SIMULTANEOUSLY DISSIPATIVE OPERATORS
AND THE INFINITESIMAL MOORE EFFECT
IN INTERVAL SPACES

A.N.Gorban1, Yu.I.Shokin2, V.I.Verbitskii1

1Krasnoyarsk Computing Center
Russian Academy of Sciences, Siberian Branch
Krasnoyarsk-36, 660036, Russian Federation
E-mail<EMAIL_ADDRESS>
2Institute of Computational Technologies
Russian Academy of Sciences, Siberian Branch
Novosibirsk-90, 630090, Russian Federation

> Abstract. In solving a system of ordinary differential equations by an
> interval method the approximate solution at any considered moment of time
> $t$ represents a set (called interval) containing the exact solution at the
> moment $t$. The intervals determining the solution of a system are often
> expanded in the course of time irrespective of the method and step used. The
> phenomenon of interval expansion, called the Moore sweep effect, essentially
> decreases the efficiency of interval methods. In the present work the
> notions of the interval and the Moore effect are formalized and the
> Infinitesimal Moore Effect (IME) is studied for autonomous systems on
> positively invariant convex compact. With IME the intervals expand along any
> trajectory for any small step, and that means that when solving a system by
> a stepwise interval numerical method with any small step the interval
> expansion takes place for any initial data irrespective of the applied
> method. The local conditions of absence of IME in terms of Jacobi matrices
> field of the system are obtained. The relation between the absence of IME
> and simultaneous dissipativity of the Jacobi matrices is established, and
> some sufficient conditions of simultaneous dissipativity are obtained. (The
> family of linear operators is simultaneously dissipative, if there exists a
> norm relative to which all the operators are dissipative.)

INTRODUCTION

For solving systems of ordinary differential equations different classes of
numerical methods with guaranteed error estimation including interval methods
are used. In solving a system by an interval method the approximate solution
at any considered moment of time $t$ represents a set (called interval)
containing the exact solution at the moment $t$. The detailed account of
interval methods can be found in monographs by R.Moore [1] and S.A.Kalmykov,
Yu.I.Shokin, Z.Kh.Yuldashev [2].

As a rule, all kinds of rectangular parallelepipeds with sides parallel to
coordinate axes [1,2] are used as intervals, less frequently – ellipsoids [3],
balls of fixed norm [4,5] etc.

One of shortcomings of stepwise interval methods is the following. The
intervals determining the solution of a system are often expanded in the
course of time irrespective of the method and step used. The simplest example
of strong expansion of intervals during a short time, belonging to R.Moore, is
given in [1]. The phenomenon of interval expansion, called the Moore sweep
effect (or simply the Moore effect), essentially decreases the efficiency of
interval methods. Earlier the Moore effect was investigated only for some
particular systems and particular intervals [1].

In the present work the notions of the interval and the Moore effect are
formalized and the Moore effect is studied for autonomous systems on
positively invariant convex compact.

Formally, one can get rid of the interval expansion for any globally stable
system (i.e. such a system, any solution of which is stable according to
Lyapunov). To demonstrate that, let consider a smooth autonomous system:

$\frac{dx}{dt}=f(x)$ $None$

on the positively invariant compact $B\subset\mathbb{R}^{n}$. Construct a
metric $\rho$ on the set $B$, assuming for any $x\in B,y\in B$:

$\rho(x,y)=\sup_{t\geq 0}\|x(t)-y(t)\|,$

where $x(t),\ y(t)$ are the solutions of the system (1) with the initial
conditions $x(0)=~{}x,$ $y(0)=~{}y$. This metric is constricting for (1), i.e.
for any pair $x(t)$, $y(t)$ of the solutions of (1) with the initial
conditions in $B$

$\rho(x(t),y(t))\leq\rho(x(s),y(s))\ \mbox{at}\ t\geq s.$

The metric $\rho$ is topologically equivalent to norm if and only if the
system (1) is globally stable in $B$. If one considers as intervals all balls
of the metric $\rho$, then in a definite sense the Moore effect is absent.
That is, there is no interval expansion when constructing the exact interval
solution with any step $h>0$. The exact interval solution of $X(t)$ is defined
in the following way: $X(0)=X_{0}$, where $X_{0}$ is the initial interval with
the centre at the point $x(0)=x_{0}$; $X((n+1)h)$ is the minimal interval with
the centre at the point $x((n+1)h)$ containing $T_{h}X(nh)$, where $T_{t}$ is
the transformation of the phase flow of (1) during the time $t\geq 0$. Indeed,
the radius $X((n+1)h)$ does not exceed the radius $X(nh)$ at any $n$.

If the system is not globally stable, then metric is not topologically
equivalent to the norm. It means that small, in usual sense, intervals became
large in the metric $\rho$. This circumstance makes one refuse from
consideration of similar metrics. Moreover, if the system (1) is absolutely
unstable (for example, a system with mixing), then there is no reasonable way
to get rid of the Moore effect.

The decsribed method of elimination the Moore effect for globally stable
system is non-constructive. This can be demonstrated as follows: for
constructing the constricting metric $\rho$ one must know all exact solutions
of the system (1). But then it is unreasonable to solve the system
numerically. We must have constructively verifiable conditions of absence of
the Moore effect and a way of construction of corresponding intervals. This is
what we deal with in the present paper. The conditions of absence of the Moore
effect are of local character and formulated in terms of Jacobi matrices of
the system. Except that the causes of frequent appearance of the Moore effect
will be pointed out.

##  1 Interval spaces and the Moore effect

###  1.1 Interval Spaces

Before starting to study the Moore effect, it is necessary to define what we
mean by intervals. Generalizing known constructions, give the following
definition.

Definition 1. We call the family $\mathbb{J}$ of convex compacts in
$\mathbb{R}^{n}$ the interval space (and its elements – intervals), if it
satisfies the following conditions:

a) $\mathbb{J}$ is closed with respect to multiplication by non-negative
scalars:

$\mbox{if}\ W\in\mathbb{J},\ \alpha\geq 0,\ \mbox{then}\ \alpha W=\\{\alpha
x\mid x\in W\\}\in\mathbb{J};$

b) $\mathbb{J}$ is closed with respect to intersection:

$\mbox{if}\ W_{1}\in\mathbb{J},\ W_{2}\in\mathbb{J},\ \mbox{then}\ W_{1}\cap
W_{2}\in\mathbb{J};$

c) $\mathbb{J}$ is closed according to Hausdorff (i.e. in the Hausdorff
metric);

d) if $W\in\mathbb{J},\ W\neq\\{0\\}$, then $0\in riW$.

Remind [6] that the Hausdorff metric on the set of all compacts in
$\mathbb{R}^{n}$ is introduced as follows:

$\rho_{H}(x,y)=\max\\{\max_{x\in X}\ \min_{y\in Y}\ \|x-y\|,\ \max_{y\in Y}\
\min_{x\in X}\ \|x-y\|\\},$

where $x,y$ are the compacts in $\mathbb{R}^{n}$, $\|.\|$ is a fixed norm in
$\mathbb{R}^{n}$. All Hausdorff metrics in $\mathbb{R}^{n}$ are equivalent.

Further on by $\lim_{Hi\to\infty}W_{i}$ we denote the Hausdorff limit of the
sequence $\\{W_{i}\\}_{i=1}^{+\infty}$ at $i\to\infty$.

Give several examples of interval spaces.

Example 1. $\mathbb{J}$ is the set of all convex compacts symmetric with
respect to $0$. It satisfies all the properties from a) to d).

Example 2. $\mathbb{J}$ is the set of all symmetric with respect to $0$
rectangular parallelepipeds (including non-singular), i.e. sets of the form

$\\{x=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}:\ |x_{k}|\leq a_{k}\
(k=1,\ldots,n)\\},$

where $a_{k}\geq 0\ (k=1,\ldots,n).$ It satisfies the properties a), b) and
d).

Let now ${\\{W_{i}\\}}_{i=1}^{\infty}\subset\mathbb{J},\
\lim_{Hi\to\infty}W_{i}=W$ with $a_{k}^{(i)}$ being $a_{k}$, corresponding to
$W_{i}$.

If $\rho_{H}(W_{i},W)<\varepsilon,\ \mbox{then}\ W\subset
W_{i}+P_{\varepsilon},\ W_{i}\subset W+P_{\varepsilon},\ \mbox{where}\
P_{\varepsilon}=\\{x\in\mathbb{R}^{n}:\ |x_{k}|\leq\varepsilon\
(k=1,\ldots,n)\\}$ (here a norm in the definition of the Hausdorff metric is
the $l^{\infty}$-norm). Then for any $x\in W$

$|x_{k}|\leq a_{k}^{(i)}+\varepsilon\ \ (k=1,\ldots,n)$

is true and for any $x\in W_{i}$

$|x_{k}|\leq\underline{\lim}_{i\to\infty}a_{k}^{(i)};$
$\lim_{i\to\infty}a_{k}^{(i)}\leq\overline{\lim}_{i\to\infty}a_{k}^{(i)},$

i.e. there exist the limits

$\tilde{a}_{k}=\lim_{i\to\infty}a_{k}^{(i)}\ \ (k=1,\ldots,n).$

If $x\in W$, then

$|x_{k}|\leq\lim_{i\to\infty}a_{k}^{(i)}\ \ (k=1,\ldots,n).$ $None$

Let $b_{i}=\min_{1\leq k}(a_{k}^{(i)}/\tilde{a}_{k})$. If for some
$x\in\mathbb{R}^{n}$ the inequalities (2) are satisfied, then
$\lim_{i\to\infty}b_{i}=1$.

Obviously, $x^{(i)}=b_{i}x\in W_{i}$, i.e. there exists such a subsequence of
$\\{x^{(i)}\\}_{i=1}^{\infty}$ that $x^{(i)}\in W_{i}$,
$x=\lim_{i\to\infty}x_{(i)}$. Hence $W=\\{x\in\mathbb{R}^{n}\ :\
|x_{k}|\leq\tilde{a}_{k}\ (k=1,\ldots,n)\\}$, i.e. $W\in\mathbb{J}$ and the
property c) is also satisfied.

In constructing interval methods of solving different problems it is, as a
rule, the considered interval space that is made use of [1,2].

Example 3. Let $\|.\|$ be a norm in $\mathbb{R}^{n}$, $\mbox{
}_{r}=\\{x\in\mathbb{R}^{n}:\|x\|\leq r\\}$, where $r\geq 0$. Let

$\mathbb{J}=\\{\mbox{ }_{r}|\ r\geq 0\\},$

i.e. J is the set of all closed balls (further on we omit the word ”closed”)
of the norm. All the properties from a) to d) are satisfied. These interval
spaces are used, for example, in [4,5].

Example 4. The construction of example 3 can be generalized as follows. Let
${\|.\|}_{1},\ldots,{\|.\|}_{m}$ be the finite set of norms in
$\mathbb{R}^{n},$

$\mbox{ }_{r_{k}}^{(k)}=\\{x\in\mathbb{R}^{n}:{\ \|x\|}_{k}\leq r_{k}\ \
(k=1,\ldots,m)\\}$

where $r_{k}\geq 0\ \ (k=1,\ldots,m)$ and

$W_{r_{1},\ldots,r_{m}}=\bigcap_{1\leq k\leq m}\mbox{ }_{r_{k}}^{(k)}.$

Let $\mathbb{J}=\\{W_{r_{1},\ldots,r_{m}}:\quad r_{k}\geq 0\ \
(k=1,\ldots,m)\\}.$ Obviously, J possess the properties a), b), and d).

Note that the same element of $\mathbb{J}$ can be associated with different
sets of $\\{r_{k}\\}$. To demonstrate that, let ${m=2},\sup_{x\neq
0}({\|x\|}_{2}/{\|x\|}_{1})=C.$ Then $\mbox{ }_{1}^{(1)}=W_{1,C^{\prime}}$,
where $C^{\prime}$ is any number not less than $C$. Also, even if one of
$r_{k}$ is equal to $0$, then

$W_{r_{1},\ldots,r_{m}}=\\{0\\}.$

To each compact $W\subset\mathbb{R}^{n}$ can be juxtaposed the set

${\\{r_{k}(W)\\}}_{k=1}^{m}:\ \ r_{k}(W)=\max_{x\in W}{\|x\|}_{k}\ \
(k=1,\ldots,m).$

If $W\in\mathbb{J}$ then $W=W_{r_{1}}\cap\ldots\cap W_{r_{m}}.$

Let now the sequence ${\\{W_{i}\\}}_{i=1}^{\infty}$ converge according to
Hausdorff to the compact $W$, with $W_{i}\in\mathbb{J}$ for all $i$. Similarly
to example 2, from the inclusions

$W_{i}\subset W+\mbox{ }_{\varepsilon}^{(k)},\ \ W\subset W_{i}+\mbox{
}_{\varepsilon}^{(k)}$

satisfied for each $\varepsilon>0$ for all $i>i_{0}(\varepsilon)$ derive the
existence of the limits:

$\tilde{r}_{k}=\lim_{i\to\infty}r_{k}(W_{i})\ \ (k=1,\ldots,m)$

and conclude that

$W=W_{\tilde{r}_{1},\ldots,\tilde{r}_{m}},$

i.e. $W\in\mathbb{J}$, and the property c) is satisfied.

Example 5. Let $Q$ be a compact convex body without symmetry centre (for
instance, a triangle in $\mathbb{R}^{2}$), $0\in intQ$. Assume

$\mathbb{J}=\\{\alpha Q:\ \alpha\geq 0\\}.$

J possesses the properties from a) to d).

Remark 1. Example 5 can be generalized. For this purpose it is necessary to
consider compact convex bodies $Q_{1},\ldots,Q_{m},$ the interior of each of
them contains $0$, and to take as $\mathbb{J}$ a family of all the sets of the
form $\bigcap_{1\leq k\leq m}{\alpha}_{k}Q_{k}$ where
${\alpha}_{k}~{}\geq~{}0\ \ (k=1,\ldots,m).$

###  1.2 Dissipative Operators

In this section the properties of the operators dissipative with respect to
compact are studied. First, let remind some notations.

The affine envelope of the convex set $W$ is denoted by $AffW$, the relative
interior $W$ (the interior of $W$ in $AffW$) is denoted by $riW$, the relative
boundary of $W$ (the boundary of $W$ in $AffW$) is denoted by $r\partial W$.
For the boundary of the set $X$ we use the notation $\partial X$, $intX$ – for
the interior of $X$, $coX$ – for the convex envelope of $X$. By the sum of the
sets of $X$ and $Y$ from ${\mathbb{R}}^{n}$ we mean the set $\\{x+y:\ x\in
X,y\in Y\\}$, by I – the unit operator.

Let introduce a new notion.

