# Characterization of the dissipative mappings and their application to
perturbations of dissipative-Hamiltonian systems

Mohit Kumar Baghel Department of Mathematics, Indian Institute of Technology
Delhi, Hauz Khas, 110016, India. Email: {maz188260,
<EMAIL_ADDRESS>MB acknowledges the support of institute Ph.D.
fellowship by IIT Delhi, India. PS acknowledges the support of the DST-Inspire
Faculty Award (MI01807-G) by Government of India and FIRP project
(FIRP/Proposal Id - 135) by IIT Delhi, India  Nicolas Gillis Department of
Mathematics and Operational Research, University of Mons, Rue de Houdain 9,
7000 Mons, Belgium. Email<EMAIL_ADDRESS>NG acknowledges the
support by ERC starting grant No 679515, the Fonds de la Recherche
Scientifique - FNRS and the Fonds Wetenschappelijk Onderzoek - Vlaanderen
(FWO) under EOS project O005318F-RG47.  Punit Sharma∗

###### Abstract

In this paper, we find necessary and sufficient conditions to identify pairs
of matrices $X$ and $Y$ for which there exists $\Delta\in{\mathbb{C}}^{n,n}$
such that $\Delta+\Delta^{*}$ is positive semidefinite and $\Delta X=Y$. Such
a $\Delta$ is called a dissipative mapping taking $X$ to $Y$. We also provide
two different characterizations for the set of all dissipative mappings, and
use them to characterize the unique dissipative mapping with minimal Frobenius
norm. The minimal-norm dissipative mapping is then used to determine the
distance to asymptotic instability for dissipative-Hamiltonian systems under
general structure-preserving perturbations. We illustrate our results over
some numerical examples and compare them with those of Mehl, Mehrmann and
Sharma (Stability Radii for Linear Hamiltonian Systems with Dissipation Under
Structure-Preserving Perturbations, SIAM J. Mat. Anal. Appl. 37 (4):
1625-1654, 2016).

##  1 Introduction

In this paper, we consider the dissipative mapping problem. Given
$X,Y\in{\mathbb{C}}^{n,m}$, the dissipative mapping problem can be divided in
three subproblems:

  1. 1)

Existence: find necessary and sufficient conditions on matrices $X$ and $Y$
for the existence of matrices $\Delta\in{\mathbb{C}}^{n,n}$ such that
$\Delta+\Delta^{*}$ is positive semidefinite and $\Delta X=Y$.

  2. 2)

Characterization: characterize all such dissipative matrices taking $X$ to
$Y$.

  3. 3)

Minimal norm: characterize all solutions to the dissipative mapping problem
that have minimal norm. In this paper, we focus on the Frobenius norm, which
is the standard in the literature.

The dissipative mapping problem belongs to a wider class of mapping problems:
for $X,Y\in{\mathbb{C}}^{n,m}$, the structured mapping problems require to
find $\Delta\in\mathcal{S}\subseteq{\mathbb{C}}^{n,n}$ such that $\Delta X=Y$,
where $\mathcal{S}$ stands for the structure of the mapping. The mapping
problems have been extensively studied in [12] and [1, 2] for the structures
that are associated with orthosymmetric scalar products. These include
symmetric, skew-symmetric, Hermitian, skew-Hermitian, Hamiltonian, skew-
Hamiltonian, Hermitian, skew-Hermitian, persymmetric, and per-skew symmetric
matrices. If $\mathcal{S}\subseteq{\mathbb{R}}^{n,n}$, then it is the real
structured mapping problem. In particular, for given
$X,Y\in{\mathbb{C}}^{n,m}$, the real dissipative mapping problem is to find
$\Delta\in{\mathbb{R}}^{n,n}$ such that $\Delta+\Delta^{T}$ is positive
semidefinite and $\Delta X=Y$.

The structured mapping problems occur and are useful in solving various
distance problems related to structured matrices and matrix polynomials, see
for example [5, 6, 16, 17, 18] and the references therein. A mapping problem
closely related to the dissipative mapping is the positive semidefinite (PSD)
mapping, when $\mathcal{S}$ is the cone of Hermitian positive semidefinite
matrices. The PSD mapping problem has been recently solved and used in [16,
17] to derive formulas for structured distances to instability for linear
time-invariant dissipative Hamiltonian (DH) systems. DH systems have the form

$\dot{x}(t)=(J-R)Qx(t),$ (1.1)

where $x(t)$ is the state vector at time $t$, $J^{*}=-J\in\mathbb{F}^{n,n}$,
$R^{*}=R\in\mathbb{F}^{n,n}$ is positive semidefinite and is referred to as
the dissipation matrix, and $Q=Q^{*}\in\mathbb{F}^{n,n}$ is positive definite
and describes the energy of the system via the function $x\mapsto x^{*}Qx$.
When the coefficient matrices in (1.1) are complex, that is,.
$\mathbb{F}={\mathbb{C}}$, then it is called a complex DH system, and when the
coefficient matrices in (1.1) are real, that is, $\mathbb{F}={\mathbb{R}}$,
then it is called a real DH system. Such systems are special cases of port-
Hamiltonian systems; see for example [8, 20, 21, 16, 17]. A linear time-
invariant (LTI) control system $\dot{x}=Ax$ with $A\in{\mathbb{C}}^{n,n}$ is
called asymptotically stable around its equilibrium point at the origin if (i)
for any $\epsilon>0$, there exists $\delta_{1}>0$ such that if
$\|x(t_{0})\|<\delta_{1}$, then $\|x(t)\|<\epsilon$, for all $t>t_{0}$, and
(ii) there exists $\delta_{2}>0$ such that if $\|x(t_{0})\|<\delta_{2}$, then
$x(t)\rightarrow 0$ as $t\rightarrow\infty$. It is called stable if only (i)
holds. An equivalent algebraic characterization of stability is given in terms
of spectral conditions on $A$: the system $\dot{x}=Ax$ is called stable if all
eigenvalues of the matrix $A$ are in the closed left half of the complex plane
and those on the imaginary axis are semisimple. It is asymptotically stable if
all eigenvalues of $A$ are in the open left half of the complex plane.

Any LTI stable system can be represented in the form (1.1) of a DH system [7].
DH systems are always stable and remain stable as long as the perturbations
preserve the DH structure. Still, they may have eigenvalues on the imaginary
axis, i.e., they are not necessarily asymptotically stable. Therefore it is
useful to know the _distance to asymptotic instability_ for DH systems [16,
17] defined as the smallest norm of the perturbation that makes system lose
its asymptotic stability. Studying this question is an essential topic in many
applications, like in power system and circuit simulation, see, e.g., [14, 15,
13], and multi-body systems, see, e.g., [9, 22]. This is also useful in some
specific applications, e.g., the analysis of disk brake squeal [16, Example
1.1], of mass-spring-damper dynamical systems [17, Example 4.1], and of
circuit simulation and power system modelling [16, Example 1.2], where the
interest is in studying the stability or the instability under perturbation of
the matrices $J$, $R$ and $Q$. In the DH modelling of physical systems, the
matrix $J$ describes the energy flux among the system’s energy storage
elements, and $R$ represents the damping effects in the system. Thus
perturbing $J$ and $R$ together, or only one at a time, is of particular
interest.

In [16], the authors derived various distances to asymptotic instability for
DH systems (1.1) while focusing on perturbations that affect only one matrix
from $\\{J,R,Q\\}$ at a time. Similarly in [17], the authors derived real
distances to instability for real DH systems while perturbing only the
dissipation matrix $R$. The framework suggested in [16] depends heavily on
reformulating the instability radius problem in terms of an equivalent problem
of minimizing the generalized Rayleigh quotient of two positive semidefinite
matrices. This reformulation was achieved using minimal norm skew-Hermitian
mappings in case only $J$ is perturbed, and PSD mappings in case only $R$ or
$Q$ is perturbed. However, we note that the framework suggested in [16] does
not work if we allow perturbations in the DH system that affect more than one
matrix from $\\{J,R,Q\\}$ at a time. Analyzing the stability of the system
(1.1) when more than one matrix from $\\{J,R,Q\\}$ is perturbed is one of the
main motivations of our work.

In this paper, we focus on perturbations of DH systems that affect both $J$
and $R$ simultaneously. More precisely, we find (see Section 5) that the
minimal-norm solution to the dissipative mapping problem can be a necessary
tool in computing the structured distance to asymptotic instability for DH
systems (1.1) when both $J$ and $R$ are simultaneously perturbed.

###  1.1 Contributions and outline of the paper

In Section 2, we present some preliminary results that will be needed to solve
the dissipative mapping problem. For the solutions to the dissipative mapping
problem, we present two different characterizations in terms of three matrix
variables $K,G,Z$ with symmetry and semidefinite structures. Both
characterizations have an advantage over each other. The first
characterization (see Section 3) results in a straightforward computation of
the minimal-norm solutions to the dissipative mapping problem, but the matrix
variables $K,G,Z$ are highly constrained. We also derive necessary and
sufficient conditions for solving the real dissipative mapping problem and
compute solutions that are of minimal Frobenius norm. The second
characterization (see Section 4) has the advantage of having a simple form in
terms of matrix variables $K,G$, and $Z$. On the other hand, it is unclear how
to find minimal-norm solutions via this second characterization. The minimal-
norm dissipative mapping is used in Section 5 for studying the structured
distance to instability for DH systems (1.1) for simultaneous perturbations of
$J$ and $R$. In Section 6, we present numerical examples comparing these
distances to instability with those of [16] where perturbations affect only
one of the matrices $J$, $R$, or $Q$.

#### Notation

In the following, we denote the identity matrix of size $n\times n$ by
$I_{n}$, the spectral norm of a matrix or a vector by $\|\cdot\|$, and the
Frobenius norm by ${\|\cdot\|}_{F}$. The Moore-Penrose pseudoinverse of a
matrix or a vector $X$ is denoted by $X^{\dagger}$, and
$\mathcal{P}_{X}=I_{n}-XX^{\dagger}$ denotes the orthogonal projection onto
the null space of $n\times n$ matrix $X^{*}$. For a square matrix $A$, its
Hermitian and skew-Hermitian parts are respectively denoted by
$A_{H}=\frac{A+A^{*}}{2}$ and $A_{S}=\frac{A-A^{*}}{2}$. For
$A=A^{*}\in\mathbb{F}^{n,n}$, where
$\mathbb{F}\in\\{{\mathbb{R}},{\mathbb{C}}\\}$, we denote $A\succ 0$ ($A\prec
0$) and $A\succeq 0$ ($A\preceq$) if $A$ is Hermitian positive definite
(negative definite) and Hermitian positive semidefinite (negative
semidefinite), respectively, and $\Lambda(A)$ denotes the set of all
eigenvalues of the matrix $A$. For a given matrix $A\in{\mathbb{C}}^{n,m}$, we
use the term SVD to denote the standard singular value decomposition of $A$,
and reduced SVD for the decomposition $A=U_{1}\Sigma_{1}V_{1}^{*}$ in which
$\Sigma_{1}$ is a square diagonal matrix of size equal to rank of $A$, $r$,
with the nonzero singular values of $A$ on its diagonal, and
$U_{1}\in{\mathbb{C}}^{n,r}$ and $V_{1}\in{\mathbb{C}}^{m,r}$ have orthonormal
columns.

##  2 Preliminaries

In this section, we discuss some basic results from the literature and derive
some elementary lemmas that will be necessary to solve the dissipative mapping
problem.

Let us start with two well-known lemmas for positive semidefinite matrices.

###### Lemma 2.1

[11] Let $P\in{\mathbb{C}}^{n,n}$ and $X\in{\mathbb{C}}^{n,m}$. If $P\succeq
0$, then $X^{*}PX\succeq 0$.

###### Lemma 2.2

[4] Let the integer $s$ be such that $0<s<n$, and
$R=R^{*}\in{\mathbb{C}}^{n,n}$ be partitioned as
$R=\left[\begin{array}[]{cc}B&C^{*}\\\ C&D\end{array}\right]$ with
$B\in{\mathbb{C}}^{s,s}$, $C\in{\mathbb{C}}^{n-s,s}$ and
$D\in{\mathbb{C}}^{n-s,n-s}$. Then $R\succeq 0$ if and only if

  1. 1)

$B\succeq 0$,

  2. 2)

$\operatorname{null}(B)\subseteq\operatorname{null}(C)$, and

  3. 3)

$D-CB^{\dagger}C^{*}\succeq 0$, where $B^{\dagger}$ denotes the Moore-Penrose
pseudoinverse of $B$.

Next, we state a result [1, Theorem 2.2.3] for skew-Hermitian mappings in
terms that allow a direct use in deriving the second characterization for
dissipative mappings.

###### Theorem 2.3

Let $X,Y\in{\mathbb{C}}^{n,k}$ and define
$\mathcal{S}:=\\{\Delta\in{\mathbb{C}}^{n,n}:~{}\Delta^{*}=-\Delta,\,\Delta
X=Y\\}$. Then $\mathcal{S}\neq\emptyset$ if and only if
$(X^{*}Y)^{*}=-X^{*}Y$. Further, if $\mathcal{S}\neq\emptyset$, then

$\mathcal{S}=\left\\{YX^{\dagger}-(YX^{\dagger})^{*}+(X^{\dagger})^{*}X^{*}YX^{\dagger}+\mathcal{P}_{X}Z\mathcal{P}_{X}:~{}Z\in{\mathbb{C}}^{n,n},\,Z^{*}=-Z\right\\}.$

The next two lemmas will be used in deriving real dissipative matrices taking
a complex $X\in{\mathbb{C}}^{n,m}$ to a complex $Y\in{\mathbb{C}}^{n,m}$.

###### Lemma 2.4

[3, Lemma 3.3] Let $A,B\in{\mathbb{C}}^{n,p}$. Then
$[A~{}\overline{A}][B~{}\overline{B}]^{\dagger}$ is a real matrix.

###### Lemma 2.5

Let $X,Y\in\mathbb{C}^{n,k}$ be such that
$M=([X~{}\overline{X}]^{*}[Y~{}\overline{Y}]+[Y~{}\overline{Y}]^{*}[X~{}\overline{X}])^{-1}$
exists. Then $[X~{}\overline{X}]M[Y~{}\overline{Y}]^{*}$ is a real matrix.

