# Positive Invertibility of Matrices and Exponential Stability of Linear Stochastic Systems with Delay

Authors: Ramazan Kadiev; Arcady Ponosov
Journal: International Journal of Differential Equations
 (2022)

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## Abstract

The work addresses the exponential moment stability of solutions of large systems of linear differential Itô equations with variable delays by means of a modified regularization method, which can be viewed as an alternative to the technique based on Lyapunov or Lyapunov-like functionals. The regularization method utilizes the parallelism between Lyapunov stability and input-to-state stability, which is well established in the deterministic case, but less known for stochastic differential equations. In its practical implementation, the method is based on seeking an auxiliary equation, which is used to regularize the equation to be studied. In the final step, estimation of the norm of an integral operator or verification of the property of positivity of solutions is performed. In the latter case, one applies the theory of positive invertible matrices. This report contains a systematic presentation of how the regularization method can be applied to stability analysis of linear stochastic delay equations with random coefficients and random initial conditions. Several stability results in terms of positive invertibility of certain matrices constructed for general stochastic systems with delay are obtained. A number of verifiable sufficient conditions for the exponential moment stability of solutions in terms of the coefficients for specific classes of Itô equations are offered as well.


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## Body

## 1. Introduction

Stability analysis of stochastic delay equations is quite popular due to its numerous applications. It is therefore impossible to give a more or less extensive overview of the topic. Some of the results are summarized in the monograph [1]; others can be found in more recent works of the author of this monograph as well as in many other publications. Indisputably, the main method of stability analysis is based on Lyapunov functions and their generalizations. However, this method may not be applicable in certain situations or may give too restrictive conditions, both in the deterministic and stochastic cases. It is particularly challengeable to use the Lyapunov framework in the case of complicated delays or/and random coefficients and initial conditions (see, e.g., discussion in the recent paper [2]). Yet, many applications require such kinds of models, e.g., in mathematical finance, especially when modeling stock prices, interest rates, or volatilities (see, e.g., [3] and the references therein), stochastic control theory [4–6], semi-Markov systems [7], epidemic models [8], and many others.Stability analysis used in this paper is based an alternative approach utilizing a regularization technique. This approach has been successfully used by several authors in the theory and applications of deterministic and stochastic delay equations. Its theoretical foundations are presented in the monograph [9], and since then numerous applications of the method have been investigated in a number of papers. Among the recent ones are the publications [10] (the Mackey–Glass equation), [11, 12] (control theory), [13] (neural networks), [14] (hyperjerk systems), and [15] (the stochastic pantograph equation).The regularization method has its roots in the theory of the input-to-state stability, which usually implies the Lyapunov stability (see, e.g., [16]). As soon as this relationship between the two types of stability is established, the algorithm starts with choosing a simpler equation (called a reference equation) that is assumed to already have the required stability properties. Resolved and substituted into the original equation, the reference equation produces a new integral equation of the form x−Θx=fx. If the latter equation is solvable (for instance, if Θ<1), then stability of the original equation is proven. Thus, the method is similar to Lyapunov’s direct method, but instead of seeking a Lyapunov function(al), one aims to first find a suitable reference equation possessing the prescribed asymptotic property. The reference equation is then used to regularize the original equation. The method proven to be particularly efficient when constructing Lyapunov function(al) may be technically difficult (random coefficients and delays, distributed delays, unbounded delays, complicated noises, and so on).In [13, 14, 17], the regularization method was combined with the estimation technique based on positive invertible matrices. This led to new, verifiable stability conditions in the case of differential equations with variable, in particular, distributed delays. In general, the approach based on ℳ–matrices gives better stability results as the technique utilizing the norm estimation (see, e.g., [15, 17]). In the above publications, verifiable stability conditions were formulated in terms of the coefficients of the systems in question.The present paper is a continuation of the authors’ work started in [16, 17]. Minding future applications of the method (see Section 8), we offer a general framework combining the regularization approach from [9] with the theory of positive invertible matrices. After that, we demonstrate how this scheme can be applied to (rather complicated) systems of Itô equations with variable delays and random coefficients, i.e., in the situations where the Lyapunov-like functionals may be difficult to construct. A motivation to study such systems goes back to several sophisticated stochastic models used, for instance, in population dynamics. Thus, equations for the aggregated state variables derived from the McKendrick–von Foerster equation for structured populations always contain random, distributed delays, random coefficients, and random initial conditions [14], so that the Lyapunov framework may be difficult to apply.For the sake of simplicity, we chose to describe the framework for the case of linear equations and exponential stability, as it was done in the monograph [9]. Even if the most realistic models are nonlinear, we stress that the methods of the present paper, primarily developed for the linear case, can be directly used to study local stability of stochastic nonlinear equations. Moreover, the regularization technique can be applied to global stability of nonlinear deterministic and stochastic delay equations. This has been clearly shown in many publications based on the regularization method (see, e.g., the monograph [9] and the references therein as well as the recent publications [10, 11, 13, 14, 17]).The presentation of the regularization method will be incomplete without mentioning the recent publications [18–20], where many profound stability results were obtained. The paper [20] deserves a special attention, as the authors regularize, in our terminology, nonlinear stochastic equations by means of nonlinear reference equations with the known stability properties. This approach is close to the one used in the present paper, but does not exploit ℳ–matrices. It seems to be a fruitful idea to combine both techniques to obtain more general stability results and to cover more applications.The paper is organized as follows. We start with some preliminary information and the notation (Section2). Then, we introduce the system of stochastic delay equations we intend to study as well as the main assumptions on it (Section 3). In Section 4, we offer a brief description of the regularization method and its adjustment based on the input-to-state stability and the theory of positive invertible matrices. We remark that the framework described in this section goes far beyond the tasks of the present paper. It can be also used in the case of arbitrary semi-martingales, unbounded delays, nonlinear systems, etc. We chose this level of generality because we intend to apply the framework to many other equations and models in the future. Section 5 contains formulation and the proof of the main result of the paper. It provides sufficient conditions (in terms of the coefficients) of exponential 2p-stability p≥1 of the system introduced in Section 3. In Section 6, the main result is specified for some particular cases of the general system. We believe that even in the case of deterministic equations, we produced some new results. Section 7 contains some examples illustrating the stability conditions offered in the previous section. Finally, in Section 8, we provide a short summary and describe some open problems.