Definition 2. The linear operator $A$ in the space ${\mathbb{R}}^{n}$ is
called dissipative with respect to the family of sets
$\\{W_{\nu}\\}\subset{\mathbb{R}}^{n}$ if every set $W_{\nu}$ is positively
invariant with respect to the system

$\frac{dx}{dt}=Ax.$ $None$

In other words, every $W_{\nu}$ is invariant with respect to the semi-group of
the operators $\exp(At)\ (t\geq 0).$

Below we consider operators dissipative with respect to families of convex
compacts. In particular, the operator is dissipative with respect to the
families of all balls of some norm (for this, dissipativity with respect to
only one ball is sufficient) if and only if $\|\exp(At)\|\leq 1$ at all $t\geq
0$. Thus, in this case we come to the known definition of dissipativity with
respect to the norm [7].

The set of all operators dissipative with respect to $\\{W_{\nu}\\}$ is
denoted by $K(\\{W_{\nu}\\})$.

Remark 2. If an operator is dissipative with respect to the family of compacts
and the interior of at least one of them is not empty, then it is dissipative
with respect to some norm.

Indeed, any symmetric with respect to $0$ compact convex body is a ball of
some norm (see, for example, [7]). Choose as a ball the following set:

$S=co\\{W\cup(-W)\\}$ $None$

where $W$ is any set of the considered family of $\\{W_{\nu}\\}$, for which
$intW\neq\emptyset$.

However, if $W$ is a compact and the operator is dissipative with respect to
the norm whose ball is $S$ (4), then it does not yet mean that the operator is
dissipative with respect to $W$ (see also example 8).

Remark 3. From the invariance of a family of compacts with respect to the
linear operator follows the invariance of the Hausdorff closure (i.e. closure
in the Hausdorff metric) of this family. Therefore from dissipativity of the
operator with respect to the family of compacts follows the dissipativity with
respect to Hausdorff closure of this family.

Let $W$ be a convex compact in ${\mathbb{R}}^{n}$ with $0\in riW$. In this
case $AffW$ is a linear subspace, and if the operator $A$ is dissipative with
respect to $W$, then $AffW$ is invariant with respect to $A$. Introduce the
following functional on the subspace $L(W)$ of the space $L({\mathbb{R}}^{n})$
(of linear operators in ${\mathbb{R}}^{n}$), consisting of the operators, with
respect to which $AffW$ is invariant:

${\mu}_{W}(A)=\sup_{x\in W}{\mu}_{W}(Ax).$ $None$

Here ${\mu}_{W}$ is the Minkovski functional of the set $W$ (defined, for
example, in [8]) in the subspace $AffW$.

It is easy to see that $A\in K(W)$ if and only if

${\mu}_{W}(\exp(At))\leq 1$

for all $t\geq 0$.

In particular, the operator $A\in L(W)$ is strongly dissipative with respect
to convex compact $W$ if exists such $\varepsilon>0$ that
${\mu}_{W}(\exp(At))\leq\exp(-\varepsilon t)$ at all $t\geq 0$. In general,
the operator $A$ is strongly dissipative with respect to convex compact $W$ if
and only if $A+\varepsilon I\in K(W)$ for some $\varepsilon>0$.

If $W$ is a ball of the norm $\|.\|$, then strong dissipativity with respect
to $W$ means the existence of such $\varepsilon>0$ that
$\|\exp(At)\|\leq\exp(-\varepsilon t)$ for all $t\geq 0$. We come to the
definition of stable dissipativity with respect to the norm [11, 12, 19].

Introduce in $L({\mathbb{R}}^{n})$ the following functional:

${\gamma}_{W}(A)=\lim_{h\to+0}\frac{{\mu}_{W}(I+hA)-1}{h}$

In the case, when $W$ is a ball of some norm (i.e. ${\mu}_{W}$ is a norm),
arrive at the known definition of the logarithmic Lozinsky norm [9, 10].

Lemma 1. The operator $A\in L({\mathbb{R}}^{n})$ is dissipative (strongly
dissipative) with respect to $W$, if and only if the inequality
${\gamma}_{W}(A)\leq 0\ \ ({\gamma}_{W}(A)<0)$ is satisfied.

Proof. Sufficiency. The following inequality is obtained in [9]

$\|\exp(At)\|\leq\exp(\gamma(A)t)$

where $\gamma(A)$ is the Lozinsky norm of the operator $A$, corresponding to
the norm $\|.\|$. By literal repetition of the reasonings from [9] (with a
substitution of the norm by Minkovski functional), one can obtain the
inequaliny

${\mu}_{W}(\exp(At))\leq\exp({\gamma}_{W}(A)t)$

for all $t\geq 0$, from which immediatelly follows the sufficiency.

Necessity. Evidently,

${\mu}_{W}(\exp(At))={\mu}_{W}(I+At)+o(t)\ (t\to 0).$

Therefore,

${\gamma}_{W}(A)=\lim_{h\to+0}\frac{{\mu}_{W}(e^{Ah})-1}{h}.$

Let $\varepsilon\geq 0$. If ${\mu}_{W}(\exp(At))\leq\exp(-\varepsilon t)$ at
all $t\geq 0$, then

${\gamma}_{W}(A)\leq\lim_{h\to+0}\frac{\exp(-\varepsilon
h)-1}{h}=-\varepsilon,$

which proves the necessity. The lemma is proved.

Assign a relatively open convex cone $Q_{x}(W)$ to every point $x\in r\partial
W$ according to the rule: $y\in Q_{x}(W)$ if and only if there exists such
$\varepsilon>0$ that

$x+\varepsilon y\in riW.$

Lemma 2. For strong dissipativity of $A$ with respect to convex compact $W$ it
is necessary and sufficient that for every point $x\in r\partial W$ the
inclusion

$Ax\in Q_{x}(W)$

be true. For dissipativity of $A$ with respect to $W$ it is necessary and
sufficient that for every point of $X\in r\partial W$ the inclusion

$Ax\in\overline{Q_{x}(W)}$

be true.

Proof. Note that the operator $A$ is strongly dissipative with respect to $W$
if and only if there exists such $t_{0}>0$ that ${\mu}_{W}(I+At_{0})<1$.
Indeed, the existence of such $t_{0}$ for a strongly dissipative operator
follows immediately from the negativeness of ${\gamma}_{W}(A)$. Conversely, if
${\mu}_{W}(I+At_{0})<1$, then there exists such ${\varepsilon}>0$ that
${\mu}_{W}(I+(A+\varepsilon I)t_{0})<1$. But then ${\gamma}_{W}(A+\varepsilon
I)\leq 0$, the operator $(A+\varepsilon I)$ is dissipative. It means that $A$
is strongly dissipative.

If the operator $A$ is strongly dissipative with respect to $W$, then,
according to the above, for each $x\in r\partial W$ there exists such
$t_{x}>0$ that $(I+t_{x}A)x\in riW$. It means that the vector $Ax$ belongs to
to the cone $Q_{x}(W)$.

Conversely, let the latter condition be satisfied. According to the hypothesis
of the theorem and convexity of $W$, for each $x\in r\partial W$ there exists
the only positive number $s=s(x)$ such that $(I+sA)x\in r\partial W$. Show
that $s_{0}=\inf_{x\in r\partial W}s(x)>0$. Let it be not so. Then there
exists such a subsequence ${\\{x_{n}\\}}_{n=1}^{+\infty}$ that
$\lim_{n\to\infty}s(x_{n})=0$. Choose from $\\{x_{n}\\}$ a converging
subsequence $\\{{x_{n}}^{\prime}\\}$. Let
$\tilde{x}=\lim_{n\to\infty}{x_{n}}^{\prime}$. For every $n\in N$ and for
every $\varepsilon>0$

$[I+(s({x_{n}}^{\prime})+\varepsilon)A]{x_{n}}^{\prime}\notin W.$

Passing to the limit, obtain

$(I+\varepsilon A)\tilde{x}\notin riW$

which contradicts the hypothesis of the theorem.

Thus, $s_{0}>0$. For any $t_{0}\in(0;s_{0})$ is true ${\mu}_{W}(I+At_{0})<1$,
i.e. the operator $A$ is strongly dissipative.

If $A\in K(W)$, then for any $\varepsilon>0$ we have $AX-\varepsilon x\in
Q_{x}(W)$ (for any $x\in r\partial W$), i.e. $Ax\in\bar{Q}_{x}$. Conversely,
if $Ax\in\bar{Q}_{x}$, then $Ax-\varepsilon x\in Qx$ at any $\varepsilon>0$,
and $A$ represents a limit point of the family of dissipative operators, i.e.
$A\in K(W)$. The lemma is proved.

Remark 4. Immediately from the Krein-Milman theorem [8] follows that it is
sufficient to require from the lemma conditions that inclusions be satisfied
not for all points $x\in r\partial W$, but for extremal points of $W$ only. In
particular, if $W$ is a polyhedron, then it is sufficient to test its vertices
only. Thus, to elucidate the question about dissipativity (strong
dissipativity) of the operator with respect to the polyhedron, one should test
only the fulfilment of finite number of linear inequalities.

Remark 5. In the proof lemma 2 we have used the obvious fact: the closure of
the set $K(W)$.

One more fact follows directly from lemma 2.

Lemma 3. The set $K(W)$ is a closed convex cone. The cone of all strongly
dissipative with respect to $W$ operators coincides with $riK(W)$ and with
$\bigcup_{\varepsilon>0}(K(W)-\varepsilon I)$. If $\\{W_{\nu}\\}$ is a family
of convex compacts with $0\in riW_{\nu}$ for all $\nu$, then
$K(\\{W_{\nu}\\})$ is a closed convex cone.

Remark 6. If $intW=\emptyset$, then $intK(W)=\emptyset$. Indeed, if $AffW$ is
invariant with respect to the operator $A_{1}$, then $A+\varepsilon
A_{1}\notin K(W)$ at $\varepsilon\neq 0$. If $intW\neq\emptyset$, then
$intK(W)$ is also non-empty and coincides with $riK(W)$.

Definition 4. The operator $A\in K(W)$ is called stable (or roughly)
dissipative with respect to $W$, if $A\in intK(W)$.

Definition 4 generalize the definition of the stable dissipativity with
respect to the norm [11, 19].

Pass to the consideration of operators dissipative with respect to interval
spaces. Let find out for which interval spaces $\mathbb{J}$ the interior of
the cone $K(\mathbb{J})$ is not empty.

Let $V$ be a set of all compact convex bodies in $\mathbb{R}_{n}$. Fix some
norm $\|.\|$ in $\mathbb{R}^{n}$ and assume

$d(W)=\min_{x\in\partial W}{\|x\|}.$

Lemma 4. The function $d(W)$ is continuous according to Hausdorff on the set
$V$.

Proof. First note that if $X\in V,\ Y\in V$, then ${\rho}_{H}(\partial
X,\partial Y)\leq{\rho}_{H}(X,Y)$. Indeed, let
${\rho}_{H}(X,Y)\leq\varepsilon$. Then $X\subset Y+S_{\varepsilon}$ where
$S_{\varepsilon}=\\{x\in\mathbb{R}^{n}:\ \|x\|\leq\varepsilon\\}$. Let,
further on, there exists such $y_{0}\in(\partial Y)\cap X$ that $y_{0}\notin
S_{\varepsilon}+\partial X$. Construct at the point $y_{0}$ a tangent
hyperplane $L$ to $Y$. Let $l$ be the direction of the external normal to
$\partial Y$ at the point $y_{0}$ orthogonal to $L$. Draw a ray from the point
$y_{0}$ in the direction of $l$ to the point $x_{0}$ of crossing with
$\partial X$. Construct such a ball $S$ of the norm $\|.\|$ with the centre at
the point $x$ that $y_{0}\in\partial S$. The radius of $S$ is larger than
$\varepsilon$ and $S\cap Y=\\{y_{0}\\}$. Thus, if one constructs a ball
$S^{\prime}\subset S$ of the radius $\varepsilon$ with the centre at $x_{0}$,
then

$S^{\prime}\cap Y=0.$

But then $x_{0}\notin Y+S_{\varepsilon}$, i.e. $X\not\subset
Y+S_{\varepsilon}$ what is contrary to the assumption.

The existence of such $y_{0}\subset\partial Y$ that $y_{0}\notin
X\cup(\partial X+S_{\varepsilon})$ is also impossible, since then $y_{0}\notin
X+S_{\varepsilon}$, i.e. $Y\not\subset X+S_{\varepsilon}$. Consequently,
$\partial Y\subset S_{\varepsilon}+\partial X$, and that means $d(X)\leq
d(Y)+\varepsilon$. Similarly, $d(Y)\leq d(X)+\varepsilon$. It means that
$|d(X)-d(Y)|\leq\varepsilon$, and the function $d(W)$ is continuous on $V$.
The lemma is proved.

Lemma 5. For non-emptiness of $intK(\mathbb{J})$ it is necessary and
sufficient for all the elements of the interval space $\mathbb{J}$, exept
$\\{0\\}$, to posess non-empty interior.

Proof. Necessity. Follows immediately from remark 6.

Sufficiency. Show that under the conditions of the theorem the inclusion

$-I\in intK(\mathbb{J})$ $None$

takes place.

To each point $x\ (\|x\|=1)$ we assign the set $W_{x}$ according to the rule:

$W_{x}=\bigcap_{W\ni x,W\in\mathbb{J}}W.$

According to the conditions b) and c) from definition 1, $W_{x}\in\mathbb{J}$.
The set

$\tilde{W}=\overline{\bigcup_{\|x\|=1}W_{x}}$

is compact. Indeed, $\tilde{W}$ is contained in any element of $\mathbb{J}$
containing unit ball of the norm $\|.\|$; such an element exists due to non-
emptiness of the interior of all intervals (exept $\\{0\\}$) and the property
a) from definition 1. Note that Hausdorff closure of the family $\\{W_{x}:\
\|x\|=1\\}$ represents a compact in the Hausdorff metric, contained in
$\mathbb{J}$ (this follows from compactness according to Hausdorff of the
family of all compact subsets of the compact [6]). From the property a)
(definition 1) follows (by virtue of lemma 4) the existence of such
$\varepsilon>0$ that $d(W_{x})\geq\varepsilon$ for all such $x$ that $\|x\|=1$
(indeed, $d(W_{x})>0$, since $0\in intW_{x}$).

Thus, there exists such $\varepsilon>0$ that for all $x\ (\|x\|=1)$ the
inclusion

$Ax\in intW_{x}$

is true if $\|A\|<\varepsilon$.

In other words, $Ax-x\in Q_{x}(W_{x})$ if $\|A\|<\varepsilon,\ \|x\|=1$ (see
lemma 1). The more so, as $Ax-x\in Q_{x}(W)$ for all $W\in\mathbb{J}\ (W\ni
x,\|A\|<\varepsilon)$ at all such $x$ that $\|x\|=1$. But then $Ax-x\in
Q_{\alpha x}(\alpha W)$ for all $\alpha>0,\|A\|<\varepsilon$. Hence, $A-I\in
K(\mathbb{J})$, i.e. (6) is satisfied. The lemma is proved.