Proof. Let $Q=\begin{bmatrix}O&I_{k}\\\ I_{k}&O\end{bmatrix}$, where
$O\in{\mathbb{C}}^{k,k}$ is the zero matrix. Then

$\displaystyle\overline{[X~{}\overline{X}]M[Y~{}\overline{Y}]^{*}}=\overline{[X~{}\overline{X}]Q^{2}MQ^{2}[Y~{}\overline{Y}]^{*}}=\overline{([X~{}\overline{X}]Q)(QMQ)(Q[Y~{}\overline{Y}]^{*})}=[X~{}\overline{X}]\overline{QMQ}[Y~{}\overline{Y}]^{*},$

because $[X~{}\overline{X}]Q=[\overline{X}~{}X]$ and
$Q[Y~{}\overline{Y}]^{*}=[\overline{Y}~{}Y]^{*}$. Thus we prove the assertion
by showing that $\overline{QMQ}=M$. This follows from the following
derivations:

$\displaystyle\overline{QMQ}$
$\displaystyle=\overline{Q([X~{}\overline{X}]^{*}[Y\overline{Y}]+[Y~{}\overline{Y}]^{*}[X~{}\overline{X}])^{-1}Q}$
$\displaystyle=Q\overline{([X~{}\overline{X}]^{*}[Y~{}\overline{Y}]+[Y~{}\overline{Y}]^{*}[X~{}\overline{X}])^{-1}}Q$
$\displaystyle=Q\overline{([X~{}\overline{X}]^{*}[Y~{}\overline{Y}]+[Y~{}\overline{Y}]^{*}[X~{}\overline{X}])}^{-1}Q$
$\displaystyle=Q([\overline{X}~{}X]^{*}[\overline{Y}~{}Y]+[\overline{Y}~{}Y]^{*}[\overline{X}~{}X])^{-1}Q$
$\displaystyle=\big{(}Q[\overline{X}~{}X]^{*}[\overline{Y}~{}Y]Q+Q[\overline{Y}~{}Y]^{*}[\overline{X}~{}X]Q\big{)}^{-1}$
$\displaystyle=\big{(}[X~{}\overline{X}]^{*}[Y~{}\overline{Y}]+[Y~{}\overline{Y}]^{*}[X\overline{X}]\big{)}^{-1}$
$\displaystyle=M.$

We close the section by discussing some simple mapping results that will be
necessary to compute the structured distances to instability in Section 5.

###### Lemma 2.6

Let $x\in{\mathbb{C}}^{q}$ and $y\in{\mathbb{C}}^{r}$. Then

$\displaystyle\inf\left\\{{\|\Delta_{1}\|}_{F}^{2}+{\|\Delta_{2}\|}_{F}^{2}:~{}\Delta_{1},\Delta_{2}\in{\mathbb{C}}^{r,q},(\Delta_{1}-\Delta_{2})x=y\right\\}=\inf\left\\{\frac{{\|\Delta\|}_{F}^{2}}{2}:~{}\Delta\in{\mathbb{C}}^{r,q},\Delta
x=y\right\\}.$ (2.1)

Proof. First, let $\Delta\in{\mathbb{C}}^{r,q}$ such that $\Delta x=y$, and
set $\Delta_{1}=\Delta/2$ and $\Delta_{2}=-\Delta/2$. Then
$(\Delta_{1}-\Delta_{2})x=\Delta x=y$, and
${\|\Delta_{1}\|}_{F}^{2}+{\|\Delta_{2}\|}_{F}^{2}=\frac{{\|\Delta\|}_{F}^{2}}{4}+\frac{{\|\Delta\|}_{F}^{2}}{4}=\frac{{\|\Delta\|}_{F}^{2}}{2}$.
This implies “$\leq$” in (2.1). Conversely, let
$\Delta_{1},\Delta_{2}\in{\mathbb{C}}^{r,q}$ be such that
$(\Delta_{1}-\Delta_{2})x=y$, and set $\Delta=\Delta_{1}-\Delta_{2}$. Then
$\Delta x=y$ and

$\displaystyle{\|\Delta\|}_{F}^{2}={\|\Delta_{1}-\Delta_{2}\|}_{F}^{2}$
$\displaystyle\leq$
$\displaystyle\left({\|\Delta_{1}\|}_{F}+{\|\Delta_{2}\|}_{F}\right)^{2}$
$\displaystyle=$
$\displaystyle{\|\Delta_{1}\|}_{F}^{2}+{\|\Delta_{2}\|}_{F}^{2}+2{\|\Delta_{1}\|}_{F}{\|\Delta_{2}\|}_{F}$
$\displaystyle\leq$ $\displaystyle
2\left({\|\Delta_{1}\|}_{F}^{2}+{\|\Delta_{2}\|}_{F}^{2}\right).$

Thus
$\frac{{\|\Delta\|}_{F}^{2}}{2}\leq{\|\Delta_{1}\|}_{F}^{2}+{\|\Delta_{2}\|}_{F}^{2}$.
This implies “$\geq$” in (2.1).

###### Lemma 2.7

Let $x,y\in{\mathbb{C}}^{r}$. Then

$\displaystyle\inf\left\\{{\|\Delta_{1}\|}_{F}^{2}+{\|\Delta_{2}\|}_{F}^{2}:~{}\Delta_{1},\Delta_{2}\in{\mathbb{C}}^{r,r},\Delta_{1}^{*}=-\Delta_{1},\Delta_{2}\preceq
0,(\Delta_{1}-\Delta_{2})x=y\right\\}$
$\displaystyle=\inf\left\\{{\|\Delta\|}_{F}^{2}:~{}\Delta\in{\mathbb{C}}^{r,r},\Delta+\Delta^{*}\succeq
0,\Delta x=y\right\\}.$ (2.2)

Proof. The idea behind the proof is similar to Lemma 2.6. Let
$\Delta\in{\mathbb{C}}^{r,r}$ be such that $\Delta+\Delta^{*}\succeq 0$ and
$\Delta x=y$. Set $\Delta_{1}=\Delta_{S}$ and $\Delta_{2}=-\Delta_{H}$. Then
clearly $\Delta_{1}^{*}=-\Delta_{1}$, $\Delta_{2}\preceq 0$ since
$\Delta+\Delta^{*}\succeq 0$, and $(\Delta_{1}-\Delta_{2})x=\Delta x=y$. Also
${\|\Delta_{1}\|}_{F}^{2}+{\|\Delta_{2}\|}_{F}^{2}={\left\|\Delta_{S}\right\|}_{F}^{2}+{\left\|\Delta_{H}\right\|}_{F}^{2}={\|\Delta\|}_{F}^{2}$.
This implies “$\leq$” in (2.7). Conversely, let
$\Delta_{1},\Delta_{2}\in{\mathbb{C}}^{r,r}$ be such that
$\Delta_{1}^{*}=-\Delta_{1}$, $\Delta_{2}\preceq 0$, and
$(\Delta_{1}-\Delta_{2})x=y$. Now set $\Delta=\Delta_{1}-\Delta_{2}$. Then
clearly $\Delta x=y$ and $\Delta+\Delta^{*}=-2\Delta_{2}\succeq 0$, since
$\Delta_{2}\preceq 0$. Further we have that
${\|\Delta\|}_{F}^{2}={\left\|\Delta_{S}\right\|}_{F}^{2}+{\left\|\Delta_{H}\right\|}_{F}^{2}={\|\Delta_{1}\|}_{F}^{2}+{\|\Delta_{2}\|}_{F}^{2}$.
This proves “$\geq$” in (2.7).

###### Lemma 2.8

[16, Theorem 2.8] Let $B\in{\mathbb{C}}^{n,r}$ with $\text{{\rm rank}}(B)=r$,
$x\in{\mathbb{C}}^{r}\setminus\\{0\\}$,
$y\in{\mathbb{C}}^{n}\setminus\\{0\\}$, and let $\Delta\in{\mathbb{C}}^{r,r}$.
Then $B\Delta x=y$ if and only if $\Delta x=B^{\dagger}y$ and
$BB^{\dagger}y=y$.

##  3 First characterization of dissipative mappings

In this section, we derive the first characterization of dissipative mappings.
This characterization allows us to find the minimal Frobenius norm solutions
to the dissipative mapping problem which is shown to be unique. This minimal-
norm solution will turn out to be a necessary tool in computing the structured
stability radius in Section 5. For given $X\in{\mathbb{C}}^{n,m}$ and
$Y\in{\mathbb{C}}^{n,m}$, we define the set of dissipative mappings from $X$
to $Y$ as follows

$\mathbb{S}(X,Y):=\\{\Delta\in{\mathbb{C}}^{n,n}~{}:~{}\Delta+\Delta^{*}\succeq
0,~{}\Delta X=Y\\}.$

We will need the following lemma in characterizing the set of all solutions to
the dissipative mapping problem.

###### Lemma 3.1

Let $X,Y\in{\mathbb{C}}^{n,m}$. Suppose that $\text{{\rm rank}}(X)=p$ and
consider the reduced singular value decomposition
$X=U_{1}\Sigma_{1}{V}_{1}^{*}$ with $U_{1}\in{\mathbb{C}}^{n,p}$,
$\Sigma_{1}\in{\mathbb{C}}^{p,p}$ and $V_{1}\in{\mathbb{C}}^{m,p}$. If
$X^{*}Y+Y^{*}X\succeq 0$, then
$U_{1}^{*}\left(YX^{\dagger}+{(YX^{\dagger})}^{*}\right)U_{1}\succeq 0$.

Proof. Since $X=U_{1}\Sigma_{1}{V}_{1}^{*}$, we have
$X^{\dagger}=V_{1}\Sigma_{1}^{-1}U_{1}^{*}$. Thus

$\displaystyle X^{*}Y+Y^{*}X\succeq 0$ $\displaystyle\iff$ $\displaystyle
V_{1}\Sigma_{1}U_{1}^{*}Y+Y^{*}U_{1}\Sigma_{1}V_{1}^{*}\succeq 0$
$\displaystyle\Longrightarrow$ $\displaystyle
V_{1}^{*}\left(V_{1}\Sigma_{1}U_{1}^{*}Y+Y^{*}U_{1}\Sigma_{1}V_{1}^{*}\right)V_{1}\succeq
0\quad(\because\,\text{using\,Lemma}\,\ref{psdsimilar})$ $\displaystyle\iff$
$\displaystyle\Sigma_{1}U_{1}^{*}YV_{1}+V_{1}^{*}Y^{*}U_{1}\Sigma_{1}\succeq
0\hskip 42.67912pt(\because V_{1}^{*}V_{1}=I_{p})$ $\displaystyle\iff$
$\displaystyle\Sigma_{1}^{-1}\left(\Sigma_{1}U_{1}^{*}YV_{1}+V_{1}^{*}Y^{*}U_{1}\Sigma_{1}\right)\Sigma_{1}^{-1}\succeq
0$ $\displaystyle\iff$ $\displaystyle
U_{1}^{*}YV_{1}\Sigma_{1}^{-1}+\Sigma_{1}^{-1}V_{1}^{*}Y^{*}U_{1}\succeq 0$
$\displaystyle\iff$ $\displaystyle
U_{1}^{*}\left(YV_{1}\Sigma_{1}^{-1}U_{1}^{*}+U_{1}\Sigma_{1}^{-1}V_{1}^{*}Y^{*}\right)U_{1}\succeq
0$ $\displaystyle\iff$ $\displaystyle
U_{1}^{*}\left(YX^{\dagger}+{(YX^{\dagger})}^{*}\right)U_{1}\succeq 0,$

which completes the proof.

###### Theorem 3.2

Let $X,Y\in{\mathbb{C}}^{n,m}$, and suppose that ${\rm rank}(X)=p$. Then
$\mathbb{S}(X,Y)\neq\emptyset$ if and only if $YX^{\dagger}X=Y$ and
$X^{*}Y+Y^{*}X\succeq 0$. Moreover, if $\mathbb{S}(X,Y)\neq\emptyset$, then

  1. 1)

Characterization: Let $X=U\Sigma V^{*}$ be the the singular value
decomposition of $X$ with $U=\left[U_{1}~{}U_{2}\right]$, where
$U_{1}\in{\mathbb{C}}^{n,p}$. Then

$\displaystyle\mathbb{S}(X,Y)=\Big{\\{}U\left[\begin{array}[]{cc}U_{1}^{*}YX^{\dagger}U_{1}&U_{1}^{*}(YX^{\dagger})^{*}U_{2}+U_{1}^{*}ZU_{2}\\\
U_{2}^{*}YX^{\dagger}U_{1}&U_{2}^{*}(K+G)U_{2}\end{array}\right]U^{*}:$ (3.3)
$\displaystyle
Z,K,G\in{\mathbb{C}}^{n,n}~{}\text{satisfying}~{}\eqref{eq:cond1}-\eqref{eq:cond3}\Big{\\}},$
(3.4)

where

$\displaystyle G^{*}=-G,\quad K\succeq 0,$ (3.5) $\displaystyle{\rm
null}\left(U_{1}^{*}(YX^{\dagger}+(YX^{\dagger})^{*})U_{1}\right)\subseteq{\rm
null}\left(U_{2}^{*}(2YX^{\dagger}+Z^{*})U_{1}\right),$ (3.6) $\displaystyle
K-\frac{1}{8}(2YX^{\dagger}+Z^{*}){\left(XX^{\dagger}YX^{\dagger}+(YX^{\dagger})^{*}XX^{\dagger}\right)}^{\dagger}(2YX^{\dagger}+Z^{*})^{*}\succeq
0.$ (3.7)

  2. 2)

Minimal norm mapping:

$\inf_{\Delta\in\mathbb{S}(X,Y)}{\|\Delta\|}_{F}^{2}\;=\;2{\|YX^{\dagger}\|}_{F}^{2}-{\rm
trace}\left((YX^{\dagger})^{*}XX^{\dagger}(YX^{\dagger})\right),$ (3.8)

where the infimum is uniquely attained by the matrix
$\mathcal{H}:=YX^{\dagger}-{(YX^{\dagger})}^{*}\mathcal{P}_{X}$, which is
obtained by setting $K=0$, $G=0$, and $Z=-2U_{1}^{*}(YX^{\dagger})^{*}U_{2}$
in (3.3).

Proof. First suppose that $\Delta\in\mathbb{S}(X,Y)$, i.e.,
$\Delta+\Delta^{*}\succeq 0$ and $\Delta X=Y$. Then $YX^{\dagger}X=\Delta
XX^{\dagger}X=\Delta X=Y$. By Lemma 2.1

$X^{*}Y+Y^{*}X=X^{*}\Delta X+X^{*}\Delta^{*}X=X^{*}(\Delta+\Delta^{*})X\succeq
0,$

since $\Delta+\Delta^{*}\succeq 0$. Conversely, suppose that $X$ and $Y$
satisfy $YX^{\dagger}X=Y$ and $X^{*}Y+Y^{*}X\succeq 0$. Then the matrix
$\mathcal{H}$ satisfies $\mathcal{H}X=YX^{\dagger}X=Y$ and

$\displaystyle\mathcal{H}+{\mathcal{H}}^{*}$ $\displaystyle=$ $\displaystyle
YX^{\dagger}-{(YX^{\dagger})}^{*}\mathcal{P}_{X}+{(YX^{\dagger})}^{*}-\mathcal{P}_{X}{YX^{\dagger}}$
$\displaystyle=$
$\displaystyle{(YX^{\dagger})}^{*}XX^{\dagger}+XX^{\dagger}(YX^{\dagger})\quad\quad(\because\mathcal{P}_{X}=I_{n}-XX^{\dagger})$
$\displaystyle=$ $\displaystyle
XX^{\dagger}\left({(YX^{\dagger})}^{*}+YX^{\dagger}\right)XX^{\dagger}\quad\quad(\because
X^{\dagger}XX^{\dagger}=X^{\dagger},~{}(XX^{\dagger})^{*}=XX^{\dagger})$
$\displaystyle=$ $\displaystyle
U_{1}U_{1}^{*}\big{(}{(YX^{\dagger})}^{*}+YX^{\dagger}\big{)}U_{1}U_{1}^{*}\quad\quad(\because
XX^{\dagger}=U_{1}U_{1}^{*})$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}U_{1}&U_{2}\end{array}\right]\left[\begin{array}[]{cc}U_{1}^{*}\big{(}{(YX^{\dagger})}^{*}+YX^{\dagger}\big{)}U_{1}&0\\\
0&0\end{array}\right]\left[\begin{array}[]{cc}U_{1}&U_{2}\end{array}\right]^{*}$
$\displaystyle\succeq$ $\displaystyle 0,$

where the last identity follows by using Lemma 3.1, since
$X^{*}Y+Y^{*}X\succeq 0$, and by Lemma 2.1.