## 2. Preliminaries

LetΩ,ℱ,ℱtt≥0,P be a stochastic basis, where Ω is set of elementary probability events, ℱ is a σ-algebra of all events on Ω, ℱtt≥0 is a right continuous family of σ-subalgebras of ℱ, and P is a probability measure on ℱ; all the above σ-algebras are assumed to be complete w.r.t. P, i.e., containing all subsets of zero measure.The following notational agreements are used throughout the paper:(i)
R=−∞,∞ is the set of all real numbers.(ii)
colx1,…,xn∈Rn is a column vector.(iii)
. is an arbitrary yet fixed norm in Rn, . being the associated matrix norm.(iv)
E¯ is the n×n-identity matrix.(v)
μ is the Lebesgue measure on 0,+∞.(vi)
.X is the norm in a normed space X.(vii)
p is an arbitrary real number satisfying 1≤p<∞.(viii)
ℬI is the σ-algebra of all Borel subsets of an interval I⊂R.(ix)
ℬ2,…,ℬm is the standard m−1-dimensional Brownian motion (i.e., the scalar Brownian motions ℬl are independent).(x)
Zt≔colt,ℬ1t,…,ℬmt (only used in Section 4).(xi)
The expectation (the integral w.r.t. the measureP) is denoted by E.(xii)
kn is the linear space of all n–dimensional, ℱ0–measurable random values.(xiii)
kqn=α:α∈kn,αkqn≡Eαq1/q<∞.(xiv)
cp is the constant from inequality (1) used in the stability conditions.Recall that aℬI⊗ℱ–measurable stochastic process ξt=ξt,ω, t∈I, is called ℱt–adapted if ξt,⋅ is ℱt-measurable for all t∈I.Am×m-matrix B=biji,j=1m is said to be nonnegative, resp. positive if bij≥0 resp. bij>0 for all i,j=1,…,m.The following definition is crucial for what follows.Definition 1.
A matrixΓ=γiji,j=1n is called a (nonsingular) ℳ–matrix if γij≤0 for i,j=1,…,n, i≠j, and all the principal minors of the matrix Γ are positive.
According to [21], a matrix B is a nonsingular ℳ–matrix if bij≤0 for all i,j=1,…,m, i≠j, and there exist positive numbers ξi, i=1,…,m, such that one of the following conditions is fulfilled:(1)
ξibii>∑j=1,i≠jmξjbij, i=1,…,m.(2)
ξjbjj>∑i=1,i≠jmξibij, j=1,…,m.
In particular, ifξi=1, i=1,…,m, in the first of the above conditions, then we obtain the class of strictly diagonally dominant matrices. In [21], one can find plenty of characterizations of nonsingular ℳ–matrices.
One of the most important properties of nonsingularℳ–matrices states the following.
The inverse of a nonsingularℳ–matrix is positive. In what follows we will always silently assume that any ℳ–matrix is nonsingular.
The next two lemmas contain inequalities to be used in this paper.Lemma 1.
(1)E∫0tfsdℬls2p1/2p≤cpE∫0tfs2dsp1/2p,for any ℱt-adapted stochastic process fs0≤s≤t, any t>0, and any component ℬls1≤l≤m of the Brownian motion ℬ.Proof.
Inequality (1) follows directly from inequality (5) in [22, p.65], where one can find explicit formulas for cp.Lemma 2.
Letgs be a scalar function that is square integrable on 0,∞ and fs be an ℱt-adapted stochastic process satisfying sups≥0Efs2p1/2p<∞. Then,(2)supt≥0E∫0tgsds2p1/2p≤supt≥0∫0tgsdssupt≥0Eft2p1/2p,and(3)supt≥0E∫0tgs2fs2dsp1/2p≤supt≥0∫0tgs2ds1/2supt≥0Eft2p1/2p.Proof.
We only prove inequality (2), as inequality (3) can be justified similarly.(4)supt≥0E∫0tgsds2p1/2p≤supt≥0E∫0tgsfsds2p1/2p≤supt≥0E∫0tgs2p−1/2pgs1/2pfsds2p1/2p≤supt≥0E∫0tgsds2p−1∫0tgsfs2pds1/2p≤supt≥0∫0tgsds2p−1∫0tgsEfs2pds1/2p≤supt≥0∫0tgsdssupt≥0Eft2p1/2p.