Thus, we have shown that under the conditions of lemma 5 $K(\mathbb{J})$ is a
convex solid cone.

Definition 5. The operator is stable dissipative with respect to the interval
space $\mathbb{J}$ if it belongs to $intK(\mathbb{J})$.

For stable dissipative operators the remark 2 is true: if an operator is
stable dissipative with respect to the family of compacts and the interior of
at least one of them is not empty, then it is stable dissipative with respect
to some norm.

###  1.3 The Moore Effect for Autonomous Systems

The results of the previous section can be applied to the study of the Moore
sweep effect. First give the exact definition of what we understand by the
Moore effect.

Let in the vicinity of a compact convex body $B\subset\mathbb{R}^{n}$ be given
a smooth autonomous system

$\frac{dx}{dt}=f(x)$ $None$

with $B$ positively invariant with respect to (7), and let $x(0)$ be
determined inexactly, namely

$x(0)\in x_{0}+W_{0},$

where $x_{0}\in B,\ W_{0}\in\mathbb{J},\ x_{0}+W_{0}\in B,\ \mathbb{J}$ is
some interval space (see definition 1).

Remark 7. Irrespective of particular numerical method (i.e. dealing with the
exact solution of the initial value problem for (7) with the initial
conditions $x(0)=x_{0}$) a stepwise interval solution with step $h>0$ can be
described as follows.

Let $T_{h}$ be the transformation of the phase flow of (7) during the time $t$
(shift over time $t$), $W_{0}\in\mathbb{J}$ is the initial interval (its sense
is an uncertainty in initial data). Assume

$X_{0}=x_{0}+W_{0},$ $X_{m+1}=T_{(m+1)h}x_{0}+W_{m+1},$
$W_{m+1}=\bigcap_{W\supset W_{m+1}(h),W\in\mathbb{J}}W,$
$W_{m+1}(h)=T_{h}(T_{mh}x_{0}+W_{m})-T_{(m+1)h}x_{0}.$

The sequence $\\{X_{m}\\}_{m=0}^{+\infty}$ is the exact stepwise interval
solution of (7).

Definition 6. The absence of infinitesimal Moore effect (IME) means that for
any $h>0$ the sequence ${\\{W_{m}\\}}_{m=0}^{+\infty}$ is enclosed:
$W_{m}\supset W_{m+1}$ for all $m$, i.e. the obtained intervals do not expand.

With IME the intervals expand along any trajectory (7) for any small step, and
that means that when solving a system by a stepwise interval numerical method
with any small step the interval expansion takes place for any initial data
irrespective of the applied method (since it is true even for exact
solutions).

Generalizing the construction [10] for norms, introduce the following
functional:

$N_{W}(x,y)=\lim_{h\to+0}\frac{{\mu}_{W}(x+hy)-{\mu}_{W}(x)}{h}.$

Literally (with substitution of the norm for Minkovski functional) repeating
the reasonings from [10] (pp.127, 426), come to the following statemets.

Statement 1. If $x(t)$ with values in $\mathbb{R}^{n}$ is differentiable on
connected subset $T$ of the real axis, and $W$ is a convex compact $(0\in
riW)$, then the function ${\mu}_{W}(x(t))$ is almost everywhere differentiable
on $T$ and the derivative (where it exists) coincides with the right-hand
derivative, equal to $N_{W}(x(t),\dot{s}(t))$. The right-hand derivative of
${\mu}_{W}(x(t))$ exists everywhere on $T$ except the right-hand end.

Statement 2.

${\gamma}_{W}(A)=\sup_{x\in W}N_{W}(x,AX).$

By $f^{\prime}(x)$ further on we denote the mapping derivative of $f$.

The main part of further results on IME can be obtained from the following
theorem.

Theorem 1. Let in the region $U\subset R^{n}$ be given a smooth autonomous
system (7), $B\subset U$ be positively invariant with respect to (7) compact
convex body. IME is absent for compact $B$, system (7) and interval space
$\mathbb{J}$ if and only if

$f^{\prime}(x)\in K(\mathbb{J})$ $None$

for all $x\in B$, i.e. for any $x\in B$ the Jacoby matrix of system (7) in the
point $x$ is strongly dissipative with respect to $\mathbb{J}$.

Proof. Sufficiency. Let $W\in\mathbb{J}$. Consider two solutions
$x_{1}(t),x_{2}(t)$ of system (7) with initial conditions from B. Denote
$\Delta(t)=x_{1}(t)-x_{2}(t)$. Using statements 1 and 2 and the theorem on
finite increment, estimate the derivative of ${\mu}_{W}(\Delta(t))$:

$\frac{d}{dt}{\mu}_{W}(\Delta(t))=N_{W}(\Delta(t),d\Delta(t)/dt)\leq$
$\leq\sup_{0\leq\Theta\leq
1}N_{W}(\Delta(t),f^{\prime}(x_{c}(t))\Delta(t))\leq$
$\leq\sup_{0\leq\Theta\leq
1}{\gamma}_{W}(f^{\prime}(x_{c}(t))){\mu}_{W}(\Delta(t)),$

where $x_{c}(t)=x_{1}(t)+\Theta(x_{2}(t)-x_{1}(t)),\ 0\leq\Theta\leq 1$ for
all $t\geq 0$. By (8) and statement 1 we obtain

$\frac{d}{dt}{\mu}_{W}(\Delta(t))\leq 0.$

Since the latter inequality holds for all $t\geq 0$ and for all
$W\in\mathbb{J}$, in systen (7) on $B$ IME with respect to $\mathbb{J}$ is
absent.

Necessity. Let $W\in\mathbb{J},\ x_{0}\in intB,\ t_{0}\geq 0,\ y\in AffB,\
y\neq 0$. There exists such $h_{0}>0$ that $x_{0}+h_{0}y\in B$. Due to
smoothness of system (7) there exist and are unique the solutions
$x_{1}(t),x_{2}(t)$ of the initial value problem for (7) with the initial
conditions $x_{1}(t_{0})=x_{0},\ x_{2}(t_{0})=x_{0}+h_{0}y$. Assume
$\Delta(t)=x_{1}(t)-x_{2}(t)$. Then

$\frac{d}{dt}\ln{{\mu}_{W}}(\Delta(t)){|}_{t=t_{0}}=$
$=N_{W}\biggl{(}\frac{\Delta(t_{0})}{{\mu}_{W}(\Delta(t_{0}))},f^{\prime}(x_{c})\frac{\Delta(t_{0})}{{\mu}_{W}(\Delta(t_{0}))}\biggr{)}=$
$=N_{W}\biggl{(}\frac{y}{{\mu}_{W}(y)},f^{\prime}(x_{c})\frac{y}{{\mu}_{W}(y)}\biggr{)},$

where $x_{c}=x_{0}+\Theta h_{0}y,\ \ 0<\Theta<1$.

By virtue of absence of IME

$\frac{d}{dt}\ln{\mu}_{W}(\Delta(t))\leq 0$

for all $t\geq 0$. Since if $x_{0}+h_{0}y\in b$, then:
(a) $x_{0}+hy\in B$ for all $h\in[0;h_{0}]$,
(b) a set of those $h\in[0;h_{0}]$, for which

$N_{W}\biggl{(}\frac{y}{{\mu}_{W}(y)},f^{\prime}(x)\frac{y}{{\mu}_{W}(y)}\biggr{)}\leq
0,$

is dense on the segment $[0;h_{0}],$
and (c) due to its closureness coincides with this segment.

By virtue of arbitrarity of the choice of $x_{0}$ for any $x_{0}\in intB,$
$t\geq 0,$ $y\in AffB$, $y\neq 0$ the inequality

$N_{W}\biggl{(}\frac{y}{{\mu}_{W}(y)},f^{\prime}(x)\frac{y}{{\mu}_{W}(y)}\biggr{)}\leq
0$

is satisfied. It holds also for any $x\in B,t\geq 0,y\in AffB,y\neq 0$. Hence,
from lemma 1 and statement 2 immediately follows dissipativity of
$f^{\prime}(x)$ with respect to $\mathbb{J}$ for all $x\in B$. The theorem is
proved.

Definition 7. The family of linear operators $\\{A_{\alpha}\\}$ is called
simultaneously dissipative, if there exists a norm relative to which all the
operators are dissipative.

Simultaneously dissipative operators were studied in detail in [11, 12,
17-22].

From theorem 1, example 3, and remark 2 we obtain the following theorem.

Theorem 2. For existence of interval space in which at least one interval
posesses non-empty interior and with respect to which in system (7) there is
no IME on $B$, it is necessary and sufficient for the family
$\\{f^{\prime}(x):x\in B\\}$ to be simultaneously dissipative.

Thus, the problem of existence of the interval space, with respect to which
IME is absent, is reduced to the problem of simultaneous dissipativity of
Jacobi matrices. As sought for space one can choose a set of all balls of that
norm relative to which all Jacobi matrices are dissipative. This norm is
constricting for (7) on $B$ (i.e. the distance between two solutions with
initial conditions from B will not expand with time). Hence, all systems
without IME (with respect to some interval space) on $B$ are globally stable
in $B$ (see introduction).

Bellow by $C^{1}(B)$ we denote the Banach space of smooth mappings of $B$ in
$\mathbb{R}^{n}$ with the norm

${\|f\|}_{C^{1}(b)}=\max_{x\in B}\|f(x)\|+\sum_{k=1}^{n}\max_{x\in
B}\biggl{\|}\frac{\partial f}{\partial x_{k}}\biggr{\|}$

where $\|.\|$ is a fixed norm in $\mathbb{R}^{n}$.

Further on, speaking about properties of autonomous systems, we mean the
properties of the vector fields generating them.

Immediately from lemma 3 and theorem 1 the following statement can be
obtained.

Theorem 3. The set of systems on $B$ without IME with respect to $\mathbb{J}$
is closed convex cone in $C^{1}(B)$.

For this cone we use the notation $F_{B}(\mathbb{J})$.

Further on, speaking about the vicinity of an autonomous system in $C^{1}(B)$
we mean a part of the vicinity, consisting only of those systems for which the
set $B$ is positively invariant.

Let us study under what conditions the interior of the cone
$F_{B}(\mathbb{J})$ is non-empty.

Theorem 4. For non-emptiness of $intF_{B}(\mathbb{J})$ in $C^{1}(B)$ it is
necessary and sufficient for all elements of $\mathbb{J}$, exept $\\{0\\}$, to
possess non-empty interior.

Proof. Necessity. Let exist such a set $W\in\mathbb{J}$ that $intW=0$.
Consider any system (7) without IME with respect to $\mathbb{J}$ on $B$. Since
$intB\neq 0$, there exist two different concentrical balls $S_{1}$ and $S_{2}$
of usual $l^{2}$-norm, belonging to $intB$ with $S_{1}\subset S_{2}$.
Construct such a function $g\in C_{\infty}(\mathbb{R}^{n})$ that $g(x)=1$ for
all $x\in S_{1}$ and $g(x)=0$ at all $x\notin S_{2}$. Since
$AffW\neq\mathbb{R}^{n}$, one can construct a linear operator $A\in
L(\mathbb{R}^{n})$ mapping $AffW$ into such a subspace $E_{0}\neq\\{0\\}$ that
$(AffW)\cap E_{0}=\\{0\\}$.

Consider the system

$\frac{dx}{dy}=f(x)+\varepsilon g(x)AX,$ $None$

where $\varepsilon>0$ is arbitrary. The set $B$ is positively invariant with
respect to (9), since the vector field generating (9) coincides with $f$ in
the vicinity of $\partial B$. On the other hand, there exist Jacobi matrices
(9) relative to which $AffW$ is not invariant, i.e. in (9) exist IME with
respect to $\mathbb{J}$ on $B$. Since in any vicinity of $f$ there is at least
one vector field, generating (9), then

$intF_{B}(\mathbb{J})=0.$

Sufficiency. Consider the system $dx/dt=-x$. It is a system on $B$ without IME
with respect to $\mathbb{J}$. Furthermore, if all elements of $\mathbb{J}$,
except $\\{0\\}$, possess non-empty interior, then by lemma 5 the matrix of
the system is stable dissipative with respect to $\mathbb{J}$ (see definition
5).

Consider the system:

$\frac{dx}{dy}=-x+v(x),$ $None$

where ${\|v\|}_{C_{1}(B)}<\varepsilon$ with $\varepsilon$ chosen so that

$A-I\in K(\mathbb{J})$

if $\|A\|<\varepsilon$ (see the proof of lemma 5). Then all Jacobi matrices
(10) are dissipative with respect to $\mathbb{J}$, and if $v$ is choosen so
that $B$ is positively invariant with respect to (10), then in (10) IME is
absent (by theorem 1). The theorem is proved.

Thus, $intF_{B}(\mathbb{J})\neq 0$ if and only if $intK(\mathbb{J})\neq 0$.

It is easy to see that in the proof of sufficiency in theorem 4 one can
instead of the system $dx/dy=-x$ consider any system whose Jacobi matrices are
stable dissipative with respect to $\mathbb{J}$.

Remark 8. By analogy with the space $C^{1}(B)$ one can construct the Banach
spaces $C^{k}(B)\ (k\in N)$ with the norm

${\|f\|}_{C^{k}(B)}=\sum_{|\alpha|=0}^{k}\max_{x\in B}{\|(D^{\alpha}f)(x)\|},$

where $\alpha=(\alpha_{1},\ldots,\alpha_{n})$ is a multiindex:

$|\alpha|=\alpha_{1}+\ldots+\alpha_{n},D^{\alpha}f=\frac{{\partial}^{|\alpha|}f}{\partial
x_{1}^{\alpha_{1}}\ldots\partial x_{n}^{\alpha_{n}}}$

and the metric space $C^{\infty}(B)$ with the system of seminorms

$\\{\max_{x\in B}\|(D_{\alpha}f)(x)\|:\ |\alpha|\leq m\\}_{m=0}^{+\infty}.$

Small $C^{k}$-additions $(1\leq k\leq+\infty)$ are small and in the
$C^{1}$-norm. Therefore, for $C^{k}$-smooth systems under the conditions of
theorem 4 the interior of $F_{B}(\mathbb{J})$ is non-empty and in $C^{k}(B)$.
As shows the proof of theorem 4 (necessity), if conditions of the theorem are
not satisfied,then the interior of $F_{B}(\mathbb{J})$ in $C^{k}(B)$ is empty.

Let clarify what autonomous system without IME in specific interval spaces
looks like.

Theorem 5. Any system without IME with respect to $\mathbb{J}$ from example 1
has the form:

$\frac{dx}{dt}=ax+c,$

where $a\leq 0,\ C\in\mathbb{R}^{n}$ is a constant vector.