Next, we prove (3.3). First suppose that $\Delta\in\mathbb{S}(X,Y)$, i.e.,
$\Delta+\Delta^{*}\succeq 0$ and $\Delta X=Y$. Let
$\Sigma=\left[\begin{array}[]{cc}\Sigma_{1}&0\\\ 0&0\end{array}\right]$ and
$V=\left[\begin{array}[]{cc}V_{1}&V_{2}\end{array}\right]$, where
$V_{1}\in{\mathbb{C}}^{m,p}$, $V_{2}\in{\mathbb{C}}^{m,m-p}$, and
$\Sigma_{1}\in{\mathbb{C}}^{p,p}$ such that $X=U_{1}\Sigma_{1}V_{1}^{*}$ is
the reduced SVD of $X$. Then $X^{*}=V_{1}\Sigma_{1}U_{1}^{*}$ and
$X^{\dagger}=V_{1}\Sigma_{1}^{-1}U_{1}^{*}$. Consider $\Delta=UU^{*}\Delta
UU^{*}$ and $\widetilde{\Delta}=U^{*}\Delta
U=\widetilde{\Delta}_{H}+\widetilde{\Delta}_{S}$, where

$\widetilde{\Delta}_{H}=U^{*}\Delta_{H}U=\left[\begin{array}[]{cc}H_{11}&H_{12}\\\
H_{12}^{*}&H_{22}\end{array}\right]\quad\text{and}\quad\widetilde{\Delta}_{S}=U^{*}\Delta_{S}U=\left[\begin{array}[]{cc}S_{11}&S_{12}\\\
-S_{12}^{*}&S_{22}\end{array}\right].$

Clearly, ${\|\Delta\|}_{F}={\|\widetilde{\Delta}\|}_{F}$ and also
$\Delta_{H}\succeq 0$ $\Longleftrightarrow$ $\widetilde{\Delta}_{H}\succeq 0$.
As $\Delta X=Y$, we have

$\displaystyle U^{*}\Delta UU^{*}X=U^{*}Y$ $\displaystyle\Longrightarrow$
$\displaystyle\widetilde{\Delta}\left[\begin{array}[]{c}U_{1}^{*}\\\
U_{2}^{*}\end{array}\right]X=\left[\begin{array}[]{c}U_{1}^{*}Y\\\
U_{2}^{*}Y\end{array}\right]$ $\displaystyle\Longrightarrow$
$\displaystyle\left[\begin{array}[]{cc}H_{11}+S_{11}&H_{12}+S_{12}\\\
H_{12}^{*}-S_{12}^{*}&H_{22}+S_{22}\end{array}\right]\left[\begin{array}[]{c}\Sigma_{1}V_{1}^{*}\\\
0\end{array}\right]=\left[\begin{array}[]{c}U_{1}^{*}Y\\\
U_{2}^{*}Y\end{array}\right].$

This implies that

$\left(H_{11}+S_{11}\right)\Sigma_{1}V_{1}^{*}=U_{1}^{*}Y,$ (3.12)

and

$\left(H_{12}^{*}-S_{12}^{*}\right)\Sigma_{1}V_{1}^{*}=U_{2}^{*}Y.$ (3.13)

Thus from (3.12), we have
$H_{11}+S_{11}=U_{1}^{*}YV_{1}\Sigma_{1}^{-1}=U_{1}^{*}YX^{\dagger}U_{1}$,
since $X^{\dagger}=V_{1}\Sigma_{1}^{-1}U_{1}^{*}$ and
$X^{\dagger}U_{1}=V_{1}\Sigma_{1}^{-1}$. This implies that

$H_{11}=U_{1}^{*}\left(\frac{(YX^{\dagger})+(YX^{\dagger})^{*}}{2}\right)U_{1}\quad\text{and}\quad
S_{11}=U_{1}^{*}\left(\frac{(YX^{\dagger})-(YX^{\dagger})^{*}}{2}\right)U_{1}.$
(3.14)

Note that since $X^{*}Y+Y^{*}X\succeq 0$, in view of Lemma 3.1, we have that
$H_{11}\succeq 0$. Similarly, from (3.13) we have
$H_{12}^{*}-S_{12}^{*}=U_{2}^{*}YV_{1}\Sigma_{1}^{-1}=U_{2}^{*}YX^{\dagger}U_{1}$.
This implies that

$H_{12}=U_{1}^{*}(YX^{\dagger})^{*}U_{2}+S_{12},$ (3.15)

where $S_{12}\in{\mathbb{C}}^{p,n-p}$ is a matrix variable. Thus from (3.14)
and (3.15), $\widetilde{\Delta}$ has the form

$\widetilde{\Delta}=\left[\begin{array}[]{cc}H_{11}+S_{11}&H_{12}+S_{12}\\\
H_{12}^{*}-S_{12}^{*}&H_{22}+S_{22}\end{array}\right]=\left[\begin{array}[]{cc}U_{1}^{*}YX^{\dagger}U_{1}&U_{1}^{*}(YX^{\dagger})^{*}U_{2}+2S_{12}\\\
U_{2}^{*}YX^{\dagger}U_{1}&H_{22}+S_{22}\end{array}\right],$ (3.16)

where $H_{22},S_{22}\in{\mathbb{C}}^{n-p,n-p}$ and
$S_{12}\in{\mathbb{C}}^{p,n-p}$ are such that $\widetilde{\Delta}_{H}\succeq
0$ and $\widetilde{\Delta}_{S}=-\widetilde{\Delta}_{S}^{*}$. That means, in
view of Lemma 2.2, $S_{22}$ satisfies that $S_{22}^{*}=-S_{22}$, and $H_{22}$
and $S_{12}$ satisfy the following constraints: $H_{22}\succeq 0$,

$H_{22}-(U_{2}^{*}YX^{\dagger}U_{1}+S_{12}^{*}){\left(\frac{U_{1}^{*}(YX^{\dagger}+(YX^{\dagger})^{*})U_{1}}{2}\right)}^{\dagger}(U_{2}^{*}YX^{\dagger}U_{1}+S_{12}^{*})^{*}\succeq
0,$ (3.17)

and

$\text{null}\left(\frac{U_{1}^{*}(YX^{\dagger}+(YX^{\dagger})^{*})U_{1}}{2}\right)\subseteq\text{null}\left(U_{2}^{*}YX^{\dagger}U_{1}+S_{12}^{*}\right).$
(3.18)

Thus from (3.16), we have

$\displaystyle\Delta$ $\displaystyle=$ $\displaystyle
U\widetilde{\Delta}U^{*}=\left[\begin{array}[]{cc}U_{1}&U_{2}\end{array}\right]\left[\begin{array}[]{cc}U_{1}^{*}YX^{\dagger}U_{1}&U_{1}^{*}(YX^{\dagger})^{*}U_{2}+2S_{12}\\\
U_{2}^{*}YX^{\dagger}U_{1}&H_{22}+S_{22}\end{array}\right]\left[\begin{array}[]{c}U_{1}^{*}\\\
U_{2}^{*}\end{array}\right].$ (3.24)

By setting $Z=2U_{1}S_{12}U_{2}^{*}$, $K=U_{2}H_{22}U_{2}^{*}$,
$G=U_{2}S_{22}U_{2}^{*}$, and by using the fact that
$U_{1}U_{1}^{*}+U_{2}U_{2}^{*}=UU^{*}=I_{n}$, $U_{1}U_{1}^{*}=XX^{\dagger}$,
$U_{1}^{*}U_{1}=I_{p}$, and $U_{2}^{*}U_{2}=I_{n-p}$, we obtain that

$\Delta=U\left[\begin{array}[]{cc}U_{1}^{*}YX^{\dagger}U_{1}&U_{1}^{*}(YX^{\dagger})^{*}U_{2}+U_{1}^{*}ZU_{2}\\\
U_{2}^{*}YX^{\dagger}U_{1}&U_{2}^{*}(K+G)U_{2}\end{array}\right]U^{*},$

where $G$, $Z$, and $K$ satisfy the conditions (3.5)–(3.7). This proves
$``\subseteq"$ in (3.3).

For the other inclusion in (3.3), let

$A=U\left[\begin{array}[]{cc}U_{1}^{*}YX^{\dagger}U_{1}&U_{1}^{*}(YX^{\dagger})^{*}U_{2}+U_{1}^{*}ZU_{2}\\\
U_{2}^{*}YX^{\dagger}U_{1}&U_{2}^{*}(K+G)U_{2}\end{array}\right]U^{*},$

where $G$, $Z$, and $K$ satisfy the conditions (3.5)–(3.7), which can be
written as

$\displaystyle
A=YX^{\dagger}+(YX^{\dagger})^{*}\mathcal{P}_{X}+XX^{\dagger}Z\mathcal{P}_{X}+\mathcal{P}_{X}K\mathcal{P}_{X}+\mathcal{P}_{X}G\mathcal{P}_{X}.$
(3.25)

Clearly $AX=Y$ since $YX^{\dagger}X=Y$ and $\mathcal{P}_{X}X=0$. Also
$A+A^{*}\succeq 0$. Indeed,

$U^{*}AU=\left[\begin{array}[]{cc}U_{1}^{*}YX^{\dagger}U_{1}&U_{1}^{*}(YX^{\dagger})^{*}U_{2}+U_{1}^{*}ZU_{2}\\\
U_{2}^{*}YX^{\dagger}U_{1}&U_{2}^{*}(K+G)U_{2}\end{array}\right],$

and thus

$U^{*}(A+A^{*})U=\left[\begin{array}[]{cc}U_{1}^{*}\left(YX^{\dagger}+(YX^{\dagger})^{*}\right)U_{1}&2U_{1}^{*}(YX^{\dagger})^{*}U_{2}+U_{1}^{*}ZU_{2}\\\
2U_{2}^{*}YX^{\dagger}U_{1}+U_{2}^{*}Z^{*}U_{1}&2U_{2}^{*}KU_{2}\end{array}\right]\succeq
0,$

because of Lemma 2.2, since $K$, $Z$, $G$ satisfy (3.5)–(3.7). This completes
the proof of (3.3).

Suppose that $\mathbb{S}(X,Y)\neq\emptyset$ and let
$\Delta\in\mathbb{S}(X,Y)$, then from (3.16) $\Delta$ satisfies

${\|\Delta\|}_{F}^{2}={\|\widetilde{\Delta}\|}_{F}^{2}={\left\|\left[\begin{array}[]{cc}U_{1}^{*}YX^{\dagger}U_{1}&U_{1}^{*}(YX^{\dagger})^{*}U_{2}+2S_{12}\\\
U_{2}^{*}YX^{\dagger}U_{1}&H_{22}+S_{22}\end{array}\right]\right\|}_{F}^{2},$

where $S_{22}\in{\mathbb{C}}^{n-p,n-p}$ satisfies that $S_{22}^{*}=-S_{22}$,
and $H_{22}\in{\mathbb{C}}^{n-p,n-p}$ and $S_{12}\in{\mathbb{C}}^{p,n-p}$
satisfy (3.17) and (3.18). This implies that

$\displaystyle{\|\Delta\|}_{F}^{2}$ $\displaystyle=$
$\displaystyle{\|U_{1}^{*}YX^{\dagger}U_{1}\|}_{F}^{2}+{\|U_{2}^{*}YX^{\dagger}U_{1}\|}_{F}^{2}+{\|U_{1}^{*}(YX^{\dagger})^{*}U_{2}+2S_{12}\|}_{F}^{2}+{\|H_{22}+S_{22}\|}_{F}^{2}$
(3.26) $\displaystyle=$
$\displaystyle{\|YX^{\dagger}U_{1}\|}_{F}^{2}+{\|U_{1}^{*}(YX^{\dagger})^{*}U_{2}+2S_{12}\|}_{F}^{2}+{\|H_{22}\|}_{F}^{2}+{\|S_{22}\|}_{F}^{2},$

where the last equality follows as
${\|U_{1}^{*}YX^{\dagger}U_{1}\|}_{F}^{2}+{\|U_{2}^{*}YX^{\dagger}U_{1}\|}_{F}^{2}={\|UYX^{\dagger}U_{1}\|}_{F}^{2}={\|YX^{\dagger}U_{1}\|}_{F}^{2}$
because $\|\cdot\|_{F}$ is unitary invariant, and by the fact that for any
square matrix $A=A_{H}+A_{S}$ we have
${\|A\|}_{F}^{2}={\|A_{H}\|}_{F}^{2}+{\|A_{S}\|}_{F}^{2}$. Further, from
(3.26) we have that

${\|\Delta\|}_{F}^{2}\geq{\|YX^{\dagger}U_{1}\|}_{F}^{2}+{\|U_{1}^{*}(YX^{\dagger})^{*}U_{2}+2S_{12}\|}_{F}^{2},$

where the lower bound is attained by setting $H_{22}=0$ and $S_{22}=0$. Thus
any $\Delta\in\mathbb{S}(X,Y)$ satisfies

${\|\Delta\|}_{F}^{2}\geq{\|YX^{\dagger}U_{1}\|}_{F}^{2}+{\|U_{1}^{*}(YX^{\dagger})^{*}U_{2}+2S_{12}\|}_{F}^{2},$

where $S_{12}$ satisfies (3.17) and (3.18) with $H_{22}=0$, but the only
$S_{12}$ that satisfies (3.17) with $H_{22}=0$ is
$S_{12}=-U_{1}^{*}(YX^{\dagger})^{*}U_{2}$ because
$U_{1}^{*}\left((YX^{\dagger})+(YX^{\dagger})^{*}\right)U_{1}\succeq 0$. Hence
by setting $H_{22}=0$, $S_{22}=0$, and
$S_{12}=-U_{1}^{*}(YX^{\dagger})^{*}U_{2}$ in (3.16), we obtain the unique
matrix

$\displaystyle\Delta$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}U_{1}&U_{2}\end{array}\right]\left[\begin{array}[]{cc}U_{1}^{*}YX^{\dagger}U_{1}&-U_{1}^{*}(YX^{\dagger})^{*}U_{2}\\\
U_{2}^{*}YX^{\dagger}U_{1}&0\end{array}\right]\left[\begin{array}[]{c}U_{1}^{*}\\\
U_{2}^{*}\end{array}\right]$ $\displaystyle=$
$\displaystyle(U_{1}U_{1}^{*}+U_{2}U_{2}^{*})YX^{\dagger}U_{1}U_{1}^{*}-U_{1}U_{1}^{*}(YX^{\dagger})^{*}U_{2}U_{2}^{*}$
$\displaystyle=$ $\displaystyle
YX^{\dagger}XX^{\dagger}-XX^{\dagger}(YX^{\dagger})^{*}\mathcal{P}_{X}$
$\displaystyle=$ $\displaystyle
YX^{\dagger}-(YX^{\dagger})^{*}\mathcal{P}_{X}$ $\displaystyle=$
$\displaystyle\mathcal{H}$

which implies that (3.8) holds.