## 3. The Main System and Formulation of the Problem

We will study exponential stability of the following system of Itô delay differential equations(5)dxt=−∑j=1m1A1jtxh1jtdt+∑i=2m∑j=1miAijtxhijtdℬitt≥0,equipped with the initial conditions(6)xt=φtt<0, (7)x0=b.Remark 1.
According to [9], we separate the initial conditions for t<0 and t=0, as we do not require the continuity of the function φt. This function is assumed to be only bounded and measurable, so that changing its value at countably many points does not change the solution of equation (5). On the other hand, it can be easily checked by examples that changing the value of x0=b usually changes the solution of equation (5). That is, the space of initial functions φ and the space of initial values b have different topologies.
The following assumptions are put on (5)–(7) throughout the paper:(1)
b=colb1,..,bn is an n-dimensional ℱ0–measurable random variable belonging to the space b∈kn.(2)
φ=colφ1,…,φn is an n-dimensional ℱ0–measurable stochastic process with essentially bounded trajectories, defined on the interval −σ,0, where σ=maxτij:i=1,…,m,j=1,…,mi.(3)
x=colx1,…,xn is an unknown n-dimensional stochastic process defined for t≥−σ; it is ℱt-adapted for t≥0 and ℱ0-measurable for −σt<0.(4)
Aij=aslijs,l=1n are n×n –matrices for i=1,…,m, j=1,…,mi, and the entries of the matrices A1j, j=1,…,m1, are scalar, ℱt-adapted stochastic processes with almost surely Lebesgue integrable trajectories, while the entries of the matrices Aij, i=2,…,m, j=1,…,mi, are scalar, ℱt-adapted stochastic processes with almost surely square integrable trajectories.(5)
hij, i=1,…,m, j=1,…,mi, are Borel measurable scalar functions defined on 0,∞ satisfying the inequalities 0≤t−hijt≤τijt∈0,∞,i=1,…,m,j=1,…,miμ–everywhere for some constants τij, i=1,…,m, j=1,…,mi.Remark 2.
It can be proven that under the assumptions 1–5, the initial value problem ((5)–(7)) has a unique (up to the natural equivalence of stochastic processes) continuous, ℱt-adapted solution xt,b,φ.Definition 2.
We say that equation (5) is exponentially q–stable 0<q<∞ with respect to the initial data, i.e., the initial value x0 and the “prehistory” function   φ, if there are positive numbers K,λ such that all solutions xt,b,φ of the initial value problem ((5)–(7)) satisfy(8)Ext,b,φq1/q≤Kexp−λtEbq1/q+esssupt<0Eφtq1/qt≥0.
In the next section, we describe the method of our analysis. We remark that this description is very general and goes far beyond the particular case of equation (5).