Proof. Let $A\in K(\mathbb{J})$. All the segments symmetrical with respect to
$0$ belong to $\mathbb{J}.$ Every such a segment has the form
$\\{y\in\mathbb{R}^{n}|y=ax,|a|\leq 1\\}$ for some $x\in\mathbb{R}^{n}.$ The
cone $Q_{x}$ (see lemma 2) for each segment consists of vectors of the form
$ax$, where $a<0$. Thus, every non-zero vector $x\in\mathbb{R}^{n}$ is
eigenvector of the operator $A$, corresponding to non-positive eigenvalue.
Thus:

$K(\mathbb{J})=\\{aI|\ a\leq 0\\}.$ $None$

Let now a system without IME have the form

$\left\\{\begin{array}[]{rcl}\displaystyle{dx_{1}\over
dt}&=&f_{1}(x_{1},\ldots,x_{n});\\\ &\cdots&\\\ \displaystyle{dx_{n}\over
dt}&=&f_{n}(x_{1},\ldots,x_{n}).\end{array}\right.$

According to (11) and theorem 1

$\frac{\partial f_{i}}{\partial x_{j}}\equiv 0\ (i\neq j);$ $None$
$\frac{\partial f_{1}}{\partial x_{1}}\equiv\frac{\partial f_{2}}{\partial
x_{2}}\equiv\ldots\equiv\frac{\partial f_{n}}{\partial x_{n}}\leq 0.$ $None$

From (12) follows that $f_{k}$ depends only on $x_{k}\ (k=1,\ldots,n)$. It
means that ${\partial f_{k}}/{\partial x_{k}}$ also depends only on $x_{k}$,
i.e. by virtue of (13) ${\partial f_{k}}/{\partial x_{k}}=const\
(k=1,\ldots,n)$. Then

$\frac{\partial f_{1}}{\partial x_{1}}\equiv\frac{\partial f_{2}}{\partial
x_{2}}\equiv\ldots\equiv\frac{\partial f_{n}}{\partial x_{n}}\equiv a\leq 0$

and the system has the form:

$\frac{dx}{dt}=ax+c,$

where $a\leq 0,\ c=const$. The theorem is proved.

Thus, whatever nonlinear (or even linear with non-scalar matrix) system we
consider, if we take as $\mathbb{J}$ the interval space of example 1 (or any
wider space), IME will be present in the system. From theorem 5 also follows
that any dissipative with respect to all norms operator has the form $aI$,
where $a\leq 0$ (see also remark 3).

Example 6. Consider $\mathbb{J}$ from example 2. $\mathbb{J}$ contains all
symmetrical with respect to $0$ segments of coordinate axes (thus, the
conditions of theorem 4 are not satisfied, i.e. $intF_{B}(\mathbb{J})=0$). Let
$A\in K(\mathbb{J})$. Reasoning like in proof of theorem 5, conclude that all
coordinate axes are eigenspaces of the operator $A$, corresponding to non-
positive eigenvalues. In other words, the matrix of the operator $A$ is
diagonal and non-positive. On the other hand, by virtue of lemma 2, all such
operators belong to $K(\mathbb{J})$. Thus, systems without IME with respect to
$\mathbb{J}$ on $B$ have the form

$\left\\{\begin{array}[]{ccc}\displaystyle{dx_{1}\over
dt}&=&f_{1}(x_{1});\\\\[11.38109pt] &\cdots&\\\ \displaystyle{dx_{n}\over
dt}&=&f_{n}(x_{n})\end{array}\right.$

where

$\frac{\partial f_{k}}{\partial x_{k}}\leq 0\ (k=1,\ldots,n)$

for all $x\in B$.

From the considered example follows that when using standard intervals
(rectangular parallelepipeds) IME will be observed in almost all systems in
$\mathbb{R}^{n}\ \mbox{if}\ n\neq 1.$

The systems without IME with respect to $\mathbb{J}$ from example 3 on $B$
represent all systems for which the norm $\|.\|$ is constricting in $B$ (see
the text after theorem 2).

Remark 9. Note that testing of dissipativity (stable dissipativity) of the
operator with respect to the norm is equivalent to non-positiveness
(negativeness) of the corresponding Lozinsky norm. For some norms an explicit
form of corresponding Lozinsky norm is known (see, for example, [9] or [10,
p.463-465]). In particular, for the Euclidean norm the Lozinsky norm of the
operator $A$ coincides with the largest eigenvalue of the operator
$(A^{*}+A)/2$. The Lozinsky norm of the operator $A$ represented by the matrix
$(a_{ij})_{i,j=1}^{n}$ with respect to $l^{1}-$ and $l^{\infty}$-norms is
given by the formulae, respectively:

$\max_{1\leq i\leq n}(\mbox{Re}\ a_{ii}+\sum_{j\neq i}|a_{ji}|);$ $\max_{1\leq
i\leq n}(\mbox{Re}\ a_{ii}+\sum_{j\neq i}|a_{ij}|).$

In remark 9 it is assumed that the operator $A$ acts in the space $C^{n}$. The
definitions and used here properties of dissipative operators in complex
spaces are analogous to those in real ones.

Example 7. Let $\mathbb{J}$ be the interval space from example 4. Then

$K(\mathbb{J})=\bigcap_{1\leq k\leq m}K_{{\|.\|}_{k}},$

where $K_{{\|.\|}_{k}}$ is the cone of all operators dissipative with respect
to the norm $K_{{\|.\|}_{k}}$. It follows from the closure under intersection
of the family of all positively invariant sets of an autonomous system.
Similarly,

$intK(\mathbb{J})=\bigcap_{1\leq k\leq m}intK_{{\|.\|}_{k}}.$

Let, for example, $\|.\|_{1}$ be $l^{\infty}$-norm, $\|.\|_{2}$ be
$l^{2}$-norm in $\mathbb{R}^{2}$. Then the conditions of stable dissipativity
of the operator $A$ with the matrix $(a_{ij})_{i,j=1}^{2}$ with respect to
$\mathbb{J}$ according to remark 9 are of the form:

$\left\\{\begin{array}[]{lcl}4a_{11}a_{22}&>&(a_{12}+a_{21})^{2};\\\\[11.38109pt]
a_{11}+|a_{12}|&<&0;\\\\[11.38109pt] |a_{21}|+a_{22}&<&0.\end{array}\right.$

Thus, for the system of the form

$\left\\{\begin{array}[]{rcl}\displaystyle{dx_{1}\over
dt}&=&f_{1}(x_{1},x_{2});\\\\[11.38109pt] \displaystyle{dx_{2}\over
dt}&=&f_{2}(x_{1},x_{2})\end{array}\right.$ $None$

if the inequalities

$\left\\{\begin{array}[]{lcl}4\displaystyle{{\partial f_{1}}\over{\partial
x_{1}}}\cdot\displaystyle{{\partial f_{2}}\over{\partial
x_{2}}}&>&\left(\displaystyle{{\partial f_{1}}\over{\partial
x_{2}}}+\displaystyle{{\partial f_{2}}\over{\partial
x_{1}}}\right)^{2};\\\\[11.38109pt] \displaystyle{{\partial
f_{1}}\over{\partial x_{1}}}+\biggl{|}\displaystyle{{\partial
f_{1}}\over{\partial x_{2}}}\biggr{|}&<&0;\\\\[11.38109pt]
\biggl{|}\displaystyle{{\partial f_{2}}\over{\partial
x_{1}}}\biggr{|}+\displaystyle{{\partial f_{2}}\over{\partial
x_{2}}}&<&0\end{array}\right.$

are satisfied and the compact convex body $B$ is positively invariant with
respect to (14), then in (14) IME with respect to $\mathbb{J}$ (from example
4) is absent on $B$. For example, such is the following system:

$\left\\{\begin{array}[]{rcl}\displaystyle{dx_{1}\over
dt}&=&-2x_{1}+x_{2};\\\\[11.38109pt] \displaystyle{dx_{2}\over
dt}&=&2x_{1}-3x_{2}\end{array}\right.$

if $B$ is the square $\\{(x_{1},x_{2}):\ |x_{1}|\leq 1,|x_{2}|\leq 1\\}$ or
the circle $\\{(x_{1},x_{2}):\ x_{1}^{2}=x_{2}^{2}\leq 1\\}$.

Example 8. Consider $\mathbb{J}$ from example 5. Let $Q$ be rectangular
triangle with vertices at the points $(-1;2);(-1;-1);(1;-1)$.

From lemma 2 and theorem 1 follows that the cone $F_{B}(\mathbb{J})$ consists
of the systems of the form (14), with respect to which the compact $B$ is
positively invariant and for which

$\left\\{\begin{array}[]{lcl}\displaystyle{{\partial f_{1}}\over{\partial
x_{1}}}+\displaystyle{{\partial f_{1}}\over{\partial
x_{2}}}&\leq&0;\\\\[11.38109pt] \displaystyle{{\partial f_{1}}\over{\partial
x_{1}}}-2\displaystyle{{\partial f_{1}}\over{\partial
x_{2}}}&\leq&0;\\\\[11.38109pt] \displaystyle{{\partial f_{2}}\over{\partial
x_{1}}}+\biggl{|}\displaystyle{{\partial f_{2}}\over{\partial
x_{2}}}\biggr{|}&\leq&0;\\\\[11.38109pt] -3\displaystyle{{\partial
f_{1}}\over{\partial x_{1}}}+6\displaystyle{{\partial f_{1}}\over{\partial
x_{2}}}-2\displaystyle{{\partial f_{2}}\over{\partial
x_{1}}}+4\displaystyle{{\partial f_{2}}\over{\partial
x_{2}}}&\leq&0;\\\\[11.38109pt] 3\displaystyle{{\partial f_{1}}\over{\partial
x_{1}}}-3\displaystyle{{\partial f_{1}}\over{\partial
x_{2}}}+2\displaystyle{{\partial f_{2}}\over{\partial
x_{1}}}-2\displaystyle{{\partial f_{2}}\over{\partial
x_{2}}}&\leq&0.\end{array}\right.$ $None$

is true.

Substituting all the inequality signs in (15) by strict ones, obtain
$intF_{B}(\mathbb{J})$. For example, the system

$\left\\{\begin{array}[]{rcl}\displaystyle{dx_{1}\over
dt}&=&-x_{1};\\\\[11.38109pt] \displaystyle{dx_{2}\over
dt}&=&-6x_{1}-4x_{2}\end{array}\right.$

belongs to $intF_{B}(\mathbb{J})$ for $B=Q$.

Corresponding ball $S$ (see remark 2) is the parallelogram with the vertices
in the points $(-1;2);(-1;-1);(1;-2);(1;1)$. From remark 2 follows that

$K(\mathbb{J})=K(Q)\subset K(S).$

One can see that $K(\mathbb{J})\neq K(S)$. For example, the operator given by
the matrix

$\left(\begin{array}[]{rr}-4&-1\\\\[11.38109pt] 2&0\end{array}\right)$

is dissipative with respect to $S$, but it is not dissipative with respect to
$\mathbb{J}$. In other words, in the systems without IME with respect to
$\\{\alpha S:\ \alpha\geq 0\\}$ (i.e. constricting according to the norm whose
ball is $\mathbb{J}$) there can be observed IME with respect to $\mathbb{J}$.

This example can be generalized as follows. Consider $\mathbb{J}$ from remark
2\. In system (7) on $B$ IME is absent with respect to $\mathbb{J}$ if the
operators $f^{\prime}(x)$ for all $x\in B$ are dissipative with respect to all
sets $Q_{k}(k=1,\ldots,m)$.

To sum up, one can say the following. When using sufficiently wide interval
spaces in almost all systems in accordance with theorem 4, IME is observed. In
particular, IME takes place almost for all systems when using standard
intervals (see example 6). Expansion of the interval space results in the
appearance of new systems with IME: thus, in using a set of all symmetrical to
$0$ convex compacts IME is absent only for linear systems with non-positive
scalar matrices. And the most impotent: the question about the existence of
interval space, with respect to which in the considered system IME is absent,
is reduced to the problem of joint dissipativity of the Jacobi matrices.
Therefore, there is no interval space with respect to which all (or even if in
some sense almost all) globally stable systems would have no IME. One has to
solve individually problems of the existence and constructing of corresponding
interval spaces for each particular system. These problems are solved
constructively very rarely.

We have treated the Moore effect in a very strong sense. The condition of
boundedness of the sequence of intervals ${\\{W_{m}\\}}_{m=0}^{+\infty}$ at
any step $h>0$ (see remark 7) is weaker (and acceptable, generally speaking,
for constructing sufficiently narrow interval solutions). This condition can
be called the condition of absence of the asymptotic Moore effect (AME). It is
the weakest from acceptable conditions, since with AME it is impossible to use
stepwise interval methods to obtain narrow interval solutions at large times.
The study of AME is still not completed. It is evident only that for a linear
autonomous system in considering the interval space from example 3 AME is
equivalent to IME. One can suggest a hypotesis: the problem of existence and
constructing of the interval space with respect to which AME is absent in the
autonomous system is reduced to the question of simultaneous dissipativity of
Jacobi matrices (and of constructing a constricting norm).

##  2 Conditions of Simultaneous Dissipativity of Operators

###  2.1 Some General Results

In the present section some conditions of simultaneous dissipativity of the
operators will be considered (see definition 7).

A definition of a simultaneous dissipativity can be generalized in such a way.

Definition 7′. A family of linear operators $\\{A_{\alpha}\\}$ is called
simultaneously stable dissipative if there exists a norm with respect to which
all operators $A_{\alpha}$ are stable dissipative.

Lemma 6. Let the space $\mathbb{R}^{n}$ be expanded into direct sum of
subspaces $E_{i}\ (i=1,\ldots,k)$ and each of them is invariant with respect
to all operators of the family $\\{A_{\alpha}\\}$. Further on, let restriction
of the family $\\{A_{\alpha}\\}$ on any $E_{i}$ be simultaneously
(sumultaneously stable) dissipative. Then $\\{A_{\alpha}\\}$ is simultaneously
(simultaneously stable) dissipative.

Proof. Let ${\|.\|}_{i}\ (i=1,\ldots,k)$ be the norms in $E_{i}$ in which the
restrictions of $\\{A_{\alpha}\\}$ on $E_{i}$ are simultaneously
(simultaneously stable) dissipative. Define the norm in $\mathbb{R}^{n}$ in
this way:

$\|x\|=\sum_{i=1}^{k}{\|x_{i}\|}_{i},$

where $x=\sum_{i=1}^{k}x_{i}$ with $x_{i}\in E_{i}\ (i=1,\ldots,k)$.

In this norm all operators $A_{\alpha}$ are simultaneously (simultaneously
stable) dissipative. The lemma is proved.

It is known [7] that for one operator the norm with respect to which it is
dissipative exists if and only if the spectrum of the operator lies in the
closed left half-plane and the boundary part is diagonalizable (i.e. Jordan
boxes corresponding to pure imaginary, including zero ones, eigenvalues are
diagonal). The norm, with respect to which the operator is stable dissipative,
exists if and only if the spectrum of the operator lies in the open left half-
plane.