The following corollary of Theorem 3.2 for the vector case ($m=1$) of the
minimal norm dissipative mapping is of particular interest for us as it will
be used in Section 5 in finding the structured stability radius for DH systems
when both $J$ and $R$ are perturbed.

###### Corollary 3.3

Let $x,y\in{\mathbb{C}}^{n}\setminus\\{0\\}$. Then
$\mathbb{S}(x,y)\neq\emptyset$ if and only if
$\mathop{\mathrm{Re}}{(x^{*}y)}\geq 0$. Moreover, if
$\mathbb{S}(x,y)\neq\emptyset$, then

$\inf_{\Delta\in\mathbb{S}(x,y)}{\|\Delta\|}_{F}^{2}\;=\;2\frac{\|y\|^{2}}{\|x\|^{2}}-\frac{|x^{*}y|^{2}}{\|x\|^{4}},$
(3.28)

where the infimum is attained by the unique matrix
$\mathcal{H}:=\frac{yx^{*}}{\|x\|^{2}}-\frac{xy^{*}}{\|x\|^{2}}+(y^{*}x)\frac{xx^{*}}{\|x\|^{4}}$.

###  3.1 Real dissipative mappings

For $X$, $Y\in{\mathbb{C}}^{n,m}$, if we consider the real dissipative mapping
problem, i.e., finding $\Delta\in{\mathbb{R}}^{n,n}$ such that
$\Delta+\Delta^{T}\succeq 0$ and $\Delta X=Y$, then the minimal Frobenius norm
solution can be easily obtained from Theorem 3.2. To see this, observe that
for a real $\Delta$, $\Delta X=Y$ if and only if
$\Delta[X~{}\overline{X}]=[Y~{}\overline{Y}]$. In the following, we show that
if there exists a complex dissipative mapping $\Delta$ satisfying
$\Delta\mathcal{X}=\mathcal{Y}$ where $\mathcal{X}=[X~{}\overline{X}]$ and
$\mathcal{Y}=[Y~{}\overline{Y}]$, then there also exists a real dissipative
mapping. Moreover, from Theorem 3.2, we can easily find a minimal norm real
dissipative mapping taking $X$ to $Y$.

###### Theorem 3.4

Let $X,Y\in{\mathbb{C}}^{n,m}$, and define
$\mathbb{S}_{{\mathbb{R}}}(X,Y):=\\{\Delta\in{\mathbb{R}}^{n,n}~{}:~{}\Delta+\Delta^{T}\succeq
0,~{}\Delta X=Y\\}$. Let $\mathcal{X}=[X~{}\overline{X}]$ and
$\mathcal{Y}=[Y~{}\overline{Y}]$. Suppose that ${\rm rank}(\mathcal{X})=p$.
Then $\mathbb{S}_{\mathbb{R}}(X,Y)\neq\emptyset$ if and only if
$\mathcal{Y}\mathcal{X}^{\dagger}\mathcal{X}=\mathcal{Y}$ and
$\mathcal{X}^{*}\mathcal{Y}+\mathcal{Y}^{*}\mathcal{X}\succeq 0$. Moreover, if
$\mathbb{S}_{\mathbb{R}}(X,Y)\neq\emptyset$ and if we consider the singular
value decomposition $\mathcal{X}=U\Sigma V^{T}$ with
$U=\left[U_{1}~{}U_{2}\right]$, where $U_{1}\in{\mathbb{C}}^{n,p}$, then the
set

$\left\\{U\left[\begin{array}[]{cc}U_{1}^{*}\mathcal{Y}\mathcal{X}^{\dagger}U_{1}&U_{1}^{*}(\mathcal{Y}\mathcal{X}^{\dagger})^{*}U_{2}+U_{1}^{*}ZU_{2}\\\
U_{2}^{*}\mathcal{Y}\mathcal{X}^{\dagger}U_{1}&U_{2}^{*}(K+G)U_{2}\end{array}\right]U^{*}:Z,K,G\in{\mathbb{R}}^{n,n}~{}\text{satisfying}~{}\eqref{realeq:cond1}-\eqref{realeq:cond3}\right\\},$

where

$\displaystyle G^{T}=-G,\quad K\succeq 0,$ (3.29) $\displaystyle{\rm
null}\left(U_{1}^{*}(\mathcal{Y}\mathcal{X}^{\dagger}+(\mathcal{Y}\mathcal{X}^{\dagger})^{T})U_{1}\right)\subseteq{\rm
null}\left(U_{2}^{*}(2\mathcal{Y}\mathcal{X}^{\dagger}+Z^{T})U_{1}\right),$
(3.30) $\displaystyle
K-\frac{1}{8}(2\mathcal{Y}\mathcal{X}^{\dagger}+Z^{T}){\left(\mathcal{X}\mathcal{X}^{\dagger}\mathcal{Y}\mathcal{X}^{\dagger}+(\mathcal{Y}\mathcal{X}^{\dagger})^{T}\mathcal{X}\mathcal{X}^{\dagger}\right)}^{\dagger}(2\mathcal{Y}\mathcal{X}^{\dagger}+Z^{T})^{T}\succeq
0,$ (3.31)

is contained in $\mathbb{S}_{\mathbb{R}}(X,Y)$. Further,

$\inf_{\Delta\in\mathbb{S}_{\mathbb{R}}(X,Y)}{\|\Delta\|}_{F}^{2}\;=\;2{\|\mathcal{Y}\mathcal{X}^{\dagger}\|}_{F}^{2}-{\rm
trace}\left((\mathcal{Y}\mathcal{X}^{\dagger})^{*}\mathcal{X}\mathcal{X}^{\dagger}(\mathcal{Y}\mathcal{X}^{\dagger})\right),$

where the infimum is attained by the matrix
$\mathcal{H}:=\mathcal{Y}\mathcal{X}^{\dagger}-{(\mathcal{Y}\mathcal{X}^{\dagger})}^{*}\mathcal{P}_{\mathcal{X}}$.

Proof. In view of Lemma 2.4, the proof follows from Theorem 3.2. Indeed, from
Theorem 3.2 there exists $\Delta\in\mathbb{S}(\mathcal{X},\mathcal{Y})$ if and
only if $\mathcal{Y}\mathcal{X}^{\dagger}\mathcal{X}=\mathcal{Y}$ and
$\mathcal{X}^{*}\mathcal{Y}+\mathcal{Y}^{*}\mathcal{X}\succeq 0$. Assuming the
latter two conditions hold true, and consider a family of mappings defined by

$\displaystyle\Delta(Z,K,G):=U\left[\begin{array}[]{cc}U_{1}^{*}\mathcal{Y}\mathcal{X}^{\dagger}U_{1}&U_{1}^{*}(\mathcal{Y}\mathcal{X}^{\dagger})^{*}U_{2}+U_{1}^{*}ZU_{2}\\\
U_{2}^{*}\mathcal{Y}\mathcal{X}^{\dagger}U_{1}&U_{2}^{*}(K+G)U_{2}\end{array}\right]U^{*},$

where $Z,K,G\in{\mathbb{R}}^{n,n}$ satisfy (3.29)-(3.31). Using Lemma 2.4, it
is easy to check that $\Delta(Z,K,G)$ is real, which implies that
$\Delta(Z,K,G)\in\mathbb{S}_{\mathbb{R}}(X,Y)$ since
$\mathbb{S}_{\mathbb{R}}(\mathcal{X},\mathcal{Y})=\mathbb{S}_{\mathbb{R}}(X,Y)$.
Further, we have that

$\displaystyle\inf_{\Delta\in\mathbb{S}(\mathcal{X},\mathcal{Y})}{\|\Delta\|}_{F}\leq\inf_{\Delta\in\mathbb{S}_{\mathbb{R}}(X,Y)}{\|\Delta\|}_{F},$
(3.33)

and the left hand side infimum in (3.33) is attained by the unique map
$\mathcal{H}=\mathcal{Y}\mathcal{X}^{\dagger}-{(\mathcal{Y}\mathcal{X}^{\dagger})}^{*}\mathcal{P}_{\mathcal{X}}$.
Observe from Lemma 2.4 that the matrix $\mathcal{H}$ is real, that is,
$\mathcal{H}\in\mathbb{S}_{\mathbb{R}}(X,Y)$. This implies that

$\inf_{\Delta\in\mathbb{S}_{\mathbb{R}}(X,Y)}{\|\Delta\|}_{F}^{2}={\|\mathcal{H}\|}_{F}^{2}=2{\|\mathcal{Y}\mathcal{X}^{\dagger}\|}_{F}^{2}-{\rm
trace}\left((\mathcal{Y}\mathcal{X}^{\dagger})^{*}\mathcal{X}\mathcal{X}^{\dagger}(\mathcal{Y}\mathcal{X}^{\dagger})\right),$
(3.34)

which completes the proof.

##  4 Second characterization of dissipative mappings

In this section, for given $X,Y\in\mathbb{C}^{n,m}\setminus{\\{0\\}}$ such
that $X^{*}Y+Y^{*}X$ is of full rank, we provide a different characterization
of dissipative mappings from $X$ to $Y$. The main advantage of this
characterization over Theorem 3.2 is that it is explicitly in terms of the
matrix variables $K,Z,G\in{\mathbb{C}}^{n,n}$ such that $K\succeq 0$ and
$G^{*}=-G$, and the matrices $K,Z,G$ do not have to satisfy other constraints
like (3.5)-(3.7). However, extracting the minimal Frobenius norm solutions
from this characterization cannot be obtained easily.

###### Theorem 4.1

Let $X,Y\in\mathbb{C}^{n,m}\setminus{\\{0\\}}$ and suppose that
$M:=(X^{*}Y+Y^{*}X)^{-1}$ exists. Then $\mathbb{S}(X,Y)\neq\emptyset$ if and
only if $X^{*}Y+Y^{*}X\succ 0$ and $YX^{\dagger}X=Y$. Moreover,

$\displaystyle\mathbb{S}(X,Y)=\\{H+\widetilde{H}(K,G,Z)\mid
K,G,Z\in\mathbb{C}^{n,n},K\succeq 0,~{}G^{*}=-G\\},$ (4.1)

where $H$ and $\widetilde{H}$ are respectively defined by

$\displaystyle
H:=\frac{1}{2}\left(YX^{\dagger}-X^{\dagger^{*}}Y^{*}\right)+\frac{1}{2}\Big{\\{}YMY^{*}+YMX^{*}YX^{\dagger}+(YX^{\dagger})^{*}XMY^{*}$
$\displaystyle+(YX^{\dagger})^{*}XMX^{*}YX^{\dagger}\Big{\\}}$ (4.2)

and

$\displaystyle\widetilde{H}(K,G,Z):=\frac{1}{2}\Big{(}{\mathcal{P}}_{X}K{\mathcal{P}}_{X}+{\mathcal{P}}_{X}G{\mathcal{P}}_{X}-{\mathcal{P}}_{X}Z^{*}XX^{\dagger}+X^{\dagger^{*}}X^{\dagger}Z{\mathcal{P}}_{X}+YMX^{*}Z{\mathcal{P}}_{X}$
$\displaystyle+(YX^{\dagger})^{*}XMX^{*}Z{\mathcal{P}}_{X}+{\mathcal{P}}_{X}Z^{*}XMY^{*}+{\mathcal{P}}_{X}Z^{*}XMX^{*}YX^{\dagger}+{\mathcal{P}}_{X}Z^{*}XMX^{*}Z{\mathcal{P}}_{X}\Big{)}.$
(4.3)

Proof. If there exists $\Delta$ such that $\Delta X=Y$ and
$\Delta+\Delta^{*}\succeq 0$, then

$X^{*}Y+Y^{*}X=X^{*}(Y+\Delta^{*}X)=X^{*}(\Delta+\Delta^{*})X.$

This implies that $X^{*}Y+Y^{*}X$ is positive definite as
$\Delta+\Delta^{*}\succeq 0$ and $X^{*}Y+Y^{*}X$ is invertible. For the
converse, let $X,Y$ be such that $X^{*}Y+Y^{*}X\succ 0$. Then the matrix $H$
in (4.1) is well defined and satisfies $HX=Y$. Also $H+H^{*}$ is a positive
semidefinite matrix of the form $BB^{*}$ for some $B\in{\mathbb{C}}^{n,n}$.
Indeed, if we let $M=M_{1}M_{1}^{*}$, where $M_{1}$ is the Cholesky factor of
$M$, then we have

$\displaystyle H+H^{*}$ $\displaystyle=$ $\displaystyle
YMY^{*}+YMX^{*}YX^{\dagger}+(YX^{\dagger})^{*}XMY^{*}+(YX^{\dagger})^{*}XMX^{*}YX^{\dagger}$
$\displaystyle=$
$\displaystyle\big{(}YM_{1}+(YX^{\dagger})^{*}XM_{1}\big{)}\big{(}YM_{1}+(YX^{\dagger})^{*}XM_{1}\big{)}^{*}$
$\displaystyle=$ $\displaystyle BB^{*},$

where $B=YM_{1}+(YX^{\dagger})^{*}XM_{1}$.