## 4. Regularization Method in Stability Analysis

In this section, we briefly describe a framework which we use to study stability properties of stochastic functional differential equations. The main idea of this approach is to convert the property of Lyapunov stability to the property of invertibility of certain operators in suitable functional spaces.In what follows, we putZt≔colt,ℬ1t,…,ℬmt. We remark, however, that the framework described below is valid for any m-dimensional semi-martingale Z (see, e.g., [16]).We consider a general linear homogeneous stochastic hereditary equation(9)dxt=VhxtdZtt≥0,equipped with two initial conditions(10)xs=φss<0, (11)x0=b.HereVh is a k-linear Volterra operator (see below) defined in certain linear spaces of vector stochastic processes, φ is an ℬ−∞,0⊗ℱ0-measurable stochastic process, and b∈kn. By k-linearity of the operator Vh we mean the following property:(12)Vhα1x1+α2x2=α1Vhx1+α2Vhx2,holding for all ℱ0-measurable, bounded, and scalar random values α1,α2 and all stochastic processes x1,x2 belonging to the domain of the operator Vh.It is easy to see that equation (5) is a particular case of equation (9).The solution of the initial value problem ((9)–(11)) will be denoted by xt,b,φ, t∈−∞,∞. Below the solution is always assumed to exist and to be unique for an appropriate choice of φs,b.We also need an adaptation of Definition1 to the case of equation (9).Definition 3.
For a given real numberq0<q<∞, we call the zero solution of equation (9)(i)
q-stable (with respect to the initial data b and φ ) if for any ϵ>0 there is δϵ>0 such that Ebq+esssups<0Eφsq<δ implies Ext,b,φq≤ϵ for all t≥0 and all ℱ0-measurable φ, b.(ii)
Exponentiallyq-stable if there exist positive constants K, λ such that the inequality(13)Ext,b,φq≤KEbq+esssups<0Eφsqexp−λt,holds true for all t≥0 and all ℱ0-measurable φ, b.
Introducing theq-norm in the space Φq of the prehistory functions φ by(14)φq=esssups<0Eφsq,and minding the norm in the space of the initial values defined in the previous section yield a shorter version of Definition 3, where the expression Ebq+esssups<0Eφsq is replaced by bkqn+φq.
To describe the regularization method, we need to represent (9) and (10) in a slightly different form. Let xt be a stochastic process 0,+∞ and x+t be a stochastic process on −∞,+∞ coinciding with xt for t≥0 and equalling 0 fort<0, while let φ−t be a stochastic process on −∞,+∞ coinciding with φt for t<0 and equalling 0 fort≥0. Then, the stochastic processx+t+φ−t, defined for t∈−∞,+∞, will be a solution of the (9)–(11) if xtt∈0,+∞ satisfies the initial value problem(15)dxt=Vxt+ftdZtt≥0,(16)x0=b,where Vxt≔Vhx+t, ft≔Vhφ−t for t≥0. Indeed, by k-linearity we have that Vhx++φ−=Vhx++Vhφ−=Vx+f yielding (15). Note that f is uniquely defined by the stochastic process φ, “the prehistory function.” Let us also observe that the initial value problem ((15) and (16)) is equivalent to the initial value problem ((9)–(11)) only for f admitting the representation f=Vhφ−, where φ′ is an arbitrary extension of the function φ to the real line −∞,∞.
LetBq be a linear subspace of the space of ℱt-adapted stochastic processes with trajectories that are almost surely essentially bounded on 0,∞. The norm in this space is defined by(17)fBqq=esssupt≥0Eftq.
As we assume the existence and uniqueness property for equation (21) for all b∈kqn and f∈Bq, we can denote the corresponding solution by xft,b. Let M stand for the space of all solutions of equation (15), and we define its linear subspace Mp by(18)Mp=x:x∈M,supt≥0Extp<∞.
The construction above produces two linear operators:(19)ℒ1:φ↦Vhφ−t,(20)ℒ2:f↦xf⋅,b.
The next theorem is crucial for the framework (see, e.g., [12]). It says that the stochastic Lyapunov stability follows from the input-to-state stability (“the stochastic Bohl–Perron theorem”).Theorem 1.
Assume that the linear operatorsℒ1:Φq⟶Bq and ℒ2:Bq⟶Mq, defined by (17) and (18), respectively, are bounded. Then, the zero solution of equation (9) is q-stable in the sense of Definition 3.We remark that the operatorℒ1 is, as a rule, bounded, so that the only challenge in application of Theorem 1 is to prove boundedness of the operator ℒ2. This can be done by the regularization method, also known as “the W-method” [9, 16]. The regularization is usually constructed with the help of an auxiliary or reference equation(21)dxt=Qxt+gtdZtt≥0,where Q is again a k-linear Volterra operator. The reference equation is, thus, similar to equation (15), but it is supposed to be “simpler” in the sense that the required stability property for this equation is already proven (see condition (2) in Theorem 2 below).Assuming the existence and uniqueness property for equation (21), we get the following “variation-of-constants” formula for its solutions:(22)xt=Utx0+Wgtt≥0,where Ut is the fundamental matrix of the associated homogeneous equation and W is the corresponding Cauchy operator.Using representation (22), we can regularize equation (15) in two ways: on the right and on the left. In this paper, we only use the left regularization to be described below. The algorithm based on the right regularization is presented in [16].Using equation (21), we rewrite equation (15) as follows:(23)dxt=Qxt+V−Qxt+ftdZtt≥0,or, if we use the representation (22), as(24)xt=Utx0+WV−Qxt+Wftt≥0.DenotingWV−Q=Θ, we obtain the operator equation(25)xt=Θxt+Utx0+Wftt≥0.Theorem 2.
Assume that equation (15) and reference equation (21) satisfy the following conditions:(1)
The linear operatorsV, Q act continuously from Mq to Bq.(2)
The Cauchy operatorW in (22) constructed for reference equation (21) is bounded as an operator from Bq to Mq.(3)
The operatorI−Θ:Mq⟶Mq has a bounded inverse.Then, the operatorℒ1:Bq⟶Mq in (20) is bounded.Theorems1 and 2 justify the regularization framework in the analysis of Lyapunov stability for stochastic linear functional differential equations. In fact, all the conditions except condition (3) in Theorem 2 are usually fulfilled, and this is easy to check. The only real challenge is therefore the invertibility of the operator I−Θ. In [16] (see also the references therein), the invertibility of this operator is verified by estimating the norm of the integral operator Θ: if ΘMq<1 in the inequality(26)xMq≤ΘMqxMq+K1x0kqn+K2fBq,then equation (6) is q-stable due to Theorem 1. If one also proves that the equation remains q-stable after the substitution yt=expλtxt for some positive λ, then equation (9) becomes exponentially q-stable.However, calculation of the normΘMq might be challengeable, especially in the vector case and in the case of random coefficients. In [13], and later in [14, 17] for the stochastic case, it was suggested to use the properties of monotone operators. The main idea was to perform all the estimates componentwise, which is much simpler, and then to check the positive invertibility of a certain matrix. Below we offer a generic description of this method.Let(27)xt=colx1t,…,xnt,x¯i=supt≥0Exitq1/q,x¯=colx¯1,…,x¯n.Suppose that after estimating each component of vector equation (23), we arrived at the vector inequality(28)Dx¯≤x0kqne¯1+fBqe¯2,where D is an n×n–matrix and e¯1, e¯2 are some column n-vectors with nonnegative components. Typically, D=E¯−T, where E¯ is the n×n identity matrix, while T and e¯i replace Θ and Kii=1,2 in the scalar inequality (26), respectively. Then, we have the following.Theorem 3.
IfD is an ℳ–matrix in the sense of Definition 1, then the operator ℒ2:Bq⟶Mq in (20) is bounded.Proof.
AsD is an ℳ–matrix, the matrix D−1 is positive, and we can rewrite (28) as(29)x¯≤D−1x0kqne¯1+fBqe¯2.
Therefore,(30)x¯≤Kx0kqn+fBq,where K=D−1maxe1,e2. As xMq≤x¯, we conclude from (30) that x∈Mq and xMq≤Kbkqn+fBq for some positive K. Thus, the operator ℒ2:Bq⟶Mq is bounded.
In the next section, the scheme just outlined is applied to equation (5), i.e., for the system of Itô equations with variable delays and random coefficients. Using the boundedness of all delay functions in equation (5), the substitution yt=expλtxt for some positive λ, and Theorems 1 and 3 for q-stability of the equation for yt, we prove exponential q-stability of equation (5) for q=2p, p≥1.
We also remark that the second Lyapunov method may be difficult to use in this case.