Several stable dissipative (in their own norms) operators not necessarily are
simultaneously dissipative. To demonstrate that, consider operators
represented by the matrices

$A_{1}=\left(\begin{array}[]{rr}-1&3\\\\[11.38109pt] 0&-1\end{array}\right);\
\ A_{2}=\left(\begin{array}[]{rr}-1&0\\\\[11.38109pt] 3&-1\end{array}\right).$

Each of them is stable dissipative in its norm (due to the location of the
spectrum). But

$A_{1}+A_{2}=\left(\begin{array}[]{rr}-2&3\\\\[11.38109pt]
3&-2\end{array}\right).$

The spectrum of the operator $(A_{1}+A_{2})$ contains the point $\lambda=1$
which does not belong to the closed left half-plane. Thus, the operator
$(A_{1}+A_{2})$ is not dissipative in any norm. By lemma 6 the operators
$A_{1}$ and $A_{2}$ are not simultaneously dissipative.

The problem to find out necessary and sufficient conditions of simultaneous
dissipativity of an arbitrary (even finite) family of operators seems to be
wery difficult. Nevertheless, one can obtain some sufficient conditions
imposing different constraints on the operators. Obtain the sufficient
condition of simultaneous dissipativity of the family generating a solvable
Lee algebra. Remind [13] that a family of matrices generates solvable Lee
algebra if and only if all elements of this family are simultaneously
reducible to triangular form (generally speaking in complex basis).

Theorem 6. Let the family $\\{A_{\alpha}\\}$ be compact and generate solvable
Lee algebra, and the spectrum of each operator $A_{\alpha}$ lies in the open
left half-plane. Then $\\{A_{\alpha}\\}$ is simultaneously stable dissipative.

Proof. First consider the case of complex space $\mathbb{C}^{n}$. Consider
matrices of the operators $A_{\alpha}$ in the basis where they are of
triangular form.

Let each matrix $A_{\alpha}$ have the form

$A_{\alpha}=\left(\begin{array}[]{cccccc}\lambda_{1}^{(\alpha)}&0&0&\cdots&0&0\\\\[11.38109pt]
\mu_{21}^{(\alpha)}&\lambda_{2}^{(\alpha)}&0&\cdots&0&0\\\\[14.22636pt]
.&.&.&\cdots&0&0\\\\[14.22636pt]
\mu_{n1}^{(\alpha)}&\mu_{n2}^{(\alpha)}&\mu_{n3}{(\alpha)}&\cdots&\mu_{n,(n-1)}^{(\alpha)}&\lambda_{n}^{(\alpha)}\end{array}\right).$

Show the existence of such a set of positive numbers $\\{c_{k}\\}_{k=1}^{n}$
that all $A_{\alpha}$ are stable dissipative in the norm

$\|z\|=\max_{1\leq k\leq n}\frac{|z_{k}|}{c_{k}}$ $None$

(here $z_{k}$ is the $k$-th coordinate of the vector $z$ in the given basis),
whose unit ball is the polycylinder

$|z_{k}|\leq c_{k}\ (k=1,\ldots,n).$ $None$

If $\\{e_{k}\\}_{k=1}^{n}$ is the considered basis, then, evidently, norm (16)
coincide with the $l^{\infty}$-norm with respect to the basis
$\\{c_{k}/e_{k}\\}_{i,j=1}^{n}$ in the norm (16):

$\mbox{Re}\ a_{ii}+\sum_{j\neq i}\frac{c_{j}}{c_{i}}|a_{ij}|<0\
(i=1,\ldots,n).$ $None$

For the matrices $A_{\alpha}$ the conditions (18) look like this:

$\left\\{\begin{array}[]{lcl}\mbox{Re}\
\lambda_{1}^{(\alpha)}&<&0;\\\\[11.38109pt] \mbox{Re}\
\lambda_{2}^{(\alpha)}+\displaystyle{c_{1}\over
c_{2}}|\mu_{21}^{(\alpha)}|&<&0;\\\\[11.38109pt] &\cdots&\\\\[11.38109pt]
\mbox{Re}\ \lambda_{n}^{(\alpha)}+\displaystyle{c_{1}\over
c_{n}}|\mu_{n1}^{(\alpha)}|+\ldots+\displaystyle{c_{n-1}\over
c_{n}}|\mu_{n,(n-1)}^{(\alpha)}|&<&0.\\\\[11.38109pt] \end{array}\right.$
$None$

Suppose $\mu=\sup_{\alpha,k\neq l}|\mu_{kl}^{(\alpha)}|;\
\lambda=-\sup_{\alpha,k}\mbox{Re}\ \lambda_{k}^{(\alpha)}$. From the
conditions of the theorem follows that $0<\lambda<+\infty,\ 0<\mu<+\infty$. To
fulfil (19) for all $A_{\alpha}$, it is sufficient that the inequalities

$(c_{1}+\ldots+c_{k-1})\mu<c_{k}\lambda\ (k=1,\ldots,n);\ c_{1}>0$ $None$

be satisfied.

Show the solvability of system (20). Let $c_{1}=1$. Choose the others $c_{k}$
so that

$c_{2}>\mu/\lambda;c_{3}>(1+c_{2})\mu/\lambda;\ldots;$
$c_{n}>(1+c_{2}+\ldots+c_{n-1})\mu/\lambda.$

Then the inequalities (20) are satisfied, i.e. all operators $A_{\alpha}$ are
stable dissipative in the norm (14).

Let now operators $A_{\alpha}$ act in the space $\mathbb{R}^{n}$. In usual way
complexify $\mathbb{R}^{n}$ and the family ${A_{\alpha}}$. Then, as it has
been described above, construct a cylinder (17). Intersection of (17) with the
initial space $\mathbb{R}^{n}$ produce a ball of the norm in which all
$A_{\alpha}$ are stable dissipative. The theorem is proved.

If instead of stable dissipative operators one considers dissipative
operators, then the analog of theorem 6 is not true, starting from real
dimension 4. Let

$A_{1}=\left(\begin{array}[]{rr}i&1\\\\[11.38109pt]
0&2i\end{array}\right);A_{2}=\left(\begin{array}[]{rr}2i&1\\\\[11.38109pt]
0&i\end{array}\right).$

Each of the operators $A_{1,2}$ is dissipative in its norm. The finite family
is compact, the matrices $A_{1}$ and $A_{2}$ generate solvable Lee algebra.
Nevertheless

$A_{1}+A_{2}=\left(\begin{array}[]{rr}3i&2\\\\[11.38109pt]
0&3i\end{array}\right).$

The only eigenvalue of the operator $(A_{1}+A_{2})$ is pure imaginary, with
the matrix of this operator representing (up to a constant factor) non-trivial
Jordan box. That means it is not dissipative in any norm, i.e. $A_{1}$ and
$A_{2}$ are not simultaneously dissipative. To obtain a real example, one has
to make the matrices $A_{1}$ and $A_{2}$ real:

$A_{1}^{R}=\left(\begin{array}[]{rrrr}0&-1&1&0\\\\[11.38109pt]
1&0&0&1\\\\[11.38109pt] 0&0&0&-2\\\\[11.38109pt]
0&0&2&0\end{array}\right);A_{2}^{R}=\left(\begin{array}[]{rrrr}0&-2&1&0\\\\[11.38109pt]
2&0&0&1\\\\[11.38109pt] 0&0&0&-1\\\\[11.38109pt] 0&0&1&0\end{array}\right).$

To keep true the statement about simultaneous dissipativity for nonstable
dissipative operators, it is sufficient to strengthen the requirement of
solvability up to nilpotency. Remind [13] that for each linear operator $A$ in
the space $E$ the operator $ad\ A$ in $L(E)$ is defined:

$(ad\ A)B=AB-BA.$

The family $\\{A_{\alpha}\\}$ generates the nilpotent Lee algebra if and only
if there exists such a number $m\in N$ that for any set of
$\\{A_{\alpha_{k}}\\}_{k=1}^{m}$ (among the elements of wich there may be the
same ones) and for all $\alpha$:

$\prod_{k=1}^{m}(ad\ A_{\alpha_{k}})A_{\alpha}=0.$ $None$

Nilpotent Lee algebra is always solvable. Commutative Lee algebra is nilpotent
(for it $m=1$) and solvable.

Theorem 7. Let the family $\\{A_{k}\\}$ be finite and generate nilpotent Lee
algebra, and for each operator $A_{k}$ exist a norm with respect to which it
is dissipative. Then $\\{A_{k}\\}$ is simultaneously dissipative.

Proof. Without loss of generality one can assume that among the operators
$A_{k}$ there are no scalar ones (if $A=aI,$ where $\mbox{Re}\ a\leq 0,$ then
$A$ is dissipative in any norm) and exists at least one operator (denote it
$A_{1}$), among eigenvalues of which there are pure imaginary (otherwise we
are under the conditions of theorem 6).

First assume that $A_{k}$ operates in $\mathbb{C}^{n}$. We prove the theorem
by induction on dimension of space. In dimension 1 the statement of the
theorem is trivial. Show that one can expand all the space $\mathbb{C}^{n}$
into a direct sum of two non-trivial subspaces invariant with respect to all
$A_{k}$. Since in both of them the conditions of the theorem (for
corresponding restrictions of $\\{A_{k}\\}$) are satisfied, then to complete
the proof one has to use lemma 6.

Let $\lambda$ be an imaginary eigenvalue of $A_{1}$; $E^{\prime}$ be the
corresponding to $\lambda$ eigen-subspace (by virtue of diagonalizability of
boundary part of $A_{1}$ it coincides with whole corresponding root subspace);
$E^{\prime\prime}$ be the sum of root subspaces corresponding to all the
others eigenvalues of $A_{1}$. Evidently, $\mathbb{C}^{n}=E^{\prime}\oplus
E^{\prime\prime}$ (the sign $\oplus$ means direct sum);
$E^{\prime}\neq\mathbb{C}^{n}$, otherwise the operator $A_{1}$ is scalar. Show
the invariance of $E^{\prime}$ and $E^{\prime\prime}$ with respect to all
$A_{k}$.

Let $x\in E^{\prime}$. Then

$A_{1}x=\lambda x.$

On the other hand, in accordance with (21) there exists such $m\in N$ that
$(ad\ A_{1})^{m}A_{k}=0$ for all $k$ and

$(A_{1}-\lambda I)^{m}A_{k}x=0.$

A more general fact is true: if $Ax=0$ and $(ad\ A)^{m}B=0,$ then $A^{m}Bx=0$.
For $m=0$ the fact is obvious. Let that be true for $m=r$. Assume

$Ax=0;(ad\ A)^{r+1}B=0.$

Then $(ad\ A)^{r}(ad\ A)B=0$, and according to the inductive hypothesis
$A^{r}(ad\ A)Bx=0.$ But $A^{r+1}Bx=A^{r}(BAx+(ad\ A)Bx)$, i.e. $A^{r+1}Bx=0$,
as was to be proved.

As a consequence of coincidence of $E^{\prime}$ with the whole root subspace,
corresponding to $\lambda,$ we have:

$(A_{1}-\lambda I)A_{k}x=0,$

i.e. $A_{1}A_{k}x=\lambda A_{k}x,A_{k}x\in E^{\prime}$.

Show now the invariance of $E^{\prime\prime}$. Let $\\{e_{j}\\}_{j=1}^{n}$ be
the Jordan basis of the operator $A_{1}$ with $E^{\prime}$ being corresponded
to the vectors $\\{e_{j}\\}_{j=j_{1}}^{j_{2}}$. One has to show that for any
$j$ less then $j_{1}$ or more then $j_{2}$ the coordinates of $A_{k}e_{j}$
with the numbers from $j_{1}$ to $j_{2}$ with respect to the assigned basis
are equal to zero. Let it be not so and exist such $j^{\prime}$ that
$e_{j^{\prime}}\in E^{\prime\prime}$, but the $j_{1}$-th coordinate
$(j_{1}\leq j_{0}\leq j_{2})$ of the vector $A_{k}e_{j^{\prime}}$ is $a\neq
0$. Write it like this:

$A_{k}e_{j^{\prime}}=\ldots=ae_{j_{0}}.$

Let $e_{j^{\prime}}$ be an eigenvector of $A_{1}$ corresponding to the
eigenvalue $\mu\neq\lambda$. Then

$(ad\
A_{1})A_{k}e_{j^{\prime}}=A_{1}(\ldots+ae_{j_{0}})-\mu(\ldots+ae_{j_{0}})=\ldots+(\lambda-\mu)ae_{j_{0}}.$

Verify that

$(ad\ A_{1})^{m}A_{k}e_{j^{\prime}}=\ldots+(\lambda-\mu)^{m}ae_{j_{0}}.$

For $m=0$ it is obvious. Let it be satisfied for $m=r$. Then

$(ad\ A_{1})^{r+1}A_{k}e_{j^{\prime}}=(ad\ A_{1})(ad\
A_{1})^{r}A_{k}e_{j^{\prime}}=$ $=A_{1}(ad\ A_{1})^{r}A_{k}e_{j^{\prime}}-(ad\
A_{1})^{r}A_{k}A_{1}e_{j^{\prime}}=$
$=A_{1}(\ldots+(\lambda-\mu)^{r}ae_{j_{0}})-\mu(ad\
A_{1})^{r}A_{k}e_{j^{\prime}}=$ $=\ldots+(\lambda-\mu)^{r}a\lambda
e_{j_{0}}-\mu(\lambda-\mu)^{r}ae_{j_{0}}=$
$=\ldots+(\lambda-\mu)^{r+1}ae_{j_{0}},$

i.e. that is true also for $m=r+1$, and, hence, for all $m\in N$.

Thus,

$(ad\ A_{1})^{m}A_{k}e_{j^{\prime}}=\ldots+(\lambda-\mu)^{m}ae_{j_{0}}\neq 0$

for any $m\in N$, which contradicts (21).

Let now $e_{j^{\prime}}$ be a root (but not eigen) vector, corresponding to
the eigenvalue $\mu$, with the $j_{0}$-th coordinate of the vector
$A_{k}e_{j^{\prime}-1}$ equal to $0$. Then

$(ad\
A_{1})A_{k}e_{j^{\prime}}=A_{1}(\ldots+ae_{j_{0}})-A_{k}(e_{j^{\prime}-1}+\mu
e_{j^{\prime}})=\ldots+(\lambda-\mu)ae_{j_{0}}.$

Analogously

$(ad\ A_{1})^{m}A_{k}a_{j^{\prime}}=\ldots+(\lambda-\mu)^{m}ae_{j_{0}}\neq 0$

for any $m\in N$, which contradicts (21).

Since the sequence of basis vectors belonging to the root subspace begins with
the eigenvector, the required statement for complex space is proved.