Next we prove $``\subseteq"$ in (4.1). For this, let $\Delta\in\mathbb{S}$,
that is, $\Delta X=Y$ and $\Delta+\Delta^{*}\succeq 0$. By [23, Lemma 1.3],
there exists $Z\in\mathbb{C}^{n,n}$ such that

$\Delta=YX^{\dagger}+Z{\mathcal{P}}_{X}.$ (4.4)

Also, we can write $\Delta$ as

$\Delta=\frac{\Delta+\Delta^{*}}{2}+\frac{\Delta-\Delta^{*}}{2}.$ (4.5)

Now since $\Delta+\Delta^{*}\succeq 0$ we have $\Delta+\Delta^{*}=A^{*}A$ for
some $A\in\mathbb{C}^{n,n}$. This implies that $(\Delta+\Delta^{*})X=A^{*}AX$
and thus $\Delta X+\Delta^{*}X=A^{*}AX$. By setting $W:=AX$, we obtain
$Y+\Delta^{*}X=A^{*}W$ because $\Delta X=Y.$ This yields the mappings $AX=W$
and $A^{*}W=Y+\Delta^{*}X.$ The matrices $X$, $Y$ and $W$ satisfy

$X^{*}(Y+\Delta^{*}X)=X^{*}Y+X^{*}\Delta^{*}X=X^{*}(\Delta+\Delta^{*})X,=X^{*}A^{*}AX=W^{*}W,$
(4.6)

$YX^{\dagger}X=Y$ and $WX^{\dagger}X=W$. Therefore, from [17, Theorem 2.1],
$A$ can be written as

$A=WX^{\dagger}+\left(\left(Y+\Delta^{*}X\right)W^{\dagger}\right)^{*}-\left((Y+\Delta^{*}X)W^{\dagger}\right)^{*}XX^{\dagger}+\big{(}I_{n}-WW^{\dagger}\big{)}R_{1}\mathcal{P}_{X},$
(4.7)

for some $R_{1}\in\mathbb{C}^{n,n}$. This can be further simplified as

$A=((Y+\Delta^{*}X)W^{\dagger})^{*}+(I_{n}-WW^{\dagger})R_{1}\mathcal{P}_{X},$
(4.8)

using the fact that
$((Y+\Delta^{*}X)W^{\dagger}))^{*}XX^{\dagger}=WX^{\dagger}$, since
$(Y+\Delta^{*}X)^{*}X=W^{*}W$. Thus from (4.8), we have

$\displaystyle A^{*}A$ $\displaystyle=$
$\displaystyle\left(\big{(}(Y+\Delta^{*}X)W^{\dagger}\big{)}^{*}+\big{(}I_{n}-WW^{\dagger}\big{)}R_{1}\mathcal{P}_{X}\big{)}^{*}\big{(}\big{(}(Y+\Delta^{*}X)W^{\dagger}\big{)}^{*}+\big{(}I_{n}-WW^{\dagger}\big{)}R_{1}\mathcal{P}_{X}\right)$
(4.9) $\displaystyle=$
$\displaystyle(Y+\Delta^{*}X)M(Y+\Delta^{*}X)^{*}+\mathcal{P}_{X}R_{1}^{*}\big{(}I_{n}-WW^{\dagger}\big{)}\big{(}I_{n}-WW^{\dagger}\big{)}R_{1}\mathcal{P}_{X}.$

Define
$K:=R_{1}^{*}\big{(}I_{n}-WW^{\dagger}\big{)}\big{(}I_{n}-WW^{\dagger}\big{)}R_{1}$,
then $K\succeq 0$, and from (4.9) we have

$\displaystyle
A^{*}A=(Y+\Delta^{*}X)M(Y+\Delta^{*}X)^{*}+\mathcal{P}_{X}K\mathcal{P}_{X}.$
(4.10)

Now we consider the skew-Hermitian part $\Delta-\Delta^{*}$ of $\Delta$. Let
$S=\Delta-\Delta^{*}$. Then $SX=\Delta X-\Delta^{*}X=Y-\Delta^{*}X$. Again
since

$X^{*}(Y-\Delta^{*}X)=X^{*}\Delta
X-X^{*}\Delta^{*}X=X^{*}(\Delta-\Delta^{*})X=-(Y^{*}-X^{*}\Delta)X=-(Y-\Delta^{*}X)^{*}X,$

by Theorem 2.3, we have

$\displaystyle
S=(Y-\Delta^{*}X)X^{\dagger}-((Y-\Delta^{*}X)X^{\dagger})^{*}\mathcal{P}_{X}+\mathcal{P}_{X}G\mathcal{P}_{X},$
(4.11)

for some skew-Hermitian matrix $G\in\mathbb{C}^{n,n}$. By inserting
$\Delta=YX^{\dagger}+Z\mathcal{P}_{X}$ from (4.4) and
$W^{\dagger}(W^{\dagger})^{*}=((Y+\Delta^{*}X)^{*}X)^{-1}=(X^{*}Y+Y^{*}X)^{-1}$
in (4.10) and (4.11), we respectively obtain

$\displaystyle
A^{*}A=YMY^{*}+YMX^{*}YX^{\dagger}+YMX^{*}Z\mathcal{P}_{X}+(YX^{\dagger})^{*}XMY^{*}+(YX^{\dagger})^{*}XMX^{*}YX^{\dagger}$
$\displaystyle+(YX^{\dagger})^{*}XMX^{*}Z\mathcal{P}_{X}+\mathcal{P}_{X}Z^{*}XMY^{*}+\mathcal{P}_{X}Z^{*}XMX^{*}YX^{\dagger}+\mathcal{P}_{X}Z^{*}XMX^{*}Z\mathcal{P}_{X}+\mathcal{P}_{X}K\mathcal{P}_{X},$

and

$\displaystyle
S=YX^{\dagger}-X^{\dagger^{*}}Y^{*}XX^{\dagger}-\mathcal{P}_{X}Z^{*}XX^{\dagger}-X^{\dagger^{*}}Y^{*}\mathcal{P}_{X}+X^{\dagger^{*}}X^{\dagger}Z\mathcal{P}_{X}+\mathcal{P}_{X}G\mathcal{P}_{X}.$
(4.13)

By inserting $\Delta+\Delta^{*}=A^{*}A$ from (4) and $\Delta-\Delta^{*}=S$
from (4.13) in (4.5), and by separating the terms with and without matrices
$G$, $K$, $Z$, the matrix $\Delta$ can be written as

$\Delta=H+\widetilde{H}(K,G,Z).$

This proves $``\subseteq"$ in (4.1).

Now, let us prove $``\supseteq"$ in (4.1). Suppose,
$\Delta=H+\widetilde{H}(K,G,Z),$ where $H$ is defined in (4.1) and
$\widetilde{H}(K,G,Z)$ is defined in (4.1) for some matrices
$G,K,Z\in{\mathbb{C}}^{n,n}$ such that $K\succeq 0$ and $G^{*}=-G$. Then it is
easy to check that $\Delta X=Y$ since $H$ and $\widetilde{H}$ satisfy $HX=Y$
and $\widetilde{H}(K,G,Z)X=0$. Also

$\displaystyle\Delta+\Delta^{*}=\big{(}H+\widetilde{H}(K,G,Z)\big{)}+\big{(}H+\widetilde{H}(K,G,Z)\big{)}^{*}\succeq
0.$ (4.14)

Indeed,

$\displaystyle\big{(}H+\widetilde{H}(K,G,Z)\big{)}+\big{(}H+\widetilde{H}(K,G,Z)\big{)}^{*}$
$\displaystyle=(H+H^{*})+(\widetilde{H}(K,G,Z)+\widetilde{H}(K,G,Z)^{*})$
$\displaystyle=YMY^{*}+YMX^{*}YX^{\dagger}+(YX^{\dagger})^{*}XMY^{*}+(YX^{\dagger})^{*}XMX^{*}YX^{\dagger}+YMX^{*}Z\mathcal{P}_{X}+$
$\displaystyle(YX^{\dagger})^{*}XMX^{*}Z\mathcal{P}_{X}+\mathcal{P}_{X}Z^{*}XMY^{*}+\mathcal{P}_{X}Z^{*}XMX^{*}YX^{\dagger}+\mathcal{P}_{X}Z^{*}XMX^{*}Z\mathcal{P}_{X}+\mathcal{P}_{X}K\mathcal{P}_{X}$
$\displaystyle=BB^{*}+\mathcal{P}_{X}K\mathcal{P}_{X},$ (4.15)

where $B=YM_{1}+(YX^{\dagger})^{*}(XM_{1})+\mathcal{P}_{X}Z^{*}XM_{1}$, and
$M=M_{1}M_{1}^{*}$ where $M_{1}$ is the Cholesky factor of $M$. The first term
in (4) is PSD being a matrix of the form $BB^{*}$ and the second term is PSD
because $K\succeq 0$. Thus (4) is PSD being the sum of two PSD matrices. Hence
$\Delta\in\mathbb{S}(X,Y)$. This shows $``\supseteq"$ in (4.1) and hence
completes the proof.

###### Remark 4.2

The assumption that $X^{*}Y+Y^{*}X$ is invertible is not necessary for the
existence of a dissipative mapping which only requires $X^{*}Y+Y^{*}X$ to be
positive semidefinite; see Theorem 3.2. This assumption is more of a technical
need for the second characterization of dissipative mappings in Theorem 4.1.
However, there are situations, such as computing the eigenpair backward error
for DH matrices, where one has to find dissipative mappings taking
$X\in{\mathbb{C}}^{n,r}$ to $Y\in{\mathbb{C}}^{n,r}$ such that $X^{*}Y+Y^{*}X$
is invertible. For example, consider a DH matrix $J-R$ with $Q=I_{n}$,
$\lambda\in{\mathbb{C}}$ which is not an eigenvalue of $J-R$, and
$\hat{X}\in{\mathbb{C}}^{n,r}$ with $\text{rank}(\hat{X})=r$. Then the
structured eigenpair backward error for the DH matrix $J-R$, for making
$\lambda$ an eigenvalue of multiplicity $r$ and the columns of $\hat{X}$ as
the corresponding eigenvectors, is the smallest norm of the perturbations
$\Delta_{J}-\Delta_{R}$ such that $(J-\Delta_{J})+(R-\Delta_{R})$ is a DH
matrix and $((J-\Delta_{J})+(R-\Delta_{R}))\hat{X}=\lambda\hat{X}$. By Lemma
2.7, the latter condition is equivalent to finding a dissipative mapping
$\Delta$ such that $\Delta\hat{X}=\hat{Y}$, where $\hat{Y}=(\lambda
I_{n}-(J-R))\hat{X}$. Such a dissipative mapping exists if and only if
$\hat{X}^{*}\hat{Y}+\hat{Y}^{*}\hat{X}=\hat{X}^{*}(\mathop{\mathrm{Re}}{(\lambda)}I_{n}+R)\hat{X}\succeq
0$. Hence if $\mathop{\mathrm{Re}}{(\lambda)}>0$, then
$\hat{X}^{*}\hat{Y}+\hat{Y}^{*}\hat{X}\succ 0$ since $R\succeq 0$.

A result analogous to Corollary 3.3 for the vector case (when $m=1$), where
the conditions on the vectors $x$ and $y$ are simpler than for Theorem 4.1, is
stated below.

###### Corollary 4.3

Let $x,y\in\mathbb{C}^{n}\setminus{\\{0\\}}$. Then
$\mathbb{S}(x,y)\neq\emptyset$ if and only if
$\mathop{\mathrm{Re}}{(x^{*}y)}\geq 0$. Moreover, if
$\mathop{\mathrm{Re}}{(x^{*}y)}>0$, then

$\displaystyle\mathbb{S}(x,y)=\\{H+\widetilde{H}(K,G,Z)\mid
K,G,Z\in\mathbb{C}^{n,n},K\succeq 0,~{}G^{*}=-G\\},$

where $H$ and $\widetilde{H}$ are respectively defined as

$H:=\frac{1}{2}\left(\frac{yx^{*}}{\|x\|^{2}}-\frac{xy^{*}}{\|x\|^{2}}\right)+\frac{\alpha}{2}\left\\{yy^{*}+(x^{*}y)\frac{yx^{*}}{\|x\|^{2}}+(y^{*}x)\frac{xy^{*}}{\|x\|^{2}}+(|x^{*}y|^{2})\frac{xx^{*}}{\|x\|^{4}}\right\\}$

and

$\displaystyle\widetilde{H}(K,G,Z):=\frac{1}{2}\Big{(}\mathcal{P}_{x}K\mathcal{P}_{x}+\mathcal{P}_{x}G\mathcal{P}_{x}+(\alpha(y^{*}x)+1)xx^{\dagger}Z\mathcal{P}_{x}+(\alpha(x^{*}y)-1)\mathcal{P}_{x}Z^{*}xx^{\dagger}$
$\displaystyle+\alpha\mathcal{P}_{x}Z^{*}xy^{*}+\alpha
yx^{*}Z\mathcal{P}_{x}+\alpha\mathcal{P}_{x}Z^{*}xx^{*}Z\mathcal{P}_{x}\Big{)},$

where $\alpha=\frac{1}{2\mathop{\mathrm{Re}}{(x^{*}y)}}$.

###### Remark 4.4

Along the lines of Section 3.1, we can also obtain a second family of real
dissipative mappings from Theorem 4.1, where the restrictions on matrix
variables $K,G,Z$ are more simplified than Theorem 3.4. Indeed, let
$X,\,Y\in{\mathbb{C}}^{n,m}$. Then for $\mathcal{X}=[X,\,\overline{X}]$ and
$\mathcal{Y}=[Y,\,\overline{Y}]$, from Theorem 4.1 whenever there exists a
complex dissipative mapping $\Delta$ satisfying
$\Delta\mathcal{X}=\mathcal{Y}$, then there also exists a real dissipative
mapping. In fact, in view of Lemmas 2.4 and 2.5, the matrix $H$ in Theorem 4.1
is real. Moreover, $H+\widetilde{H}(K,G,Z)$ with real
$K,G,Z\in{\mathbb{R}}^{n,n}$ such that $K\succeq 0$ and $G^{T}=-G$, gives a
family of real dissipative mappings from $X$ to $Y$.

##  5 Structured distance to asymptotic instability for DH systems

In this section, we study the distance to asymptotic instability for DH
systems $\dot{x}=(J-R)Qx$ defined in (1.1). DH systems are always stable, but
they are not necessarily asymptotically stable. In [16], the authors have
obtained various structured distances to asymptotic instability for complex DH
systems while perturbing $J$, $R$, or $Q$ (only one matrix at a time).
Similarly in [17], authors derived the real distances to asymptotic
instability for real DH systems while perturbing only $R$. The main tool in
the computation of stability radii in [16, 17] is minimal norm solutions to
Hermitian, skew-Hermitian and semidefinite mapping problems. To compute such
radii in the case of perturbing both $J$ and $R$ simultaneously, we will need
minimal norm solutions to complex and real dissipative mappings that map one
vector to another vector.

Therefore, in this section, we exploit the minimal norm solutions to the
dissipative mapping problem from previous sections and derive the structured
distance to asymptotic instability for DH systems while perturbing both $J$
and $R$ at a time. By following the terminology in [16], we define the
unstructured and the structured distances to asymptotic instability for DH
systems as follows:

###### Definition 5.1

Let $\mathbb{F}\in\\{{\mathbb{R}},{\mathbb{C}}\\}$. Consider a DH system of
the form (1.1) and let $B\in\mathbb{F}^{n,r}$ and $C\in\mathbb{F}^{q,n}$ be
given matrices. Then we define

  1. 1)

the unstructured stability radius $r_{\mathbb{F}}(J,R;B,C)$ of system (1.1)
with respect to general perturbations to $J$ and $R$ under the restrictions
$(B,C)$ by

$\displaystyle r_{\mathbb{F}}(J,R;B,C)=\inf\Big{\\{}$
$\displaystyle\sqrt{{\|\Delta_{J}\|}_{F}^{2}+{\|\Delta_{R}\|}_{F}^{2}}:\Delta_{J},\Delta_{R}\in\mathbb{F}^{r,q},$
(5.1) $\displaystyle\Lambda\big{(}(J-R)Q+B(\Delta_{J}-\Delta_{R})CQ\big{)}\cap
i{\mathbb{R}}\neq\emptyset\Big{\\}};$

  2. 2)

the structured stability radius $r_{\mathbb{F}}^{S_{d}}(J,R;B)$ of system
(1.1) with respect to structure preserving skew-Hermitian perturbations to $J$
and negative semidefinite perturbations to $R$ from the set

$S_{d}^{\mathbb{F}}(J,R;B)=\left\\{(\Delta_{J},\Delta_{R})\in(\mathbb{F}^{r,r})^{2}:~{}\Delta_{J}^{*}=\Delta_{J},\Delta_{R}\preceq
0,R+B\Delta_{R}B^{*}\succeq 0\right\\};$ (5.2)

by

$\displaystyle r_{\mathbb{F}}^{S_{d}}(J,R;B)=\inf\Big{\\{}$
$\displaystyle\sqrt{{\|\Delta_{J}\|}_{F}^{2}+{\|\Delta_{R}\|}_{F}^{2}}:(\Delta_{J},\Delta_{R})\in
S_{d}^{\mathbb{F}}(J,R;B),$ (5.3)
$\displaystyle\Lambda\big{(}(J-R)Q+B(\Delta_{J}-\Delta_{R})B^{*}Q\big{)}\cap
i{\mathbb{R}}\neq\emptyset\Big{\\}}.$

In the complex case ($\mathbb{F}={\mathbb{C}}$), $r_{{\mathbb{C}}}(J,R;B,C)$
and $r_{{\mathbb{C}}}^{S_{d}}(J,R;B)$ are respectively called the complex
unstructured and complex structured stability radii. Similarly, in the real
case ($\mathbb{F}={\mathbb{R}}$), $r_{{\mathbb{R}}}(J,R;B,C)$ and
$r_{{\mathbb{R}}}^{S_{d}}(J,R;B)$ are respectively called the real
unstructured and real structured stability radii.