## 5. The Main Result

The following conditions are supposed to be valid throughout this section:(C1) There exist positive numbersa¯slij,i=1,..,m,j=1,…,mi,s,l=1,…,n,, so that the coefficients of equation (5) satisfy the inequalities(31)aslijt≤a¯slij,t∈0,+∞,i=1,..,m,j=1,…,mi,s,l=1,…,n,μ×P–almost everywhere.(C2) For eachs=1,…,n, there is a nonempty subset Is⊂1,…,m1 and positive number a¯s, so that(32)∑k∈Isass1kt≥a¯s,t∈0,+∞,s=1,…,n,μ×P–almost everywhere.Then×n–matrix C is defined as follows:(33)css=1−1a¯s∑k∈Is∑j=1m1τ1ka¯ss1ka¯ss1j+cp∑k∈Is∑i=2m∑j=1miτ1ka¯ss1ka¯ssij+∑j=1,j∉Ism1a¯ss1j−cp2a¯s∑i=2m∑j=1mia¯ssij,s=1,…,n,csl=−1a¯s∑k∈Is∑j=1m1τ1ka¯ss1ka¯sl1j+cp∑k∈Is∑i=2m∑j=1miτ1ka¯ss1ka¯slij+∑j=1m1a¯sl1j−cp2a¯s∑i=2m∑j=1mia¯slij,s,l=1,…,n,s≠l.Theorem 4.
If conditions (C1) and (C2) are fulfilled and ifC is an ℳ–matrix, then equation (5) is exponentially 2p–stable in the sense of Definition 2.Proof.
We rewrite equation (5) and the initial condition (6) as a single system:(34)dx¯st=−∑j=1m1∑l=1nasl1jtx¯lh1jt+φ¯lh1jtdt+∑i=2m∑j=1mi∑l=1naslijtx¯lhijt+φ¯lhijtdℬitt≥0,s=1,…,n,where x¯st, s=1,…,n, are scalar, ℱt-adapted stochastic processes defined on −σ,∞ satisfying the condition x¯st=0 for t<0, while φ¯st, s=1,…,n, are scalar, ℱ0–measurable stochastic processes defined on −σ,∞ satisfying the conditions φ¯st=φst for t∈−σ,0 and φ¯st=0 for t∈0,+∞. Due to the assumed property of the existence and uniqueness of the solutions of the initial value problem ((34) and (7)), we can use the notation x¯t,b,φ¯ for its solution. It is straightforward to check that xt,b,φ=x¯t,b,φ¯ for t≥0, where xt,b,φ is the solution of (5)–(7).
The next step consists in introducing the new variablesyst=x¯stexpλt, where yst is defined on −σ,∞ and the number λ satisfies the inequalities 0<λ<mina¯s,s=1,…,n for all s=1,…,n. Clearly, yst=0 for t<0, and we obtain the following system:(35)dyst=λyst−∑j=1m1∑l=1nasl1jtexpλt−h1jtylh1jt+expλtφ¯lh1jtdt+∑i=2m∑j=1mi∑l=1naslijtexpλt−hijtylhijt+expλtφ¯lhijtdℬitt≥0,s=1,…,n.
Denotingηst=∑k∈Isass1ktexpλt−h1kt−λ for s=1,…,n and minding ∫h1kttdysτ=yst−ysh1kt,k∈Is, transform (35) to the system(36)dyst=−ηstyst+∑k∈Isass1ktexpλt−h1kt∫h1kttdysτ+∑k∈Isass1ktexpλtφ¯sh1kt+∑j=1m1∑l=1,l≠s,j∈Isnasl1jtexpλt−h1jtylh1jt+expλtφ¯lh1jtdt+∑i=2m∑j=1mi∑l=1naslijtexpλt−hijtylhijt+expλtφ¯lhijtdℬitt≥0,s=1,…,n.
Substituting the expression fordyst from the s-th equation in (35) into the s-th equation in (36), where s=1,…,n, leads to(37)dyst=−ηstyst+∑k∈Isass1ktexpλt−h1kt∫h1kttλysτ+∑j=1m1∑l=1nasl1jτexpλτ−h1jτylh1jτ+expλτφ¯lh1jτdτ+∑i=2m∑j=1mi∑l=1naslijτexpλτ−hijτylhijτ+expλτφ¯lhijτdℬiτ+∑k∈Isass1ktexpλtφ¯sh1kt+∑j=1m1∑l=1,l≠s,j∈Isnasl1jtexpλt−h1jtylh1jt+expλtφ¯lh1jtdt+∑i=2m∑j=1mi∑l=1naslijtexpλt−hijtylhijt+expλtφ¯lhijtdℬitt≥0,s=1,…,n.