The transfer onto the case of real space can be done in the same way as in the
proof of theorem 6 (the ball of corresponding norm in the copmlexified
$\mathbb{R}^{n}$ intersects with $\mathbb{R}^{n}$). The theorem is proved.

From theorems 6 and 7 follows, in particular, that a finite (compact)
commutative family consisting of operators dissipative (stable dissipative) in
their own norms is simultaneously dissipative (simultaneously stable
dissipative).

###  2.2 The Mass Action Law and Dissipative Mechanisms

Some constructive conditions of simultaneous dissipativity can be obtained for
finite families of operators of rank 1. The problem of the absence of Moore
effect in the system constructed in accordance with the Mass Action Law (MAL)
is reduced to the problem on simultaneous dissipativity of such operators.

MAL systems appear from mathematical description of systems of chemical and
biological kinetics and in some other problems. To the considered process is
assigned an algebraic object, called reaction mechanism and having the form:

$\alpha_{r1}A_{1}+\ldots+\alpha_{rn}A_{n}\to\beta_{r1}A_{1}+\ldots+\beta_{rn}A_{n},\
(r=1,\ldots,d).$ $None$

Speaking in terms of chemical kinetics, the reaction mechanism is a list of
stoichiometric equations of elementary reactions (22). In this case
$A_{1},\ldots,A_{n}$ are the substances taking part in the reaction;
$\alpha_{ri},\beta_{ri}$ are the non-negative integers called stoichiometric
coefficients and showing in what amount the particles of $A_{i}$ enter into
the $r$-th elementary reaction as the initial substance $(\alpha_{ri})$ or
product $(\beta_{ri})$. The following notations are accepted:
$\gamma_{ri}=\beta_{ri}-\alpha_{ri},\ \gamma_{r}$ is the vector with the
components $\gamma_{ri}\ (i=1,\ldots,n)$ – so-called stoichiometric vector of
the $r$-th elementary reaction.

In accordance with MAL [14,15], to the mechanism (22) corresponds the
following system of ordinary differential equations:

$\frac{dc_{i}}{dt}=\sum_{r=1}^{d}\gamma_{ri}w_{r}$ $None$

where $c_{i}(t)$ is the concentration of substance $A_{i}$ at the moment of
time $t\geq 0,$

$w_{r}=k_{r}(t)\prod_{j=1}^{n}c_{j}^{\alpha_{rj}}$

is the rate of the $r$-th elementary reaction, continuously depending on time.
In particular, if reaction proceeds under constant external conditions, then
$k_{r}(t)={\rm const}\ (r=1,\ldots,d)$ and $k_{r}$ is called rate constant of
the $r$-th elementary reaction. In the latter case (23) represents an
autonomous system with polynomial right sides.

Let $L$ be a linear envelope of the family $\\{\gamma_{r}\\}_{r=1}^{d}$. If
$L\neq\mathbb{R}^{n}$, then there exist such $a_{i}\ (i=1,\ldots,n)$, not all
equal to sero, that for all $r=1,\ldots,d$ the equalities

$\sum_{i=1}^{n}a_{i}\gamma_{ri}=0$

are satisfied, from which for system (23) follows that

$\sum_{i=1}^{n}a_{i}c_{i}(t)={\rm const}$ $None$

Relationships (24) are called stoichiometric conservation laws. If all $a_{i}$
are positive, then the corresponding stoichiometric law is called the positive
conservation law [15]. In MAL positive conservation laws takes place rather
often (but not always).

As it is known [15], balance polyhedrons are intersections of affine subspaces
of the form $(L+c)$, where $c$ is a constant vector, with a cone of non-
negative vectors (first orthant) in $\mathbb{R}^{n}$. Balance polyhedrons
represent positive invariant with respect to (23) convex sets (one can find
the proof of their positive invariance in [15]). If there exists at least one
of positive conservation law, they are compact.

The question arises: under what conditions does the norm exist in
$\mathbb{R}^{n}$ according to which the system (23) is constricting in all
balance polyhedrons and independent of rate constants?

Definition 8. Mechanism (22) is called dissipative, if for system (23) there
exists a norm, constricting in all balance polyhedrons irrespective of rate
constants (in other words, the constricting norm depends on the mechanism
only).

We use the notation $M_{ri}$ for the operator in $\mathbb{R}^{n}$, represented
by the matrix, in the $i$-th column of which there are components of the
vector $\gamma_{r}$, and on other places – zeros. The subspace $L$ is
invariant with respect to all $M_{ri}$ [15]. The notation $M^{\prime}_{ri}$
stays for restriction of $M_{ri}$ on $L$.

Theorem 8. Let for mechanism (22) exist at least one positive conservation
low. This mechanism is dissipative if and only if the family
$\\{M^{\prime}_{ri}:\ \alpha_{ri}>0\\}$ is simultaneously dissipative.

Proof. Sufficiency. It is known [15] that the Jacobi matrix $J_{c}$ of system
(22) at the point $c$, whose coordinates are positive, has the form

$J_{c}=\sum_{\alpha_{ri}>0}\alpha_{ri}\frac{w_{r}}{c_{i}}M_{ri}.$ $None$

Matrices $J_{c}$ belong to the convex cone produced by the family $\\{M_{ri}|\
\alpha_{ri}>0\\}$. Besides, the difference of any two solutions (23) from one
balance polyhedron belongs to the subspace $L$. Under the conditions of lemma
3 and theorem 2 obtain the existence of constricting norm in the subspace $L$.
It can be expanded onto all $\mathbb{R}^{n}$.

Necessity. Matrices $M_{ri}\ (\alpha_{ri}>0)$ belong to the closure of the
family of matrices $J_{c}$ for arbitrary non-negative vectors $c$ and rate
constants $k_{r}$. To prove this, first let consider the case when $c_{j}\
(j=1,\ldots,n)$ and $k_{r}$ are fixed and all $k_{l}(l\neq r)$ tend to zero.
In the limit in (25) only the sum for given $r$ is left. Further on, fix all
$c_{j}>0\ (j\neq i)$ and let $c_{i}$ tend to zero, changing $k_{r}$ so that
the equality $\alpha_{ri}w_{r}/c_{i}=1$ holds true. Then all the terms except
one tend to zero and in the limit we obtain $M_{ri}$.

Thus, the matrices $M^{\prime}_{ri}\ (\alpha_{ri}>0)$ belong to the closure of
the family of restrictions of the matrices $J_{c}$ on the subspace $L$. Hence,
according to lemma 3 and theorem 2 the necessity follows. The theorem is
proved.

Note that matrices $M_{ri}$ represent matrix-columns (in each matrix there is
only one non-zero column) and that means that the rank of each of them is
equal to unity. We come to the problem of simultaneous dissipativity of the
finite family of operators of rank 1.

Note that dissipative mechanisms of reactions were studied in details in [12].
In particular, some classes of dissipative mechanisms are pointed out and all
dissipative mechanisms for $n=3,\ \sum_{i=1}^{3}\alpha_{ri}\leq 3,\
\sum_{i=1}^{3}\beta_{ri}\leq 3\ (r=1,\ldots,d),\ c_{1}+c_{2}+c_{3}={\rm
const}$ enumerated.

In the next subsection are obtained necessary and sufficient conditions of
simultaneous dissipativity of the operators of rank 1 in $\mathbb{R}^{2}$
(corresponding to the case $dimL=2$) and some sufficient conditions of
simultaneous dissipativity of matrix-columns.

###  2.3  Constructive Conditions of Simultaneous Dissipativity of One-
Dimensional Operators

Before consideration of simultaneous dissipativity of operators of rank 1,
find out what can be said about dissipativity of one such operator. From
necessary and sufficient conditions (see the paragraph after lemma 6) follows
that the norm in which the given operator of rank 1 is dissipative exists if
and only if it has a negative eigenvalue.

Positive semi-trajectories of system (3) corresponding to the initial
condition $x(0)=x_{0}$ are in this case rectilinear segments parallel to the
image of $A$ and connecting $x_{0}$ with $Ker\ A$. Operator A of rank 1 is
dissipative in the given norm if and only if for any point $x\ (\|x\|=1)$
there exists such $\varepsilon>0$ that $\|x+\varepsilon Ax\|\leq 1$. It means
that the negative number belongs to the spectrum of $A$, and the image of $A$
is orthogonal to its kernel (in the given norm, the subspace $E_{2}$ is
orthogonal to $E_{1}$ if $\|x+y\|\geq\|x\|$ for any $x\in E_{1},y\in E_{2}$
[7]).

Let now be given a family $\\{M_{k}\\}_{k=1}^{m}$ of operators of rank 1 in
$\mathbb{R}^{n}$. Each of them can be represented in the form $(\cdot\
;\psi_{k})\varphi_{k}$, i.e. $M_{k}x=(x\ ;\psi_{k})\varphi_{k}$ where $(\cdot\
;\cdot)$ is the standard scalar product in $\mathbb{R}^{n}$. The vectors
$\varphi_{k}$ and $\psi_{k}$ are determined by the operator $M_{k}$
unambiguously (up to scalar factors). Let
$\lambda_{k}=(\varphi_{k};\psi_{k})$, i.e. $\lambda_{k}$ is an eigenvalue of
$M_{k}$ (either it is the only non-zero eigenvalue, or $0$, if the operator
$M_{k}$ is nilpotent). As it has already been mentioned, for simultaneous
dissipativity of $\\{M_{k}\\}$ the conditions

$\lambda_{k}<0\ (k=1,\ldots,m)$ $None$

are necessary.

Assign to each operator $M_{k}$ the projector $P_{k}$ projecting parallel to
the image of $M_{k}$ on the kernel of $M_{k}$. It is easy to see that
$P_{k}=I-M_{k}/\lambda_{k}$. By virtue of the above-mentioned condition of
dissipativity of the operator of rank 1 in the given norm the operator $M_{k}$
is dissipative in some norm if and only if $P_{k}$ is constriction in this
norm.

All $P_{k}$ can be constrictions in one norm if and only if all products of
the form $\prod_{j=1}^{q}P_{k_{j}}$ ($q\in N$ is arbitrary;
$k_{j}\in\\{1,\ldots,m\\}$ and they are not necessarily different) are jointly
bounded. We come to the following conclusion.

Lemma 7. The family $\\{M_{k}\\}_{k=1}^{m}$ of operators of rank 1 is
simultaneously dissipative if and only if the conditions (26) are satisfied
and all products of the form $\prod_{j=1}^{q}P_{k_{j}}\ (q\in N$ is arbitrary;
$k_{j}\in\\{1,\ldots,m\\}$ and they are not necessarily different) are jointly
bounded. As a constricting norm one can take

$\|x\|=\sup_{q\in N,1\leq\ k_{j}\leq
m}\\{\|x\|_{0},\bigl{\|}(\prod_{j=1}^{q}P_{k_{j}})x\bigr{\|}_{0}\\},$ $None$

where $\|.\|_{0}$ is any norm in $\mathbb{R}^{n}$.

Proof. All statements of the lemma, exept the latter, follow immediatily from
the above reasonings. Further on, if all products $\prod_{j=1}^{q}P_{k_{j}}$
are jointly bounded, then

$\sup_{q\in N,1\leq\ k_{j}\leq
m}\\{\|x\|_{0},\bigl{\|}(\prod_{j=1}^{q}P_{k_{j}})x\bigr{\|}_{0}\\}<{\infty}$

for each $x\in\mathbb{R}^{n}$. This expression possesses all properties of
norm and all operators $P_{k}$ are constrictions in such norm, i.e. all
$M_{k}$ are simultaneously dissipative. The lemma is proved.

From lemma 7 follows a simple consequence.

Corollary 1. If all $\varphi_{k}$ are collinear (images of $M_{k}$ coincide)
or all $\psi_{k}$ are collinear (kernels of $M_{k}$ coincide) and
$(\varphi_{k};\psi_{k})<0$ for all $k=1,\ldots,m$, then the operators $M_{k}\
(k=1,\ldots,m)$ are simultaneously dissipative. As corresponding constricting
norm one can take

$\sup_{q\in N,1\leq k_{j}\leq m}\\{\|x\|_{0},\|P_{k}x\|_{0}\\}.$

To demonstrate this, it is sufficient to note that in these cases

$\prod_{j=1}^{q}P_{k_{j}}=P_{k_{1}}$

or

$\ \prod_{j=1}^{q}P_{k_{j}}=P_{k_{q}},$

respectively.

Remark 10. If not all $\varphi_{k}$ are collinear, then as a norm in lemma 7
one can take

$\sup_{q\in N,1\leq k_{j}\leq
m}\bigl{\|}(\prod_{j=1}^{q}P_{k_{j}})x\bigr{\|}_{0}.$ $None$

The criterion established in lemma 7 is not constructive. Constructive
criteria of simultaneous dissipativity of finite family of operators of rank 1
in $\mathbb{R}^{n}$ have been obtained only at $n=2$ (for arbitrary $n$ there
exist sufficient conditions for one class of operators; they are given at the
end of the section). Pass to the consideration of the case $n=2$.

Consider the family $\\{M_{k}\\}_{k=1}^{m}$ of the operators of rank 1 in
$\mathbb{R}^{2}$. As before, represent each operator $M_{k}$ in the form
$(\cdot\ ;\psi_{k})\varphi_{k}$. Let first $m=2$.

Lemma 8. The operators $M_{1}=(\cdot\ ;\psi_{1})\varphi_{1}$ and
$M_{2}=(\cdot\ ;\psi_{2})\varphi_{2}$ are simultaneously dissipative in
$\mathbb{R}^{2}$ if and only if the condition

$\biggl{|}\frac{(\varphi_{1};\psi_{2})\ \cdot\
(\varphi_{2};\psi_{1})}{(\varphi_{1};\psi_{1})\cdot(\varphi_{2};\psi_{2})}\biggr{|}\leq
1$ $None$

is satisfied together with the conditions

$(\varphi_{1};\psi_{1})<0;\ (\varphi_{2};\psi_{2})<0.$

As a corresponding constricting norm one can take

$\|x\|=\max\\{\|x\|_{0},\|P_{1}x\|_{0},\|P_{2}x\|_{0},\|P_{1}P_{2}x\|_{0},\|P_{2}P_{1}x\|_{0}\\}.$
$None$

Proof. In $\mathbb{R}^{2}$ the projectors $P_{1}$ and $P_{2}$ have rank 1 and
are represented in the form

$P_{1}=(\cdot\ ;\eta_{1})\chi_{1};\ P_{2}=(\cdot\ ;\eta_{2})\chi_{2},$

where $\eta_{1},\eta_{2},\chi_{1},\chi_{2}$ are some vectors in
$\mathbb{R}^{2}$.