In order to obtain bounds for the structured stability radius
$r_{\mathbb{F}}^{S_{d}}(J,R;B)$, we also define the structured eigenvalue
backward error as follows

$\displaystyle\eta_{\mathbb{F}}^{S_{d}}(J,R;B,\lambda)=\inf\Big{\\{}$
$\displaystyle\sqrt{{\|\Delta_{J}\|}_{F}^{2}+{\|\Delta_{R}\|}_{F}^{2}}:(\Delta_{J},\Delta_{R})\in
S_{d}^{\mathbb{F}}(J,R;B),$ (5.4)
$\displaystyle\text{det}\left((J-R)Q+B(\Delta_{J}-\Delta_{R})B^{*}Q-\lambda
I_{n}\right)=0\Big{\\}},$

where $\lambda\in{\mathbb{C}}$, $(J-R)Q$ is a DH matrix, and the perturbation
set $S_{d}^{\mathbb{F}}(J,R;B)$ is as defined in (5.2). Consequently, we have
the following result.

###### Theorem 5.2

Consider an asymptotically stable DH system of the form (1.1). Then

$r_{\mathbb{F}}^{S_{d}}(J,R;B)=\inf_{w\in{\mathbb{R}}}\eta_{\mathbb{F}}^{S_{d}}(J,R;B,iw).$
(5.5)

###  5.1 Complex stability radius

Consider a complex LTI DH system of the form

$\dot{x}(t)=(J-R)Qx(t),$ (5.6)

where $J,R,Q\in{\mathbb{C}}^{n,n}$ such that $J^{*}=-J$, $R^{*}=R\succeq 0$,
and $Q^{*}=Q\succ 0$. Here, we study the complex stability radii
$r_{{\mathbb{C}}}(J,R;B,C)$ and $r_{{\mathbb{C}}}^{S_{d}}(J,R;B)$.

Inspired by the proof of [10, Proposition 2.1], we obtain the following
formula for the unstructured stability radius $r_{{\mathbb{C}}}(J,R;B,C)$.

###### Theorem 5.3

Consider an asymptotically stable DH system of the form (5.6). Let
$B\in{\mathbb{C}}^{n,r}$ and $C\in{\mathbb{C}}^{q,n}$ be given restriction
matrices. Then $r_{{\mathbb{C}}}(J,R;B,C)$ is finite if and only if
$G(w)=CQ\left(iwI_{n}-(J-R)Q\right)^{-1}B$ is not zero for some
$w\in{\mathbb{R}}$. In the latter case, we have

$r_{{\mathbb{C}}}(J,R;B,C)=\frac{1}{\sqrt{2}}\inf_{w\in{\mathbb{R}}}\frac{1}{\|G(w)\|}.$

Proof. In view of (5.1), note that for any
$\Delta_{J},\Delta_{R}\in{\mathbb{C}}^{r,q}$
$\Lambda\big{(}(J-R)Q+B(\Delta_{J}-\Delta_{R})CQ\big{)}\,\cap\,i{\mathbb{R}}\neq\emptyset$
if and only if $\text{det}\left((J-R)Q+B(\Delta_{J}-\Delta_{R})CQ-
iwI_{n}\right)=0$ for some $w\in{\mathbb{R}}$ if and only if
$\text{det}\left(I_{n}-(\Delta_{J}-\Delta_{R})CQ(iwI_{n}-(J-R)Q)^{-1}B\right)=0$
for some $w\in{\mathbb{R}}$. Using this in (5.1), we have

$\displaystyle r_{{\mathbb{C}}}(J,R;B,C)=$
$\displaystyle\inf\Big{\\{}\sqrt{{\|\Delta_{J}\|}_{F}^{2}+{\|\Delta_{R}\|}_{F}^{2}}:\Delta_{J},\Delta_{R}\in{\mathbb{C}}^{r,q},\,w\in{\mathbb{R}}$
(5.7)
$\displaystyle\text{det}\left(I_{n}-(\Delta_{J}-\Delta_{R})CQ(iwI_{n}-(J-R)Q)^{-1}B\right)=0\Big{\\}}$
$\displaystyle=\inf_{w\in{\mathbb{R}}}\inf\Big{\\{}\sqrt{{\|\Delta_{J}\|}_{F}^{2}+{\|\Delta_{R}\|}_{F}^{2}}:\Delta_{J},\Delta_{R}\in{\mathbb{C}}^{r,q},\,x\in{\mathbb{C}}^{r}\setminus\\{0\\}$
$\displaystyle(\Delta_{J}-\Delta_{R})CQ(iwI_{n}-(J-R)Q)^{-1}Bx=x\Big{\\}}.$

In view of Lemma 2.6, (5.7) becomes

$r_{{\mathbb{C}}}(J,R;B,C)^{2}=\inf_{w\in{\mathbb{R}}}\inf\Big{\\{}\frac{{\|\Delta\|}_{F}^{2}}{2}:\Delta\in{\mathbb{C}}^{r,q},\,x\in{\mathbb{C}}^{r}\setminus\\{0\\},\Delta
CQ(iwI_{n}\text{$-$}(J\text{$-$}R)Q)^{-1}Bx\text{$=$}x\Big{\\}}.\\\ $ (5.8)

Note that $r_{{\mathbb{C}}}(J,R;B,C)\geq 0$. Therefore
$r_{{\mathbb{C}}}(J,R;B,C)$ is finite if and only if the inner infimum in
(5.8) is finite for some $w\in{\mathbb{R}}$, which is true if and only if
$G(w):=CQ\left((J-R)Q-iwI_{n}\right)^{-1}B\neq 0$ for some $w\in{\mathbb{R}}$.
Indeed, if $G(w)\neq 0$ for some $w\in{\mathbb{R}}$ then for any
$x\in{\mathbb{C}}^{r}\setminus\\{0\\}$, there exists
$\tilde{\Delta}\in{\mathbb{C}}^{r,q}$ such that $\tilde{\Delta}G(w)x=x$. This
implies, using (5.8), that
$r_{{\mathbb{C}}}(J,R;B,C)\leq\frac{{\|\tilde{\Delta}\|}_{F}^{2}}{2}$. Using
the fact that ${\|\Delta\|}_{F}\geq\|\Delta\|$ for any $\Delta$ in (5.8) , we
have

$\displaystyle r_{{\mathbb{C}}}(J,R;B,C)$ $\displaystyle\geq$
$\displaystyle\inf_{w\in{\mathbb{R}}}\inf\Big{\\{}\frac{{\|\Delta\|}^{2}}{2}:\Delta\in{\mathbb{C}}^{r,q},\,x\in{\mathbb{C}}^{r}\setminus\\{0\\},\Delta
CQ((J-R)Q-iwI_{n})^{-1}Bx=x\Big{\\}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\inf_{w\in{\mathbb{R}}}\frac{1}{{\|G(w)\|}^{2}},$

where the last equality follows by slightly modifying the proof of [10,
Proposition 2.1]. In fact, we have the equality in (LABEL:eq:inside_1) because
for any $w\in{\mathbb{R}}$ the infimum in the right hand side of
(LABEL:eq:inside_1) is attained for a rank one matrix $\Delta$ for which we
have $\|\Delta\|=\frac{1}{{\|G(w)\|}}={\|\Delta\|}_{F}$ [10].

Next, using Theorem 5.2, we derive bounds for
$r_{{\mathbb{C}}}^{S_{d}}(J,R;B)$ via bounds for the backward error
$\eta_{{\mathbb{C}}}^{S_{d}}(J,R;B,\lambda)$.

###### Theorem 5.4

Consider a DH matrix $(J-R)Q$ and let $w\in{\mathbb{R}}$ be such that
$(J-R)Q-iwI_{n}$ is invertible. Let $B\in{\mathbb{C}}^{n,r}$ be of full column
rank and define $\Omega=\text{{\rm
null}}\left((I_{n}-BB^{\dagger})(iwI_{n}-(J-R)Q)\right)$. Suppose that
$\Omega\neq\emptyset$. Then

$\eta_{{\mathbb{C}}}^{S_{d}}(J,R;B,iw)^{2}\geq\inf_{x\in\Omega}\left\\{\frac{2{\|B^{\dagger}(iwI_{n}-(J-R)Q)x\|}^{2}}{{\|B^{*}Qx\|}^{2}}-\frac{|x^{*}Q(iwI_{n}-(J-R)Q)x|^{2}}{{\|B^{*}Qx\|}^{4}}\right\\}.$
(5.10)

Moreover, let the infimum in the right hand side of (5.10) be attained at
$\hat{x}\in\Omega$ and define
$\hat{\Delta}_{R}=-\hat{x}^{*}QRQ\hat{x}\frac{(B^{*}Q\hat{x})(B^{*}Q\hat{x})^{*}}{{\|B^{*}Q\hat{x}\|}^{4}}$.
If $R+B\hat{\Delta}_{R}B^{*}\succeq 0$, then equality holds in (5.10). In this
case, we have

$\sqrt{2}\,\sigma_{\min}\left(B^{\dagger}(iwI_{n}-(J-R)Q)UW^{-*}\right)\geq\eta_{{\mathbb{C}}}^{S_{d}}(J,R;B,iw)\geq\sigma_{\min}\left(B^{\dagger}(iwI_{n}-(J-R)Q)UW^{-*}\right),$
(5.11)

where the columns of $U$ form an orthonormal basis for $\Omega$ and $W$ is the
Cholesky factor of $U^{*}QBB^{*}QU$.

Proof. By definition (5.4),

$\displaystyle\eta_{{\mathbb{C}}}^{S_{d}}(J,R;B,iw)=$
$\displaystyle\inf\Big{\\{}\sqrt{{\|\Delta_{J}\|}_{F}^{2}+{\|\Delta_{R}\|}_{F}^{2}}:(\Delta_{J},\Delta_{R})\in
S_{d}^{{\mathbb{C}}}(J,R;B),x\in{\mathbb{C}}^{n}\setminus\\{0\\},$
$\displaystyle\left((J-R)Q+B(\Delta_{J}-\Delta_{R})B^{*}Q-iwI_{n}\right)x=0\Big{\\}},$
$\displaystyle=$
$\displaystyle\inf\Big{\\{}\sqrt{{\|\Delta_{J}\|}_{F}^{2}+{\|\Delta_{R}\|}_{F}^{2}}:(\Delta_{J},\Delta_{R})\in
S_{d}^{{\mathbb{C}}}(J,R;B),x\in{\mathbb{C}}^{n}\setminus\\{0\\},$
$\displaystyle B(\Delta_{J}-\Delta_{R})B^{*}Qx=(iwI_{n}-(J-R)Q)x\Big{\\}},$
$\displaystyle=$
$\displaystyle\inf\Big{\\{}\sqrt{{\|\Delta_{J}\|}_{F}^{2}+{\|\Delta_{R}\|}_{F}^{2}}:(\Delta_{J},\Delta_{R})\in
S_{d}^{{\mathbb{C}}}(J,R;B),x\in\Omega\setminus\\{0\\},$ (5.12)
$\displaystyle(\Delta_{J}-\Delta_{R})B^{*}Qx=B^{\dagger}(iwI_{n}-(J-R)Q)x\Big{\\}},$
$\displaystyle=$
$\displaystyle\inf\Big{\\{}{\|\Delta\|}_{F}:\Delta\in{\mathbb{C}}^{r,r},\Delta_{R}=-(\Delta+\Delta^{*})\preceq
0,R+B\Delta_{R}B^{*}\succeq 0,$ $\displaystyle
x\in\Omega\setminus\\{0\\},\Delta
B^{*}Qx=B^{\dagger}(iwI_{n}-(J-R)Q)x\Big{\\}},$ $\displaystyle\geq\hskip
5.69046pt\inf\Big{\\{}{\|\Delta\|}_{F}:\Delta\in{\mathbb{C}}^{r,r},\Delta_{R}=-(\Delta+\Delta^{*})\preceq
0,x\in\Omega\setminus\\{0\\},$ $\displaystyle\Delta
B^{*}Qx=B^{\dagger}(iwI_{n}-(J-R)Q)x\Big{\\}},$ (5.14)

where we have used Lemma 2.8 in (5.12) and Lemma 2.7 in (5.1). If
$\Omega\neq\emptyset$, then infimum on the right hand side of (5.14) is
finite. In fact, from Corollary 3.3 for any $x\in\Omega$ there exists
$\Delta\in{\mathbb{C}}^{r,r}$ such that $\Delta+\Delta^{*}\succeq 0$ and
$\Delta B^{*}Qx=B^{\dagger}(iwI_{n}-(J-R)Q)x$ if and only if
$\mathop{\mathrm{Re}}{\left((B^{*}Qx)^{*}B^{\dagger}(iwI_{n}-(J-R)Q)x\right)}\geq
0$. Clearly for any $x\in\Omega$, we have

$\displaystyle\mathop{\mathrm{Re}}{\left((B^{*}Qx)^{*}B^{\dagger}(iwI_{n}-(J-R)Q)x\right)}$
$\displaystyle=$
$\displaystyle\mathop{\mathrm{Re}}{\left(x^{*}QBB^{\dagger}(iwI_{n}-(J-R)Q)x\right)}$
$\displaystyle=$
$\displaystyle\mathop{\mathrm{Re}}{\left(x^{*}Q(iwI_{n}-(J-R)Q)x\right)}\quad(\because
x\in\Omega)$ $\displaystyle=$
$\displaystyle\mathop{\mathrm{Re}}{\left(x^{*}((iwQ-QJQ)+QRQ)x\right)}$
$\displaystyle=$ $\displaystyle\mathop{\mathrm{Re}}{(x^{*}QRQx)}\quad(\because
iwQ-QJQ\,\text{ is\, skew-Hermitian})$ $\displaystyle=$ $\displaystyle
x^{*}QRQx\geq 0\quad(\because R\succeq 0,~{}Q\succ 0).$