Finally, settingmst,ς=exp−∫ςtμsζdζ,s=1,…,n, and remembering initial condition (7), carry (37) over to the system(38)yst=mst,0bs+∑k∈Is∫0tmst,ςass1kςexpλς−h1kς∫h1kςςλysτdτdς+∑k∈Is∑j=1m1∑l=1n∫0tmst,ςass1kςexpλς−h1kς×∫h1kςςasl1jτexpλτ−h1jτylh1jτ+expλτφ¯lh1jτdτdς+∑k∈Is∑i=2m∑j=1mi∑l=1n∫0tmst,ςass1kςexpλς−h1kς×∫h1kςςaslijτexpλτ−hijτylhijτ+expλτφ¯lhijτdℬiτdς+∑k∈Is∫0tmst,ςass1kςexpλςφ¯sh1kςdς+∑j=1m1∑l=1,l≠s,j∈Isn∫0tmst,ςasl1jςexpλς−h1jςylh1jς+expλςφ¯lh1jςdς+∑i=2m∑j=1mi∑l=1n∫0tmst,ςaslijςexpλς−hijςylhijς+expλςφ¯lhijςdℬiςt≥0,i=1,…,n.
To obtain estimates for the solutions of (38), we will adopt the following notation:(i)
y^s=supt≥0Eyst2p1/2p, s=1,…,n.(ii)
φ^s=esssupt<0Eφst2p1/2p, s=1,…,n.(iii)
φ=esssupt<0Eφt2p1/2p.
In addition, we will use the following inequalities:(i)
esssupt≥0Eexpλtφ¯lhijt2p1/2p≤expλτijesssupt<0Eφlt2p1/2pforl=1,…,n, i=1,…,m, j=1,…,mi.(ii)
mst,ς≤exp−a¯s−λt−ς,t∈0,+∞,ς∈0,tP−almost surely for alls=1,…,n.(iii)
∫0texp−a¯s−λt−ςdς≤1/a¯s−λ,s=1,…,n.(iv)
∫0texp−2a¯s−λt−ςdς≤1/2a¯s−λ,s=1,…,n.
Now, from (38) and inequalities (1)–(3), we obtain(39)y^s≤bsk2p1+λa¯s−λ∑k∈Isa¯ss1kexpλτ1kτ1ky^s+1a¯s−λ∑k∈Is∑j=1m1∑l=1na¯ss1kexpλτ1kτ1ka¯sl1jexpλτ1jy^l+φ^l+cpa¯s−λ∑k∈Is∑i=2m∑j=1mi∑l=1na¯ss1kexpλτ1kτ1ka¯slijexpλτijy^l+φ^l+1a¯s−λ∑k∈Isa¯ss1kexpλτ1kφ^s+1a¯s−λ∑j=1m1∑l=1,l≠s,j∈Isna¯sl1jexpλτ1jy^l+φ^l+cp2a¯s−λ∑i=2m∑j=1mi∑l=1na¯slijexpλτijy^l+φ^l,s=1,…,n.
From the definition, we have thatφ^j≤φ, j=1,…,n. Hence, (39) yields(40)y^s≤bsk2p1+∑l=1nNslλy^l+Msλφs=1,…,n,where(41)Nssλ≔λa¯s−λ∑k∈Isa¯ss1kexpλτ1kτ1k+1a¯s−λ∑k∈Is∑j=1m1a¯ss1kexpλτ1kτ1ka¯ss1jexpλτ1j+cp∑k∈Is∑i=2m∑j=1mia¯ss1kexpλτ1kτ1ka¯ssijexpλτij+∑j=1,j∉Ism1a¯ss1jexpλτ1j+cp2a¯s−λ∑i=2m∑j=1mia¯ssijexpλτij,s=1,…,n,Nslλ≔1a¯s−λ∑k∈Is∑j=1m1a¯ss1kexpλτ1kτ1ka¯sl1jexpλτ1j+cp∑k∈Is∑i=2m∑j=1mia¯ss1kexpλτ1kτ1ka¯slijexpλτij+∑j=1m1a¯sl1jexpλτ1j+cp2a¯s−λ∑i=2m∑j=1mia¯slijexpλτij,s,j=1,…,n,s≠l,Msλ≔1a¯s−λ∑k∈Is∑j=1m1∑l=1na¯ss1kexpλτ1kτ1ka¯sl1jexpλτ1j+cp∑k∈Is∑i=2m∑j=1mi∑l=1na¯ss1kexpλτ1kτ1ka¯slijexpλτij+∑k∈Isa¯ss1kexpλτ1k+∑j=1m1∑l=1,l≠s,j∈Isna¯sl1jexpλτ1j+cp2a¯s−λ∑i=2m∑j=1mi∑l=1na¯slijexpλτij,s=1,…,n.
Put nowyt=coly1t,…,ynt, y¯=coly¯1,…,y¯n, and Mλ=colM1λ,…,Mnλ and let Cλ=cijλi,j=1n be an n×n –matrix with the entries given by(42)cssλ=1−Nssλ,s=1,…,n,cslλ=−Nslλ,s,l=1,…,n,s≠l.
From (40), we then deduce the following estimate:(43)Cλy¯≤bk2pne¯+φMλ,where e¯=col1,…,1. Evidently, C0=C. According to the assumptions of the theorem, C is an ℳ–matrix, so that Cλ0 is also an ℳ–matrix for small λ0>0. Thus, from (43) and Theorem 3, we obtain(44)y¯≤Kbk2pn+φ,for some constant K. Combining the substitution xt,b,φ=exp−λtyt with the inequalities supt≥0Eyt2p1/2p≤y¯ and (44), we get the estimate(45)Ext,b,φ2p1/2p≤Kexp−λtbk2pn+esssupt<0Eφt2p1/2pt∈0,∞.
Hence, equation (5) is exponentially 2p–stable in the sense of Definition 2.
The theorem is proven.Remark 3.
Whether or not the matrixC is an ℳ–matrix can be verified by calculation of its principal minors: if all of them are positive, then C is an ℳ–matrix. Otherwise, one can use the sufficient conditions described right after Definition 1.