The operators $\prod_{j=1}^{q}P_{k_{j}}$ are bounded when the spectrum of the
operator $(P_{1}P_{2})$ lies on the segment $[-1;1]$:

$|(\chi_{1};\eta_{2})\cdot(\chi_{2};\eta_{1})|\leq 1.$ $None$

In a standard orthonormalized basis $P_{k}$ acts like this:

$P_{k}x=\frac{1}{(\varphi_{k};\psi_{k})}\cdot\left(\left(\begin{array}[]{r}x^{(1)}\\\\[5.69054pt]
x^{(2)}\end{array}\right);\left(\begin{array}[]{r}\varphi_{k}^{(2)}\\\\[5.69054pt]
-\varphi_{k}^{(1)}\end{array}\right)\right)\cdot\left(\begin{array}[]{r}\psi_{k}^{(2)}\\\\[5.69054pt]
-\psi_{k}^{(1)}\end{array}\right)$

where

$\left(\begin{array}[]{r}a^{(1)}\\\\[5.69054pt] a^{(2)}\end{array}\right)$

denotes the vector with the coordinates $a^{(1)}$ and $a^{(2)}$. Hence

$(\eta_{1};\chi_{2})=\frac{(\varphi_{1};\psi_{2})}{(\varphi_{2};\psi_{2})};$
$(\chi_{1};\eta_{2})=\frac{(\varphi_{2};\psi_{1})}{(\varphi_{1};\psi_{1})},$

i.e. condition (31) takes the form (29).

To complete the proof, use lemma 7. To check a possibility of choosing
corresponding norm in the form (30), note that

$(P_{1}P_{2})^{r}P_{1}=(\eta_{1};\chi_{2})^{r}\cdot(\eta_{2};\chi_{1})^{r}\cdot
P_{1};$
$(P_{2}P_{1})^{r}P_{2}=(\eta_{1};\chi_{2})^{r}\cdot(\eta_{2};\chi_{1})^{r}\cdot
P_{2}$

for any $r\in N$. It means that with the account of (31), in (27) one can
restrict oneself to finite number of products. The lemma is proved.

Remark 11. If $\varphi_{1}$ and $\varphi_{2}$ are non-collinear, then as
required norm we can take

$\max\\{\|P_{1}x\|_{0},\|P_{2}x\|_{0},\|P_{1}P_{2}x\|_{0},\|P_{2}P_{1}x\|_{0}\\}.$

This follows from remark 10. Then the ball of the norm is determined by the
inequalities

$|(x;\eta_{1})|\leq\min\biggl{\\{}\frac{1}{\|\chi_{1}\|_{0}},\frac{1}{|(\chi_{1};\eta_{2})|\cdot\|\chi_{2}\|_{0}}\biggr{\\}};$
$|(x;\eta_{2})|\leq\min\biggl{\\{}\frac{1}{\|\chi_{2}\|_{0}},\frac{1}{|(\chi_{2};\eta_{1})|\cdot\|\chi_{1}\|_{0}}\biggr{\\}},$

i.e. it is parallelogram.

Also note that for simultaneous dissipativity of a family the dissipativity of
each operator from convex envelope of the family is insufficient. To see this,
consider the operators represented by the matrices

$M_{1}=\left(\begin{array}[]{rr}-1&1\\\\[5.69054pt] 0&0\end{array}\right);\
M_{2}=\left(\begin{array}[]{rr}0&0\\\\[5.69054pt] -2&-1\end{array}\right).$

Each of them is dissipative in its norm. It is easy to show that spectrum of
any non-trivial convex combination of $M_{1}$ and $M_{2}$ lies in open left
half-plane. Nevertheless

$\frac{(\varphi_{1};\psi_{2})\cdot(\varphi_{2};\psi_{1})}{(\varphi_{1};\psi_{1})\cdot(\varphi_{2};\psi_{2})}=-2,$

i.e. condition (29) is not satisfied.

Reasoning like in proof of lemma 8, it is easy to obtain a criterion of a
simultaneous dissipativity for arbitrary $m$. The result is a set of
conditions of the form

$(\varphi_{k};\psi_{k})<0\ (k=1,\ldots,m);$ $None$
$\biggl{|}\frac{(\varphi_{k_{1}};\psi_{k_{2}})\cdot(\varphi_{k_{2}};\psi_{k_{3}})\cdot\ldots\cdot(\varphi_{k_{q}};\psi_{k_{1}})}{(\varphi_{k_{1}};\psi_{k_{1}})\cdot(\varphi_{k_{2}};\psi_{k_{2}})\cdot\ldots\cdot(\varphi_{k_{q}};\psi_{k_{q}})}\biggr{|}\leq
1,$ $None$

where $\\{k_{j}\\}_{j=1}^{q}$ is a set of different numbers from 1 to $m$, and
inequalities (33) holds for all such sets. The number of conditions has the
order $O((m-1)!)$ and for any large $m$ testing of these conditions becomes
unrealizable. It turns out, however, that among inequalities (33) there are
dependent ones and the number of conditions can be reduced.

Theorem 9. Let the vectors $\psi_{k}\ (k=1,\ldots,m)$ lie in one half-plane
clockwise. Then the family of operators $\\{M_{k}\\}_{k=1}^{m}$ where
$M_{k}=(\cdot;\psi_{k})\varphi_{k}$ is simultaneously dissipative if and only
if the vectors $\varphi_{k}\ (k=1,\ldots,m)$ lie in one half-plane clockwise
and the conditions (32) and the followings ((34), (35)) are satisfied:

$\biggl{|}\frac{(\varphi_{k};\psi_{k+1})\cdot(\varphi_{k+1};\psi_{k})}{(\varphi_{k};\psi_{k})\cdot(\varphi_{k+1};\psi_{k+1})}\biggr{|}\leq
1$ $None$ $(k=1,\ldots,m\ \mbox{with}\
\varphi_{m+1}=-\varphi_{1};\psi_{m+1}=-\psi_{1});$
$\left\\{\begin{array}[]{rcl}\displaystyle\biggl{|}\frac{(\varphi_{1};\psi_{2})\cdot(\varphi_{2};\psi_{3})\cdot\ldots\cdot(\varphi_{m};\psi_{1})}{(\varphi_{1};\psi_{1})\cdot(\varphi_{2};\psi_{2})\cdot\ldots\cdot(\varphi_{m};\psi_{m})}\biggr{|}&\leq&1;\\\\[11.38109pt]
\displaystyle\biggl{|}\frac{(\varphi_{1};\psi_{m})\cdot(\varphi_{m};\psi_{m-1})\cdot\ldots\cdot(\varphi_{2};\psi_{1})}{(\varphi_{1};\psi_{1})\cdot(\varphi_{2};\psi_{2})\cdot\ldots\cdot(\varphi_{m};\psi_{m})}\biggr{|}&\leq&1.\end{array}\right.$
$None$

The corresponding norm can be chosen polyhedral (a norm, whose ball is
polygon).

Proof. Necessity. Let $l_{k}$ be the kernels of the operators $M_{k}$ (i.e.
straight lines orthogonal to $\psi_{k}$). Straight lines $l_{k}$ divide the
plane into $2m$ sectors. If among the vectors $\psi_{k}$ there are collinear,
then some sectors are singular, but this does not change the further
reasonings. In each sector $G$ and for each $p\in\\{1,\ldots,m\\}$

$sign(x_{1};\psi_{p})=sign(x_{2};\psi_{p})$

for all $x_{1}\in int\ G,\ \ x_{2}\in int\ G$.

Let $G_{r}$ be a sector lying between corresponding rays of straight lines
$l_{r}$ and $l_{r+1}$ (where $l_{m+1}=l_{1}$). It is enough to consider the
sectors $\\{G_{k}\\}_{k=1}^{m}$ into which one half-plane is divided, since
for sectors lying in vertical angles to $G_{k}$ the reasons are the same.

Note that by inequality (32) for each operator $M_{k}$ the projector $P_{k}$
is determined, which operates in each sector $G_{k}$ as a projector in the
direction $\upsilon_{kr}=sign(x;\psi_{k})\cdot\varphi_{k}\ (x\in G_{r})$ onto
the straight line $l_{k}$.

A norm with respect to which all $M_{k}$ are dissipative exists if and only if
there exist a convex body $Q$ symmetrical with respect to $0$ and positively
invariant with respect to all systems of the following form

$\frac{dx}{dt}=\sum_{k=1}^{m}h_{k}(t)(x;\psi_{k})\varphi_{k},$ $None$

where $h_{k}(t)$ is any function piecewise continuous and non-negative for
$t\geq 0.$ The sufficiency is evident (suppose $h_{k}(t)\equiv
1,h_{j}(t)\equiv 0$ for $j\neq k$ and come to dissipativity of $M_{k}$ with
respect to $Q$). To prove the necessity, it is sufficient to make an
estimation analogous to that made in the proof of theorem 1:

$\frac{d}{dt}\|x(t)\|_{Q}=N_{Q}\bigl{(}x(t),\sum_{k=1}^{m}h_{k}(t)M_{k}x(t)\bigr{)}\leq$
$\leq\gamma_{Q}\bigl{(}\sum_{k=1}^{m}h_{k}(t)M_{k}x(t)\bigr{)}\cdot\|x(t)\|_{Q}\leq
0.$

Here $\|.\|_{Q}$ is a norm whose unit ball is $Q$.

Since $(x;\psi_{k})\varphi_{k}=|(x;\psi_{k})|\cdot\upsilon_{kr}$ at $x\in
G_{r}$, then (36) can be rewritten as follows:

$\frac{dx}{dt}=\sum_{k=1}^{m}y_{k}(t)\upsilon_{kr}$ $None$

where $y_{k}(t)$ is piecewise continuous and non-negative for $t\geq 0$. Thus,
it is sufficient to construct such a polygon $W$ that from each point of its
boundary $\partial W$ all the vectors $\upsilon_{kr}$ are not directed into
the exterior of $W$. Then one can take

$Q=co\ \\{W\cup(-W)\\}.$

Let (37) have at least one unbounded solution, whose positive semi-trajectory
lies inside one of sectors. Then (36) has an unbounded solution, i.e. the
operators $M_{k}$ are not simultaneously dissipative.

The notation $C\\{\upsilon_{kr}\\}$ is used for a convex cone produced by
$\\{\upsilon_{kr}\\}_{r=1}^{m}$.

Let this cone concide with $\mathbb{R}^{2}$ at least in one sector $G_{r}$
(i.e. the vectors generating it do not lie in one half-plane). Then as
$y_{k}(t)$ one can choose such constants that
$\upsilon=\sum_{k=1}^{m}y_{k}\upsilon_{kr}\in G_{r}$, and then, drawing a ray
from the point $x_{0}\in intG_{r}$ in the direction of $\upsilon$, obtain a
positive semi-trajectory of unbounded solution (47) lying inside $G_{r}$.

Thus, for simultaneous dissipativity of $\\{M_{k}\\}$ it is necessary to
satisfy the conditions

$C\\{\upsilon_{kr}\\}\neq\mathbb{R}^{2}\ (k=1,\ldots,m).$ $None$

If $C\\{\upsilon_{kr}\\}$ in some sector is a half-plane, then it must contain
the vertical angle to $G_{r}$ \- $\hat{G}_{r}$ (and thus intersect with
$G_{r}$ only at zero); otherwise (37) has an unbounded solution. For each
sector $G_{r}$ consider the boundary of the cone $C\\{\upsilon_{kr}\\}$. It
consists of two directions. Show that for $G_{j}$ it is
$\upsilon_{jj},\upsilon_{(j+1),j}$. It is sufficient to show that for $j=1$.

Let $\upsilon_{1,1}$ and $\upsilon_{2,1}$ be collinear and oppositely
directed. Then to satisfy (38) it is necessary tha the other $\upsilon_{k1}$
lie on one side of the straight line, stretched on $\upsilon_{1,1}$. But if
$\upsilon_{1,1}$ and $\upsilon_{2,1}$ are non-collinear, then all other
$\upsilon_{k1}$ can be expanded in terms of the basis
$\upsilon_{1,1},\upsilon_{2,1}$.

Let, for example, $\upsilon_{3,1}=c_{1}\upsilon_{1,1}+c_{2}\upsilon_{2,1},$
and $\upsilon_{3,1}$ be collinear to one of the basis vectors (for example,
$\upsilon_{1,1}$; the case with $\upsilon_{2,1}$ is considered analogously).
Then $c_{2}=0$. If $c_{1}>0$ then $\upsilon_{3,1}\in
C\\{\upsilon_{1,1},\upsilon_{2,1}\\}$. Let $c_{1}<0$. Then to satisfy (38) in
$G_{1}$ it is necessary for $\upsilon_{1,1}$ and $\upsilon_{3,1}$ to be
boundary directions in $C\\{\upsilon_{k1}\\}$. Since
$\upsilon_{k,(l+1)}=\upsilon_{kl}$, if $k\neq l+1$, and
$\upsilon_{(l+1),(l+1)}=-\upsilon_{l,(l+1)}$, then the same directions are
boundary for $C\\{\upsilon_{k2}\\}$ as well, otherwise (38) is not satisfied
in $G_{2}$. Simultaneously $\upsilon_{2,1}\in C\\{\upsilon_{k1}\\},\
\upsilon_{2,2}=-\upsilon_{2,1}\in C\\{\upsilon_{k2}\\}$. It means
$C\\{\upsilon_{k1}\\}$ and $\\{\upsilon_{k2}\\}$ represent half-plane whose
join is all $\mathbb{R}^{2}$, what is impossible. That means $c_{1}>0$.

Let now $\upsilon_{3,1}$ be non-collinear neither to $\upsilon_{1,1}$ nor to
$\upsilon_{2,1}$. If $c_{1}<0,c_{2}<0$, then in $G_{1}$ (38) is not satisfied.
If $c_{1}<0,c_{2}>0$, then in $G_{2}$ there
$\upsilon_{3,2}=c_{1}\upsilon_{1,2}+(-c_{2})\upsilon_{2,2}$, i.e again (38) is
not satisfied. Analogous reasonings hold for the case $c_{1}>0,c_{2}<0$, i.e.
the only possible case is $c_{1}\geq 0,\ c_{2}\geq 0$ and therefore
$\upsilon_{3,1}\in C\\{\upsilon_{1,1},\upsilon_{2,1}\\}$ (where $C\\{x,y\\}$
is a convex cone, stretched on the vectors $x$ and $y$).

The case is left when the directions $\upsilon_{1,1}$ and $\upsilon_{2,1}$
coincide.

Without loss of generality one can assume non-collinearity of $\upsilon_{3,1}$
and $\upsilon_{1,1}$. Then $\upsilon_{2,2}$ and $\upsilon_{3,2}$ are boundary
directions in $C\\{\upsilon_{k2}\\}$. Consequently, $\upsilon_{1,1}\in
C\\{-\upsilon_{1,1},\upsilon_{3,1}\\}$ i.e. the directions $\upsilon_{3,1}$
and $\upsilon_{1,1}$ coincide contrarily to the assumption. It means that if
$\upsilon_{rr}$ and $\upsilon_{(r+1),r}$ are co-directed, all $\upsilon_{kr}$
are collinear, i.e. all $\varphi_{k}$ are collinear. In this case the
directions $\upsilon_{rr}$ and $\upsilon_{(r+1),r}$ are also boundary.