Thus using the minimal norm mapping from Corollary 3.3 in (5.14), we obtain

$\eta_{{\mathbb{C}}}^{S_{d}}(J,R;B,iw)^{2}\geq\inf_{x\in\Omega}\left\\{\frac{2{\|B^{\dagger}(iwI_{n}-(J-R)Q)x\|}^{2}}{{\|B^{*}Qx\|}^{2}}-\frac{|x^{*}Q(iwI_{n}-(J-R)Q)x|^{2}}{{\|B^{*}Qx\|}^{4}}\right\\}.$
(5.15)

This proves (5.10). Now suppose infimum in the right hand side of (5.15) is
attained at $\hat{x}\in\Omega$, then in view of Corollary 3.3 consider

$\displaystyle\hat{\Delta}=\frac{B^{\dagger}(iwI_{n}-(J-R)Q)\hat{x}\hat{x}^{*}QB}{{\|B^{*}Q\hat{x}\|}^{2}}+\frac{(B^{\dagger}(iwI_{n}-(J-R)Q)\hat{x})^{*}B^{*}Q\hat{x}}{{\|B^{*}Q\hat{x}\|}^{4}}(B^{*}Q\hat{x})(B^{*}Q\hat{x})^{*}$
$\displaystyle-\frac{B^{*}Q\hat{x}(B^{\dagger}(iwI_{n}-(J-R)Q)\hat{x})^{*}}{{\|B^{*}Q\hat{x}\|}^{2}}.$

Then $\hat{\Delta}B^{*}Q\hat{x}=B^{\dagger}(iwI_{n}-(J-R)Q)\hat{x}$ and
$\hat{\Delta}+\hat{\Delta}^{*}\succeq 0$. Set
$\hat{\Delta}_{R}=-\frac{\hat{\Delta}+\hat{\Delta}^{*}}{2}$ and
$\hat{\Delta}_{J}=\frac{\hat{\Delta}-\hat{\Delta}^{*}}{2}$. Then
$\hat{\Delta}_{R}\preceq 0$. Thus, if $R+B\hat{\Delta}_{R}B^{*}\succeq 0$,
then clearly $(\hat{\Delta}_{J},\hat{\Delta}_{R})\in
S_{d}^{{\mathbb{C}}}(J,R;B)$, and

${\|\hat{\Delta}_{J}\|}_{F}^{2}+{\|\hat{\Delta}_{R}\|}_{F}^{2}={\|\hat{\Delta}\|}_{F}^{2}=\frac{2{\|B^{\dagger}(iwI_{n}-(J-R)Q)\hat{x}\|}^{2}}{{\|B^{*}Q\hat{x}\|}^{2}}-\frac{|\hat{x}^{*}Q(iwI_{n}-(J-R)Q)\hat{x}|^{2}}{{\|B^{*}Q\hat{x}\|}^{4}}.$

This show the equality in (5.15) and hence in (5.14), i.e.,

$\eta_{{\mathbb{C}}}^{S_{d}}(J,R;B,iw)^{2}=\inf_{x\in\Omega}\left\\{\frac{2{\|B^{\dagger}(iwI_{n}-(J-R)Q)x\|}^{2}}{{\|B^{*}Qx\|}^{2}}-\frac{|x^{*}Q(iwI_{n}-(J-R)Q)x|^{2}}{{\|B^{*}Qx\|}^{4}}\right\\}.$
(5.16)

Now suppose $k=\text{dim}(\Omega)$ and let columns of $U\in{\mathbb{C}}^{n,k}$
forms an orthonormal basis for $\Omega$. Then $x(\neq 0)\in\Omega$ if and only
if $x=U\alpha$ for some $\alpha\in{\mathbb{C}}^{k}\setminus\\{0\\}$. Using
this in (5.16), we obtain

$\eta_{{\mathbb{C}}}^{S_{d}}(J,R;B,iw)^{2}=\inf_{\alpha\in{\mathbb{C}}^{k}\setminus\\{0\\}}\left\\{\frac{2{\|B^{\dagger}(iwI_{n}-(J-R)Q)U\alpha\|}^{2}}{{\|B^{*}QU\alpha\|}^{2}}-\frac{|\alpha^{*}U^{*}Q(iwI_{n}-(J-R)Q)U\alpha|^{2}}{{\|B^{*}QU\alpha\|}^{4}}\right\\}.$
(5.17)

Note that $B^{*}QU$ is a full rank matrix because if suppose $B^{*}QU\alpha=0$
for some $\alpha\in{\mathbb{C}}^{k}\setminus\\{0\\}$. This implies that

$0=B\hat{\Delta}B^{*}QU\alpha=(iwI_{n}-(J-R)Q)U\alpha,$

so that $iw\in\Lambda((J-R)Q)$ which is a contradiction. Thus the matrix
$U^{*}QBB^{*}QU$ is positive definite. Let $U^{*}QBB^{*}QU=WW^{*}$, where $W$
is the unique Cholesky factor of $U^{*}QBB^{*}QU$, then by using
$y=W^{*}\alpha$ in (5.17), we have

$\displaystyle\eta_{{\mathbb{C}}}^{S_{d}}(J,R;B,iw)^{2}=$
$\displaystyle\inf_{y\in{\mathbb{C}}^{k}\setminus\\{0\\}}\left\\{\frac{2{\|B^{\dagger}(iwI_{n}-(J-R)Q)UW^{-*}y\|}^{2}}{{\|y\|}^{2}}-\frac{|y^{*}W^{-1}U^{*}Q(iwI_{n}-(J-R)Q)UW^{-*}y|^{2}}{{\|y\|}^{4}}\right\\}.$
(5.18)

In view of the Cauchy-Schwarz inequality, note that for every
$y\in{\mathbb{C}}^{k}\setminus\\{0\\}$ we have

$\displaystyle\frac{2{\|B^{\dagger}(iwI_{n}-(J-R)Q)UW^{-*}y\|}^{2}}{{\|y\|}^{2}}$
$\displaystyle\geq$
$\displaystyle\frac{2{\|B^{\dagger}(iwI_{n}-(J-R)Q)UW^{-*}y\|}^{2}}{{\|y\|}^{2}}$
$\displaystyle-\frac{|y^{*}W^{-1}U^{*}Q(iwI_{n}-(J-R)Q)UW^{-*}y|^{2}}{{\|y\|}^{4}}$
$\displaystyle\geq$
$\displaystyle\frac{{\|B^{\dagger}(iwI_{n}-(J-R)Q)UW^{-*}y\|}^{2}}{{\|y\|}^{2}}.$

Using this in (5.1) yields

$\displaystyle
2\inf_{y\in{\mathbb{C}}^{k}\setminus\\{0\\}}\frac{{\|B^{\dagger}(iwI_{n}-(J-R)Q)UW^{-*}y\|}^{2}}{{\|y\|}^{2}}$
$\displaystyle\geq$ $\displaystyle\eta_{{\mathbb{C}}}^{S_{d}}(J,R;B,iw)^{2}$
$\displaystyle\geq$
$\displaystyle\inf_{y\in{\mathbb{C}}^{k}\setminus\\{0\\}}\frac{{\|B^{\dagger}(iwI_{n}-(J-R)Q)UW^{-*}y\|}^{2}}{{\|y\|}^{2}},$

and hence

$\sqrt{2}\,\sigma_{\min}\left(B^{\dagger}(iwI_{n}-(J-R)Q)UW^{-*}\right)\geq\eta_{{\mathbb{C}}}^{S_{d}}(J,R;B,iw)\geq\sigma_{\min}\left(B^{\dagger}(iwI_{n}-(J-R)Q)UW^{-*}\right).$

This completes the proof.

By using Theorem 5.4 in Theorem 5.2, we obtain a lower bound for the
structured stability radius $r_{{\mathbb{C}}}^{S_{d}}(J,R;B)$ as follows.

###### Theorem 5.5

Consider an asymptotically stable DH system of the form (5.6). Let
$B\in{\mathbb{C}}^{n,r}$ be of full column rank. For $w\in{\mathbb{R}}$,
define $\Omega_{w}:=\text{{\rm
null}}\left((I_{n}-BB^{\dagger})(iwI_{n}-(J-R)Q)\right)$. Suppose that
$\Omega_{w}\neq\emptyset$ for some $w\in{\mathbb{R}}$. Then

$r_{{\mathbb{C}}}^{S_{d}}(J,R;B)^{2}\geq\inf_{w\in{\mathbb{R}}}\inf_{x\in\Omega_{w}}\left\\{\frac{2{\|B^{\dagger}(iwI_{n}-(J-R)Q)x\|}^{2}}{{\|B^{*}Qx\|}^{2}}-\frac{|x^{*}Q(iwI_{n}-(J-R)Q)x|^{2}}{{\|B^{*}Qx\|}^{4}}\right\\}.$

###### Remark 5.6

We conclude the section with a few remarks about Theorem 5.4.

  * •

Since $\hat{\Delta}_{R}$ in Theorem 5.4 is a rank-one matrix with only one
negative eigenvalue, if $R\succ 0$ and $R+B\hat{\Delta}_{R}B^{*}$ is singular,
then from [16, Lemma 4.4] $R+B\hat{\Delta}_{R}B^{*}$ is positive semidefinite,
and this implies the equality in (5.10).

  * •

As a particular case of Theorem 5.4, i.e., when $w=0$, we obtain the distance
to singularity $d^{S_{d}}(J,R;B):=\eta^{S_{d}}(J,R;B,0)$ with respect to
structure-preserving perturbations to both $J$ and $R$ from the set
$S_{d}(J,R;B)$ which is an analogous result to [16, Theorem 6.2]. Note that in
[16], the structured distances to singularity for DH matrices were obtained
for structure-preserving perturbations to $Q$ only.

###  5.2 Real stability radius

Consider a real LTI DH system of the form

$\dot{x}(t)=(J-R)Qx(t),$ (5.19)

where $J,R,Q\in{\mathbb{R}}^{n,n}$ such that $J^{T}=-J$, $R^{T}=R\succeq 0$,
and $Q^{T}=Q\succ 0$. In this section, we discuss the real unstructured
distance $r_{{\mathbb{R}}}(J,R;B,C)$ defined by (5.1) and the real structured
distance $r_{{\mathbb{R}}}^{S_{d}}(J,R;B)$ defined by (5.3) for real DH
systems. For this, define, for a given $M\in{\mathbb{C}}^{p,m}$,

$\mu_{{\mathbb{R}},q}(M):=\left(\inf\left\\{{\|\Delta\|}_{q}:~{}\Delta\in{\mathbb{R}}^{m,p},~{}\text{det}(I_{m}-\Delta
M)=0\right\\}\right)^{-1},$ (5.20)

where $q\in\\{2,F\\}$. Let us first state a result from [19] that will be
useful in determining the unstructured distance $r_{{\mathbb{R}}}(J,R;B,C)$.

######  Theorem 5.7 ([19])

We have

$\mu_{{\mathbb{R}},2}(M)=\inf_{\gamma\in(0,1]}\sigma_{2}\left(\left[\begin{array}[]{cc}\mathop{\mathrm{Re}}{M}&-\gamma\mathop{\mathrm{Im}}{M}\\\
\gamma^{-1}\mathop{\mathrm{Im}}{M}&\mathop{\mathrm{Re}}{M}\end{array}\right]\right),$

where $\sigma_{2}(A)$ is the second largest singular value of a matrix $A$.
Furthermore, an optimal $\Delta$ that attains the value of
$\mu_{{\mathbb{R}},2}(M)$ can be chosen of rank at most two.

Let us now prove a result bounding $\mu_{{\mathbb{R}},F}(M)$ using
$\mu_{{\mathbb{R}},2}(M)$.

###### Theorem 5.8

We have

$\frac{1}{\sqrt{2}}\mu_{{\mathbb{R}},2}(M)\leq\mu_{{\mathbb{R}},F}(M)\leq\mu_{{\mathbb{R}},2}(M).$
(5.21)

Proof. By definition,

$\displaystyle\mu_{{\mathbb{R}},2}(M)^{-1}$ $\displaystyle=$
$\displaystyle\inf\left\\{{\|\Delta\|}_{2}:~{}\Delta\in{\mathbb{R}}^{m,p},~{}\text{det}(I_{m}-\Delta
M)=0\right\\}$ (5.22) $\displaystyle\leq$
$\displaystyle\inf\left\\{{\|\Delta\|}_{F}:~{}\Delta\in{\mathbb{R}}^{m,p},~{}\text{det}(I_{m}-\Delta
M)=0\right\\}$ $\displaystyle=$ $\displaystyle\mu_{{\mathbb{R}},F}(M)^{-1},$

since for any $\Delta\in{\mathbb{R}}^{m,p}$, we have
${\|\Delta\|}_{2}\leq{\|\Delta\|}_{F}$. Now suppose
$\hat{\Delta}\in{\mathbb{R}}^{m,p}$ such that
$\mu_{{\mathbb{R}},2}(M)^{-1}={\|\hat{\Delta}\|}_{2}$. Then from the first
part $\hat{\Delta}$ is of rank at most two. This implies that
$\text{det}(I_{m}-\hat{\Delta}M)=0$ and
${\|\hat{\Delta}\|}_{F}\leq\sqrt{2}{\|\hat{\Delta}\|}_{2}$, which yields that

$\mu_{{\mathbb{R}},F}(M)^{-1}\leq\sqrt{2}{\|\hat{\Delta}\|}_{2}=\mu_{{\mathbb{R}},2}(M).$
(5.23)

Hence from (5.22) and (5.23), we obtain (5.21).

As an application of the above theorem we bound the real unstructured distance
$r_{{\mathbb{R}}}(J,R;B,C)$.

###### Theorem 5.9

Consider a real asymptotically stable DH system of the form (5.19) and let
$B\in{\mathbb{R}}^{n,r}$ and $C\in{\mathbb{R}}^{q,n}$ be restriction matrices.
Then

$\frac{1}{\sqrt{2}}\,\inf_{w\in{\mathbb{R}}}\mu_{{\mathbb{R}},2}(G(w))^{-1}\leq
r_{{\mathbb{R}}}(J,R;B,C)\leq\inf_{w\in{\mathbb{R}}}\mu_{{\mathbb{R}},2}(G(w))^{-1},$

where
$\mu_{{\mathbb{R}},2}(G(w))^{-1}=\inf_{\gamma\in(0,1]}\sigma_{2}\left(\left[\begin{array}[]{cc}\mathop{\mathrm{Re}}{G(w)}&-\gamma\mathop{\mathrm{Im}}{G(w)}\\\
\gamma^{-1}\mathop{\mathrm{Im}}{G(w)}&\mathop{\mathrm{Re}}{G(w)}\end{array}\right]\right)$
and $G(w)=CQ\left(iwI_{n}-(J-R)Q\right)^{-1}B$.