## 6. Some Corollaries

In this section, we produce some sufficient conditions for exponential stability of (5).Let us start with the following particular case of equation (5):(46)dxt=−∑j=1m1A1jtxh1jtdtt≥0.We define then×n–matrix C1=csl as follows:(47)css=1−1a¯s∑k∈Is∑j=1m1τ1ka¯ss1ka¯ss1j+∑j=1,j∉Ism1a¯ss1j,s=1,…,n,csl=−1a¯s∑k∈Is∑j=1m1τ1ka¯ss1ka¯sl1j+∑j=1m1a¯sl1j,s,l=1,…,n,s≠l.From Theorem4, we readily obtain the following.Corollary 1.
If condition (C1) withi=1 and condition (C2) are fulfilled and C1 is an ℳ–matrix, then equation (46) is exponentially 2p–stable in the sense of Definition 2.In particular, for the system(48)dxt=−A11txh11tdtt≥0,we obtain the following.Corollary 2.
If condition (C1) withi=1, j=1 and condition (C2) with m1=1, Is=1s=1,…,n are fulfilled and C2=csl given by(49)css=1−a¯ss112τ11a¯s,s=1,…,n,csl=−a¯ss112τ11+a¯sl11a¯s,s,l=1,…,n,s≠l,is an ℳ–matrix, then equation (48) is exponentially 2p–stable in the sense of Definition 2.Remark 4.
Systems (46) and (48) consist of random delay equations or, if the coefficients A1j,j=1,..,m1 are all nonrandom, they become deterministic delay systems. Even in these cases, the results in the two above corollaries seem to be new.
Let us now consider the following system of stochastic delay equations:(50)dxt=−A11txh11tdt+∑i=2m∑j=1miAijtxhijtdℬitt≥0.
We define the matrixC3=csl as(51)css=1−1a¯sa¯ss1k2τ11+cp∑i=2m∑j=1miτ11a¯ss11a¯ssij−cp2a¯s∑i=2m∑j=1mia¯ssij,s=1,…,n,(52)csl=−1a¯sa¯ss11a¯sl11τ11+a¯sl11+∑i=2m∑j=1miτ11a¯ss11a¯slij−cp2a¯s∑i=2m∑j=1mia¯slij,s,l=1,…,n,s≠l.Corollary 3.
If condition (C1) and condition (C2) withm1=1, Is=1s=1,…,n are fulfilled and C3 is an ℳ–matrix, equation (50) is exponentially 2p–stable in the sense of Definition 2.In particular, we have the following.Corollary 4.
Ifn=2, condition (C1) and condition (C2) with m1=1, Is=1s=1,2 are fulfilled and c11>0,c11c22>c12c21, where csls,l=1,2 are defined in (51) and (52); then, equation (50) is exponentially 2p–stable in the sense of Definition 2.Indeed, in this case, the matrixC3 with n=2 is an ℳ–matrix, as its principal minors are positive.Corollary 5.
If condition (C1) and condition (C2) withm1=1, Is=1s=1,…,n are fulfilled and C4=csl given as(53)css=1−cp2a¯s∑i=2m∑j=1mia¯ssij,s=1,…,n,csl=−1a¯sa¯sl11−cp2a¯s∑i=2m∑j=1mia¯slij,s,l=1,…,n,s≠l,is an ℳ–matrix, equation (50) is exponentially 2p–stable in the sense of Definition 2.In particular, we obtain the following.Corollary 6.
Ifn=2, h11t=tt∈0,∞μ–almost everywhere, condition (C1) and condition (C2) with m1=1, Is=1s=1,2 are fulfilled and, finally, if(54)cp2a¯1∑i=2m∑j=1mia¯11ij>1,1−cp2a¯1∑i=2m∑j=1mia¯11ij1−cp2a¯2∑i=2m∑j=1mia¯22ij>1a¯1a¯1l11+cp2a¯1∑i=2m∑j=1mia¯1lij1a¯2a¯2l11+cp2a¯2∑i=2m∑j=1mia¯2lij,then equation (50) is exponentially 2p–stable in the sense of Definition 2.This follows from Corollary5 and the observation that the principal minors of the matrix defined by (51) and (52) are in this case positive.