We call the obtained fact the boundariness condition.

Since all $\psi_{k}$ lie clockwise in one half-plane, then it is easy to check
that in sector $G_{m}$ either all $(x;\psi_{k})\geq 0$ for all $k$ or
$(x;\psi_{k})\leq 0$ for all $k$. Thus, by virtue of (32), all $\varphi_{k}$
lie in one half-plane. From the boundariness condition follows that
$\varphi_{k}\in C\\{\varphi_{k-1},\varphi_{k+1}\\}$, i.e vectors $\varphi_{k}$
are arranged either clockwise, or anti-clockwise.

Let, for example, $\upsilon_{1,1}=\varphi_{1}$ (the case
$\upsilon_{1,1}=-\varphi_{1}$ is considered analogously). Then
$\upsilon_{2,1}=\varphi_{2}$ lies in the half-plane bounded by the straight
line stretched on $\upsilon_{1}$ and containing $\hat{S}_{1}$. Therefore the
direction from $\varphi_{1}$ to $\varphi_{2}$ in the half-plane containing all
$\varphi_{k}$ is the same as from $\psi_{1}$ to $\psi_{2}$, i.e. clockwise.

The necessity of the other conditions is obvious, since (34)-(35) is simply a
part of conditions (33).

Sufficiency. Let the family $\\{\varphi_{k}\\}_{k=1}^{m}$ be arranged
clockwise in one half-plane and the conditions (32) and (34)-(35) be
satisfied. Assume that among $\varphi_{k}$ there are non-collinear vectors,
and among $\psi_{k}$ there are no collinear ones.

The condition of clockwise arrangement of $\varphi_{k}$ in one half-plane
means that the angle (counted from $\varphi_{1}$ clockwise) between
$\varphi_{1}$ and the vectors $\varphi_{1},\varphi_{2},\ldots,\varphi_{m},$
$\varphi_{m+1}=-\varphi_{1}$ monotonously increases from $0$ to $\pi$. Taking
into account that the angle between $\varphi_{k1}$ and $(-\varphi_{k2})$ is
the angle between $\varphi_{k1}$ and $\varphi_{k2}$, taken with opposite sign,
it is easy to conclude that systems $\\{\upsilon_{kr}\\}$ (in each sector) lie
in one half-plane and are arranged clockwise (to avoid exiting from
corresponding half-plane we start counting in sector $G_{r}$ from
$\upsilon_{(r+1),r}$).

From conditions (34) follows that in each sector there is a ”convex
configuration”, i.e. there is vector $x\in G_{r}$, representable in the form

$x=-\sum_{k=1}^{m}c_{k}\upsilon_{kr},$

where all $c_{k}>0$.

It means that if from one point $\tilde{x}\in int\ G_{r}$ one draws segments
$\bar{a}$ and $\bar{b}$ in the directions of $\upsilon_{rr}$ and
$\upsilon_{(r+1),r}$ up to the crossing with $l_{r}$ and $l_{r+1}$,
respectively, then these segments together with the segments connecting $0$
with the point of crossing $\bar{a}$ with $l_{r}$ and $\bar{b}$ with
$l_{r+1}$, respectively, form a convex polygon (if $\upsilon_{rr}$ and
$\upsilon_{(r+1),r}$ are oppositely directed, it will be a triangle, and if
they are non-collinear – a quadrangle; as we have seen before they cannot be
co-directed).

Due to the same orientation of $\\{\varphi_{k}\\}$ and $\\{\psi_{k}\\}$ all
the other $\upsilon_{kr}$ are directed (from point $\tilde{x}$) into this
polygon, i.e. for any cone $C\\{\upsilon_{kr}\\}$ directions on the straight
lines $\upsilon_{r}$ and $\upsilon_{r+1}$ are boundary.

Fix now the point $x_{0}\in l_{1}\ (x_{0}\neq 0)$ on the boundary ray of
sector $G_{1}$ (actually, one can begin from any straight line $l_{k}$; we
begin from $l_{1}$). Due to the boundariness condition either direction from
$x_{0}$ on $l_{2}$ goes into sector $G_{1}$, or direction from $x_{0}$ on
$l_{m}$ goes into $\hat{G}_{m}$.

If one and only one of these statements is true, continue moving in the
corresponding direction (to the neighboring straight line) till the direction
on the neighboring straight line goes into the neighboring sector. In other
words, move from $l_{r}$ to $l_{r+1}$ in the direction parallel to
$\varphi_{r}$, if this direction goes into sector $G_{r}$ (or, into
$\hat{G}_{r-1}$, respectively). As a polygon $W$ mentioned after (37) one
should take a polygon formed by the segments which we moved along, and the
segments of those straight lines on which the movement broke (if exit on the
initial ray did not occur, in our case it is a part of $l_{1}$ corresponding
to $G_{1}$, then it is a segment connecting $x_{0}$ with $0$, and a segment of
that straight line on which the movement broke, connecting the point of
breaking with zero; if exit on the initial ray occured, then it is a segment
connecting $x_{0}$ with the point of exit).

If both statements are satisfied, then as $W$ one can take a join of two such
polygons formed in moving to both sides from $x_{0}$.

This algorithm is easy to check proceeding from boundariness conditions,
”convex configuration”, and (35) (the latter condition means that if exit on
the initial ray occured in moving in either side, then the point of exit is no
farther from the beginning of coordinates than the initial point; in
particular, if the point of exit coincides with the initial point, then the
formed polygon can be taken as $W$). The ball of the sought for norm is a
polygon.

If some of $\psi_{k}$ are collinear, then some sectors $G_{k}$ are singular.
This, however, does not change the results. The reasonings are analogous to
the case when among $\psi_{k}$ there are no collinear vectors. The only
difference here is the following: some straight lines $l_{k}$ correspond to
several directions $\\{\varphi_{j}\\}_{j=k_{0}}^{k_{1}}$. Then in constructing
$W$ one needs to move along $\varphi_{k_{0}}$.

In the case when all $\varphi_{k}$ (or all $\psi_{k}$) are collinear (see
corollary 1), all the same one can regard that $\\{\varphi_{k}\\}$ and
$\\{\psi_{k}\\}$ have the same orientation, starting from (32).

Conditions (34)-(35) are satisfied in this case. The norm can be chosen
polyhedral, if one chooses a polyhedral norm as $\|.\|_{0}$ in (30). The
theorem is proved.

Remark 12. One can obtain the arrangement of vectors $\psi_{k}$ required by
the conditions of theorem 9 by renumbering vectors and (if it is necesary)
changing signs of some of them.

Thus, the problem of simultaneous dissipativity of a family of operators of
rank 1 in $\mathbb{R}^{2}$ is solved completely. The number of conditions to
be checked now, in contrast to (33), is only of the order $O(m)$.

With theorem 9 one can study the MAL mechanism on dissipativity (and,
respectively, on the absence of IME). For example, let the mechanism be

$A_{1}\to A_{2},\ A_{1}\to A_{3},\ A_{2}\to A_{1},\ A_{2}\to A_{3},$
$3A_{2}\to A_{1}+2A_{3},\ 2A_{1}\to A_{2}+A_{3},\ 2A_{2}\to A_{1}+A_{3},$
$2A_{3}\to A_{1}+A_{2},\ 3A_{1}\to A_{2}+2A_{3},\ 3A_{2}\to 2A_{1}+A_{3},$
$A_{1}+A_{2}\to 2A_{3}.$ $None$

This mechanism possesses positive conservation law $c_{1}+c_{2}+c_{3}={\rm
const}$. The corresponding subspace is the plane

$c_{1}+c_{2}+c_{3}=0.$

Obviously, $dimL=2$, and one can use theorem 9. Writing matrices
$M^{\prime}_{ri}$ and using theorem 9, let make sure that mechanism (39) is
dissipative. The corresponding norm in the subspace $L$ has the form

$\|c\|=|c_{1}|+|c_{2}|.$

It can be expanded onto all $\mathbb{R}^{3}$, for example, in this way:

$\|c\|=|c_{1}|+|c_{2}|+|c_{1}+c_{2}+c_{3}|.$

To complete the section, consider the question of simultaneous dissipativity
of the finite family of operators of rank 1 of special form in
$\mathbb{R}^{n}$ for arbitrary $n$. Namely, we consider operators represented
by matrix-columns. Let obtain sufficient conditions of simultaneous
dissipativity of such operators.

Let the basis $\\{e_{k}\\}_{k=1}^{n}$ and the norm

$\|x\|=\sum_{k=1}^{n}p_{k}|x_{k}|$ $None$

be given in $\mathbb{R}^{n}$, where $p_{k}>0\ (k=1,\ldots,n),\ x_{k}\ $ is the
$k$-th coordinate of vector $x$ in the basis $\\{e_{k}\\}$. Norm (40)
coincides with $l^{1}$ norm with recpect to the basis $\\{e_{k}/p_{k}\\}$.
Therefore, the necessary and sufficient dissipativity conditions of the
operator $A$ represented by the matrix $(a_{ij})_{i,j=1}^{n}$ according to
remark 9 have the form

$p_{i}a_{ii}+\sum_{j\neq i}p_{j}|a_{ji}|\leq 0\ (i=1,\ldots,n).$ $None$

Let now there be a family of operators, represented by the matrix-columns
$A_{kl_{k}}$ $(k=1,\ldots,n;\ l_{k}=0,\ldots,r_{k}),$ where $A_{kl_{k}}$ is
the $l_{k}$-th matrix with non-zero $k$-th column:

$A_{kl_{k}}=\left(\begin{array}[]{ccccccc}0&\ldots&0&a_{1k}^{(l_{k})}&0&\ldots&0\\\\[11.38109pt]
0&\ldots&0&a_{2k}^{(l_{k})}&0&\ldots&0\\\\[11.38109pt]
&\ldots&&\ldots&&\ldots&\\\\[11.38109pt]
0&\ldots&0&a_{nk}^{(l_{k})}&0&\ldots&0\end{array}\right).$ $None$

Coming from (41), write dissipativity conditions of all operators in norm (40)
with some constants $p_{k}$:

$p_{k}a_{kk}^{(l_{k})}+\sum_{j\neq k}p_{j}|a_{jk}^{(l_{k})}|\leq 0\
(k=1,\ldots,n;\ l_{k}=0,\ldots,r_{k}).$ $None$

Theorem 10. If the system of linear inequalities (43) complemented by the
inequalities

$p_{k}>0\ (k=1,\ldots,n)$ $None$

has a solution, then the family of operators represented by matrices (42) is
simultaneously dissipative.

Proof. Solvability of the systems (43)-(44) means the existence of positive
constants $p_{k}\ (k=1,\ldots,n)$ for which inequalities (43) are satisfied,
and that is dissipativity condition of all operators of the family in norm
(40). Obviously, in this case the family is dissipative. The theorem is
proved.

Thus, for simultaneous dissipativity of finite family of operators represented
by matrices-columns the solvability of above written finite system of linear
inequalities proves to be sufficient. To check solvability, one can use
algorithms of linear programming [16].

Remark 13. The solution of the system (43)-(44) exists if there exists
solution of the system of $(n-d)$ linear inequalities complemented by
inequalities (44) (where $d$ is the number of those $k$ for which $r_{k}=0$;
evidently $0\leq d\leq n-1$). To prove this, assume

$a_{kk}=\max_{0\leq l_{k}\leq r_{k}}a_{kk}^{(l_{k})};\ a_{jk}=\max_{0\leq
l_{k}\leq r_{k}}|a_{jk}^{(l_{k})}|\ (j\neq k)$ $(k=1,\ldots,n).$

Consider the system

$p_{k}a_{kk}+\sum_{j\neq k}p_{j}a_{jk}\leq 0\ (k=1,\ldots,n).$ $None$

Obviously, if the set $\\{p_{k}\\}$ satisfies the system (44)-(45), then it
satisfies the system (43)-(44) as well. Numbers $k$ for which $r_{k}=0$ are
excluded. Therefore, in system (45) there are $(n-d)$ inequalities.

Remark 14. For $n=2$ theorem 10 provides necessary and sufficient conditions
of simultaneous dissipativity. To demonstrate that, note that for operator
$M_{k}$ of the considered form the vector $\psi_{k}$ (see the notation at the
beginning of the subsection) is directed along one of the coordinate axes.
Therefore (see the proof of sufficiency in theorem 9), if the family is
simultaneously dissipative, then one can choose parallelogram as a ball of the
corresponding norm, with vertices on coordinate axes, i.e. the norm is of the
form (40). In the case of arbitrary $n$ the conditions of theorem 10 are
already not necessary. To see this, let

$A_{1}=\left(\begin{array}[]{rrr}-1&0&0\\\\[11.38109pt] -1&0&0\\\\[11.38109pt]
1&0&0\end{array}\right);\
A_{2}=\left(\begin{array}[]{rrr}0&-1&0\\\\[11.38109pt] 0&-1&0\\\\[11.38109pt]
0&1&0\end{array}\right).$

The system of linear inequalities

$\left\\{\begin{array}[]{rcl}-p_{1}+p_{2}+p_{3}&\leq&0;\\\\[11.38109pt]
p_{1}-p_{2}+p_{3}&\leq&0\end{array}\right.$

has no positive solutions. Nevertheless, simultaneous dissipativity exists,
since each of the operators is dissipative in its norm and
$\varphi_{1}=\varphi_{2}$ (see corollary 1).

Conclusion

Let us resume. The infinitesimal Moore effect (IME) in the interval space for
smooth autonomous system on positively invariant convex compact is studied.
The local conditions of absence of IME in terms of Jacobi matrices field of
the system are obtained. The relation between the absence of IME and
simultaneous dissipativity of the Jacobi matrices is established, and some
sufficient conditions of simultaneous dissipativity are obtained.

On the basis of the conducted analysis the reason of weak efficiency of
interval stepwise methods is pointed out. The main reason is that to solve the
problem of absence of IME in the system and to construct corresponding
interval space one needs analysis of simultaneous dissipativity of Jacobi
matrices of system and constructing a constricting norm. The latter questions
are rarely solved constructively. Besides, in sufficiently rich interval
spaces (for example, in using standard intervals – rectangular
parallelepipeds) IME is almost always present. One should, however, remember
that the notion of the Moore effect in the work is treated sufficiently
strongly. The final conclusion on the efficiency of stepwise interval methods
can be drawn only after studying asymptotic Moore effect (AME). It should also
be noted that there may be definitions of interval spaces, different from
definition 1.

Some particular classes of systems without IME and corresponding interval
spaces are pointed out. These results can be used in solving by interval
methods particular systems from the pointed out classes.

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