

(i.e. if there is a positive diagonal matrix $\mathbf{M}=\mathrm{diag}\{m_1,\dots ,m_n\}$, such that $\mathbf{A}\mathbf{M}$ is strictly row diagonally dominant). If, in addition, $a_{ii}<0$, $i=1,\dots ,n$, then A is called negative diagonally dominant (NDD).

It is well-known (see, for example, [18], p. 382, Theorem 4.C.2) that if a complex matrix A is NDD, then A is Hurwitz stable, i.e. every $\lambda \in \sigma (\mathbf{A})$ satisfies $\mathrm{Re}(\lambda )<0$. The stability of A implies the Lyapunov asymptotic stability of a system of ODE with the system matrix A. Being a sufficient for stability condition, negative diagonal dominance is of particular interest by itself due to its connection to the properties of nonlinear systems (see [17]). Less known is the fact that negative diagonal dominance implies some special concepts of matrix and system stability, that are stronger than just stability and shows stability preservation under specific matrix (system) perturbations. Such concepts include multiplicative D-stability (see, for example, [8], [10]).

Definition 2

A matrix A $\in {\mathcal{M}}^{n\times n}$ is called (multiplicative) D-stable if $\mathrm{Re}(\lambda )<0$ for all $\lambda \in \sigma (\mathbf{DA})$, where D is any matrix from ${\mathcal{D}}^{+}$.

Matrix D-stability plays an important role in the theory of stability (see, for example, [14], [9], [12] and references therein) and has numerous applications in the economics, mathematical ecology, mechanics and other branches of science. The proof of the fact that NDD matrices are D-stable is based on the simple observance that the negative diagonal dominance implies stability and is preserved under multiplication by a positive diagonal matrix (see [8], p. 54, Observation (i)). Note, that the property of diagonal dominance can be proved (disproved) by a finite number of steps while the conditions of D-stability of A involve checking all the products of the form $\mathbf{D}\mathbf{A}$, where D runs along the infinite set of positive diagonal matrices.

In this paper, we are concerned with the problem of a matrix spectra localization inside a prescribed convex region $\mathfrak{D}\subset \mathbb{C}$ (so called $\mathfrak{D}$-stability problem) and its robust aspects. Many problems of the system dynamics lead to establishing $\mathfrak{D}$-stability of the system matrix with respect to a specified region $\mathfrak{D}\subset \mathbb{C}$ (see, for example, [5], [6]). The regions of particular interest are LMI (Linear Matrix Inequality) regions, introduced in [3], that include the shifted left half-plane, the unit disk and the conic sector around the negative direction of the real axis. Due to the rapid advance in control theory and its application, we face the problem of finding some easy-to-verify conditions, which allow us to establish:

• $\mathfrak{D}$-stability of a given matrix A;

• the preservation of $\mathfrak{D}$-stability under some specific perturbations of A, e.g. multiplication by a diagonal matrix.

Though an amount of research (see [5], [6]) is due to $\mathfrak{D}$-stability (or eigenvalue clustering in a region $\mathfrak{D}\subset \mathbb{C}$), the progress in describing the matrix classes which are $\mathfrak{D}$-stable, and, moreover, preserve $\mathfrak{D}$-stability under multiplication by a specific diagonal matrix is still very little and concerns classical regions such as the left half-plane (see [1], [4], [8]) and the unit disk (see [9]). Here, we make certain efforts to fill this gap.

The paper is organized as follows. In Section 2, we study unbounded LMI regions and their properties, focusing on the most well-known examples, such as the shifted left half-plane, the conic sector, hyperbola and parabola. We also recall the concept of $(\mathfrak{D},D)$-stability for an unbounded LMI region $\mathfrak{D}$. In Section 3, we introduce the crucial concept of this paper, namely, diagonal dominance with respect to a given LMI region $\mathfrak{D}$ (so-called diagonal $\mathfrak{D}$-dominance). We consider the particular cases of diagonal $\mathfrak{D}$-dominance, with respect to the most important LMI regions. Section 4 deals with the basic properties of diagonally $\mathfrak{D}$-dominant matrices. Section 5 gives the most important results of the paper, i.e. the implications between diagonal $\mathfrak{D}$-dominance, $\mathfrak{D}$-stability and $(\mathfrak{D},D)$-stability.