## 7. Examples

Consider the deterministic system(55)dxt=−∑j=1mAjxt−hjdtt≥0,where(56)Aj=asljs,l=1n,j=1,…,m,are the real-valued n×n–matrices and hj, j=1,…,m, are nonnegative real numbers.Example 1.
Assume that(i)
∑j=1massj=as>0,s=1,…,n.(ii)
css=1−1/as∑k=1m∑j=1mhkasskassj,s=1,…,n.(iii)
csl=−1/as∑k=1m∑j=1mhkasskaslj+∑j=1maslj,s,l=1,…,n,s≠l.
Then, from Corollary1, we obtain that equation (55) is exponentially stable with respect to initial data if the n×n–matrix C5=csl with csl defined in Example 1 is an ℳ–matrix.Example 2.
Assume that in (55), h1=0, ass1>0,s=1,…,n, and the n×n–matrix C6=csl defined by(57)css=1−1ass1∑j=2massj,s=1,…,n,csl=−1ass1∑j=1maslj,s,l=1,…,n,s≠l,is an ℳ –matrix. Then, from Corollary 1, we obtain that equation (55) is exponentially stable with respect to initial data.
It is straightforward to observe that the entriescsl,s,l=1,…,n of the matrix C6 defined in (57) satisfy the estimates(58)css>∑l=1,l≠sncsl,s=1,…,n.
Thus,C6 is an ℳ–matrix, and equation (55) becomes exponentially stable with respect to initial data.
In particular, the deterministic system(59)dxt=−Axtdtt≥0,is exponentially stable if ass1>∑l=1,l≠snasl1,s=1,…,n.Example 3.
Consider the following system of linear Itô equations with constant delays:(60)dxt=−∑j=1m1A1jxt−h1jdt+∑i=2m∑j=1miAijxt−hijdℬitt≥0,where(61)Aij=aslijs,l=1n,i=1,…,m,j=1,…,mi,are the real-valued n×n–matrices and hij, i=1,…,m,j=1,…,mi, are nonnegative real numbers.
We define then×n–matrix C7=csl by(62)css=1−1as∑k=1m1∑j=1m1h1kass1kass1j+cp∑k=1m1∑i=2m∑j=1mih1kass1kassij−cp2as∑i=2m∑j=1miassij,s=1,…,n,csl=−1as∑k=1m1∑j=1m1h1kass1kasl1j+cp∑k=1m1∑i=2m∑j=1mih1kass1kaslij+∑j=1m1asl1j−cp2as∑i=2m∑j=1miaslij,s,l=1,…,n,s≠l.
IfC7=csl is an ℳ–matrix and(63)∑j=1m1ass1j=as>0,s=1,…,n,then system (60) is exponentially 2p–stable in the sense of Definition 2.Example 4.
Consider the system(64)dxt=−A11xtdt+∑i=2m∑j=1miAijxt−hijdℬitt≥0,and define the n×n–matrix C8=csl by(65)css=1−cp2ass11∑i=2m∑j=1miassij,s=1,…,n,csl=−asl11ass11−cp2ass11∑i=2m∑j=1miaslij,s,l=1,…,n,s≠l.
From Theorem4, it follows that if C8 is an ℳ–matrix and ass11>0,s=1,…,n, then equation (64) is exponentially 2p–stable in the sense of Definition 2. In particular, C8 becomes an ℳ–matrix if(66)1−cp2ass11∑i=2m∑j=1miassij>∑l=1nasl11ass11+cp2ass11∑i=2m∑j=1miaslij,s=1,…,n.


## 8. Conclusions and Outlook

We described a general framework to study exponential stability of linear functional differential stochastic equations combining the regularization technique with the theory ofℳ–matrices. We demonstrated how this techniques can be directly applied to stochastic Itô equations with variable delays and random coefficients, i.e., in the cases where the Lyapunov-like functionals may be difficult to find. In addition to the general stability result, we provided several specific stability conditions, including those for deterministic systems with delays. The results clearly show that introducing the time delays into a system deteriorates its stability properties in most cases. However, in some special situations (see Corollaries 4–6 and Example 4), our methods provide delay-independent stability conditions.The suggested framework is by no means thought to replace the Lyapunov method and its stochastic counterparts. Rather, our method may serve as an alternative in the situations where a direct application of the Lyapunov method seems to be more difficult (some examples can be found in the introductory section). On the other hand, our approach is not yet well developed to study global stability of nonlinear differential equations, and the main reason for that is the absence of the complete theory of the stochastic input-to-state stability, which the regularization method is based upon. Therefore, the efficiency of the method should be investigated further. For this purpose, it may be appropriate to study the following problems:(i)
Stability in the first approximation. This is an important problem in many models, for instance, in connection with extinction of populations in the random environment. The techniques developed in the present paper can be directly applied to this problem.(ii)
Asymptotic stability that is not exponential stability. This property is typical for equations with unbounded delays, for instance, for the stochastic pantograph equation [15]. The regularization method in its conventional setting has been used in this case, so that it is quite realistic to assume that combining it with the theory of ℳ–matrices will not be difficult.(iii)
Extension of the analysis based onℳ–matrices using nonlinear reference equations, as it is suggested in the paper [20]. Such constructions are believed to be challengeable, as it is known from the deterministic theory of functional differential equations [9]. However, the breakthrough ideas from [20] can help to overcome some of the obstacles, and not only in the stochastic case, but also in the deterministic case.(iv)
Applications to specific time-continuous stochastic models from various fields, e.g., stochastic networks, population dynamics, and analysis of neural fields. In this case, a graphical comparison with the existing results should be an essential part of the analysis.(v)
It is highly desirable to apply the theory ofℳ–matrices to analyse time-discrete stochastic systems, especially those arising in the control theory. A popular and challengeable example is given by the Roesser model described by linear, time-delayed systems. The Lyapunov function approach, frequently used to study these systems (see, e.g., [6, 23]), may be complemented by the techniques developed in the present paper.



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