Proof. By Definition (5.1), we have

$\displaystyle r_{{\mathbb{R}}}(J,R;B,C)$
$\displaystyle=\inf\Big{\\{}\sqrt{{\|\Delta_{J}\|}_{F}^{2}+{\|\Delta_{R}\|}_{F}^{2}}:\Delta_{J},\Delta_{R}\in{\mathbb{R}}^{r,q},\Lambda\big{(}(J-R)Q+B(\Delta_{J}-\Delta_{R})CQ\big{)}\cap
i{\mathbb{R}}\neq\emptyset\Big{\\}}$
$\displaystyle=\inf\Big{\\{}\sqrt{{\|\Delta_{J}\|}_{F}^{2}+{\|\Delta_{R}\|}_{F}^{2}}:\Delta_{J},\Delta_{R}\in{\mathbb{R}}^{r,q},\,w\in{\mathbb{R}}$
$\displaystyle\text{det}\left((J-R)Q+B(\Delta_{J}-\Delta_{R})CQ-
iwI_{n}\right)=0\Big{\\}}$
$\displaystyle=\inf_{w\in{\mathbb{R}}}\inf\Big{\\{}\sqrt{{\|\Delta_{J}\|}_{F}^{2}+{\|\Delta_{R}\|}_{F}^{2}}:\Delta_{J},\Delta_{R}\in{\mathbb{R}}^{r,q},$
$\displaystyle\text{det}\left(I_{n}-(\Delta_{J}-\Delta_{R})CQ(iwI_{n}-(J-R)Q)^{-1}B\right)=0\Big{\\}}$
$\displaystyle=\inf_{w\in{\mathbb{R}}}\inf\Big{\\{}\frac{{\|\Delta\|}_{F}}{\sqrt{2}}:\Delta\in{\mathbb{R}}^{r,q},\text{det}\left(I_{n}-\Delta
CQ(iwI_{n}-(J-R)Q)^{-1}B\right)=0\Big{\\}}$ (5.24)
$\displaystyle=\frac{1}{\sqrt{2}}\inf_{w\in{\mathbb{R}}}\left(\mu_{{\mathbb{R}},F}G(w)\right)^{-1},$
(5.25)

where $G(w)=CQ\left(iwI_{n}-(J-R)Q\right)^{-1}B$. The equality (5.24) follows
from Lemma 2.6, and (5.25) follows by definition of $\mu_{{\mathbb{R}},F}G(w)$
in (5.20). Thus the result follows immediately using the inequality (5.23) in
Theorem 5.8.

We note that a result similar to Theorem 5.4 for the real backward error,
$\eta_{{\mathbb{R}}}^{S_{d}}(J,R;B)$, defined in (5.4) may be obtained by
using minimal-norm real dissipative mappings from Theorem 3.4. Thus in view of
Theorem 5.2 we can obtain a lower bound for the real structured stability
radius $r_{{\mathbb{R}}}^{S_{d}}(J,R;B)$. In the following we state this
result for $\eta_{{\mathbb{R}}}^{S_{d}}(J,R;B)$ and skip its proof as it is
similar to the proof of Theorem 5.4.

###### Theorem 5.10

Consider a real DH system of the form (5.19). Let $w\in{\mathbb{R}}$ be such
that $iwI_{n}-(J-R)Q$ is invertible. Let $B\in{\mathbb{R}}^{n,r}$ be of full
column rank and let $\Omega$ be the subset of
$\text{null}\left((I_{n}-BB^{\dagger})(iwI_{n}-(J-R)Q)\right)$ such that if
$x\in\Omega$ then $x^{*}QRQx\geq|x^{T}(iwQ+QRQ)x|$. Suppose that
$\Omega\neq\emptyset$. Then

$\eta_{{\mathbb{R}}}^{S_{d}}(J,R;B,iw)^{2}\geq\inf_{x\in\Omega}\left(2{\left\|\mathcal{Y}\mathcal{X}^{\dagger}\right\|}_{F}^{2}-\text{trace}\left((\mathcal{Y}\mathcal{X}^{\dagger})^{*}(\mathcal{X}\mathcal{X}^{\dagger})(\mathcal{Y}\mathcal{X}^{\dagger})\right)\right),$
(5.26)

where for $x\in\Omega$,
$\mathcal{Y}=\left[\begin{array}[]{cc}B^{\dagger}(iwI_{n}-(J-R)Q)x&\overline{B^{\dagger}(iwI_{n}-(J-R)Q)x}\end{array}\right]$
and
$\mathcal{X}=\left[\begin{array}[]{cc}B^{T}Qx&\overline{B^{T}Qx}\end{array}\right]$.
Moreover, let the infimum in the right hand side of (5.26) be attained at
$\tilde{x}\in\Omega$ and define
$\tilde{\Delta}_{R}:=\mathcal{\tilde{Y}}\mathcal{\tilde{X}}^{\dagger}-{(\mathcal{\tilde{Y}}\mathcal{\tilde{X}}^{\dagger})}^{*}\mathcal{P}_{\mathcal{\tilde{X}}}$,
where
$\mathcal{\tilde{Y}}=\left[\begin{array}[]{cc}B^{\dagger}(iwI_{n}-(J-R)Q)\tilde{x}&\overline{B^{\dagger}(iwI_{n}-(J-R)Q)\tilde{x}}\end{array}\right]$
and
$\mathcal{\tilde{X}}=\left[\begin{array}[]{cc}B^{T}Q\tilde{x}&\overline{B^{T}Q\tilde{x}}\end{array}\right]$.
If $R+B\tilde{\Delta}_{R}B^{T}\succeq 0$, then equality holds in (5.26).

Proof. In view of Theorem 3.4, the proof is similar to Theorem 5.4.

We close the section by noting that a result analogous to Theorem 5.5 can be
obtain for real structured stability radius $r_{{\mathbb{R}}}^{S_{d}}(J,R;B)$
by using Theorem 5.10 in Theorem 5.2.

##  6 Numerical experiments

In this section, we illustrate the significance of our distances obtained in
Theorems 5.3 and 5.5 and compare them with those of [16], where various
structured stability radii have been obtained for DH systems while perturbing
only $J$ or only $R$ one at a time. In the following, l.b. stands for “lower
bound”, $r_{{\mathbb{C}}}(R;B,C)$ (resp. $r_{{\mathbb{C}}}(J;B,C)$) denotes
the unstructured stability radius while perturbing only $R$ (resp. $J$) with
restriction matrices $B$ and $C$ [16, Theorem 3.3],
$r_{\mathbb{C}}^{\mathcal{S}}(J;B)$ denotes the structured stability radius
with respect to structure-preserving perturbations to $J$ [16, Theorem 5.1],
and $r_{{\mathbb{C}}}^{\mathcal{S}_{d}}(R;B)$ denotes the structured stability
radius with respect to structure-preserving negative semidefinite
perturbations to $R$ [16, Theorem 4.2]. We note that
$r_{{\mathbb{C}}}(R;B,C)=r_{{\mathbb{C}}}(J;B,C)$ [16, Theorem 3.3]. It is
also clear that $r_{{\mathbb{C}}}(J,R;B,C)\leq r_{{\mathbb{C}}}(R;B,C)$ and
the minimum of $r_{{\mathbb{C}}}^{\mathcal{S}}(J;B)$ and
$r_{{\mathbb{C}}}^{\mathcal{S}_{d}}(R;B)$ gives an upper bound to
$r_{{\mathbb{C}}}^{\mathcal{S}_{d}}(J,R;B)$.

All the experiments are performed in Matlab Version No. 9.1.0 (R2016b). To
compute these values (or lower bounds), we proceed as follows.

  * •

The formulas for the three radii $r_{{\mathbb{C}}}(R;B,B^{*})$,
$r_{{\mathbb{C}}}^{\mathcal{S}}(J;B)$, and
$r_{{\mathbb{C}}}^{\mathcal{S}_{d}}(R;B)$ were obtained in [16] in terms of
some non-convex optimization problems. In [16], the authors used the function
fminsearch in Matlab to solve these optimization problems which only gives a
local minimum starting from an initial guess. We instead used the GlobalSearch
solver in Matlab, with its default parameters which attempts to locate the
global solution. However, we can only guarantee that the computed solutions
are lower bounds to the exact distances.

  * •

$r_{{\mathbb{C}}}(J,R;B,B^{*})$: a formula for $r_{{\mathbb{C}}}(J,R;B,B^{*})$
is obtained in Theorem 5.3 which is $\frac{1}{\sqrt{2}}$ times the
unstructured distance $r_{{\mathbb{C}}}(R;B,B^{*})$.

  * •

$r_{{\mathbb{C}}}^{\mathcal{S}_{d}}(J,R;B)$: a lower bound for
$r_{{\mathbb{C}}}^{S_{d}}(J,R;B)$ is obtained in Theorem 5.5. We again used
the GlobalSearch solver in Matlab for this to get a good approximation for the
lower bound. We note that solving the optimization problem involved in (5.10)
is challenging, and beyond the scope of this this paper. This paper aims to
solve the dissipative mapping problem and show it is useful in engineering
applications such as DH systems. A possible future work would be to develop
more sophisticated ways to solve (5.10).

###### Example 6.1

[16, Example 7.1] Consider a prototype example of disk brake squeal problem
with the matrices $G,M,K,D$ from [16]. We consider the DH system
$\dot{x}=(J-R)Qx$ with

$J=\left[\begin{array}[]{cc}G&K+\frac{1}{2}N\\\
-(K+\frac{1}{2}N^{*})&0\end{array}\right],\quad
R=\left[\begin{array}[]{cc}D&0\\\ 0&0\end{array}\right],\quad\text{and}\quad
Q=\left[\begin{array}[]{cc}M&0\\\ 0&K\end{array}\right]^{-1}.$

We compute the various distances for the restriction matrices $B=I_{4}$ and
$C=I_{4}$, which are given in the following table:

$r_{{\mathbb{C}}}(J,R;B,B^{*})$ | $r_{{\mathbb{C}}}(R;B,B^{*})$ | l.b. to $r_{{\mathbb{C}}}^{\mathcal{S}_{d}}(J,R;B)$ | $r_{{\mathbb{C}}}^{\mathcal{S}}(J;B)$ | $r_{{\mathbb{C}}}^{\mathcal{S}_{d}}(R;B)$
---|---|---|---|---
0.0218 | 0.0308 | 0.0310 | 2.4725 | 5.6149

The lower bound $0.0310\leq r_{{\mathbb{C}}}^{\mathcal{S}_{d}}(J,R;B)$ is
obtained by Theorem 5.5, which as expected shows that the unstructured radius,
$r_{{\mathbb{C}}}(J,R;B,B^{*})=0.0218$, is smaller than the structured radius
$r_{{\mathbb{C}}}^{\mathcal{S}_{d}}(J,R;B)$. If we replace the $J$ matrix by
$J_{1}=\left[\begin{array}[]{cc}G&K\\\ -K&0\end{array}\right]$, then the
corresponding results are as follows:

$r_{{\mathbb{C}}}(J_{1},R;B,B^{*})$ | $r_{{\mathbb{C}}}(R;B,B^{*})$ | l.b. to $r_{{\mathbb{C}}}^{\mathcal{S}_{d}}(J_{1},R;B)$ | $r_{{\mathbb{C}}}^{\mathcal{S}}(J_{1};B)$ | $r_{{\mathbb{C}}}^{\mathcal{S}_{d}}(R;B)$
---|---|---|---|---
0.0826 | 0.1169 | 0.1185 | 2.2340 | 5.7971

In this case, the difference between the unstructured radius
$r_{{\mathbb{C}}}(J_{1},R;B,B^{*})$ and the structured radius
$r_{{\mathbb{C}}}^{\mathcal{S}_{d}}(J,R;B)$ is more significant. This implies
that if the magnitude of the structured perturbations to both $J$ and $R$ in
Frobenius norm is smaller than $0.1185$ than the system remains stable.

###### Example 6.2

To emphasize more on the stability radii $r_{{\mathbb{C}}}(J,R;B,B^{*})$ and
$r_{{\mathbb{C}}}^{\mathcal{S}_{d}}(J,R;B)$ obtained in this paper and to show
that these general distances are indeed different than the ones in [16], we
generate matrices $J$, $R$, $Q$, $B\in{\mathbb{C}}^{n,n}$ of different sizes
$(n\leq 9)$ randomly following a normal distribution with mean $0$ and
standard deviation $1$ (randn$(m,n)$ in Matlab) which we project on the
feasible set, that is, $J=-J^{*}$ and $R,Q\succeq 0$, so that
$\dot{x}=(J-R)Qx$ is a DH system, and all restricted stability radii were
finite. The restriction matrices $B$ and $C=B^{*}$ are chosen to be of full
rank. We compute our distances $r_{{\mathbb{C}}}(J,R;B,B^{*})$ and
$r_{{\mathbb{C}}}^{\mathcal{S}_{d}}(J,R;B)$ and compare these results with
those of [16] in Table 6.1. We see that (i) the various unstructured and
structured distances are quite a different; (ii) as expected, the stability
radii $r_{{\mathbb{C}}}(J,R;B,B^{*})$ and $r^{\mathcal{S}_{d}}(J,R;B)$ are
significantly smaller than their counterparts from [16] of perturbing only one
of $J$ and $R$; (iii) the results indicate that the lower bound to
$r_{{\mathbb{C}}}^{\mathcal{S}_{d}}(J,R;B)$ in some cases $(n=3,8,9)$ is
significantly larger than the unstructured distances (columns 2-3 in Table
6.1) and reasonably close to the upper bound (minimum of the last two columns
in Table 6.1).

Table 6.1: Various disances to instability for DH matrices size $n$ | $r_{{\mathbb{C}}}(J,R;B,B^{*})$ | $r_{{\mathbb{C}}}(R;B)$ | l.b. to $r_{{\mathbb{C}}}^{\mathcal{S}_{d}}(J,R;B)$ | $r_{{\mathbb{C}}}^{\mathcal{S}}(J;B)$ | $r_{{\mathbb{C}}}^{\mathcal{S}_{d}}(R;B)$
---|---|---|---|---|---
3 | 0.1162 | 0.1644 | 0.7353 | 7.8741 | 1.5038
4 | 0.0047 | 0.0067 | 0.5157 | 4.4828 | 620.6015
5 | 0.0550 | 0.0778 | 0.5108 | 1.2273 | 2.5896
6 | 0.0566 | 0.0801 | 0.2764 | 2.1462 | 1.1335
7 | 0.0165 | 0.0234 | 0.2421 | 0.8284 | 4.7189
8 | 0.0415 | 0.0587 | 0.9596 | 4.3467 | 1.9909
9 | 0.0632 | 0.0894 | 1.3540 | 1.7549 | 3.6928

##  7 Conclusion

In this paper, we have derived necessary and sufficient conditions for the
existence of the dissipative mappings taking $X\in{\mathbb{C}}^{n,k}$ to
$Y\in{\mathbb{C}}^{n,k}$, charactered the set of dissipative mappings, and
found the minimal Frobenius norm solution. We have then applied these results
to DH systems. In particular, we have used dissipative mappings to derive
bounds for the structured distance to asymptotic instability for both complex
and real DH systems (1.1) when both $J$ and $R$ are subject to perturbations.

The bounds computed in this paper involve solving a difficult non-convex
optimization problem. Possible future works include more sophisticated methods
to compute (5.10), and to study other applications of the dissipative
mappings.

## Acknowledgement

The authors would like to thank the anonymous reviewers for their careful
feedback that helped us improve the paper.

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