LARGE-SCALE STOCHASTIC HEREDITARY SYSTEMS UNDER MARKOVIAN STRUCTURAL PERTURBATIONS. PART III. QUALITATIVE ANALYSIS

In this final part of the work, the convergence and stability analysis of large-scale stochastic hereditary systems under random structural perturbations is investigated. This is achieved through the development and the utilization of comparison theorems in the context of vector Lyapunov-like functions and decomposition-aggregation method. The byproduct of the investigation suggests that the qualitative properties of decoupled stochastic hereditary subsystems under random structural perturbations are preserved, as long as the self-inhibitory effects of subsystems are larger than cross-interaction effects of the subsystems. Again, it is shown that these properties are affected by hereditary and random structural perturbations effects. It is further shown that the mathematical conditions are algebraically simple, and are robust to the parametric changes. Moreover, the work generates a concept of block quasimonotone nondecreasing property that is useful for the investigation of hierarchic systems. These results are further extended to the integrodifferential equations of Fredholm type.


Introduction
A variety of problems that arise in the fields of engineering, medical, social, and physical sciences under hereditary and random environmental perturbations can be modeled by a system of stochastic functional differential equations. A feasible model for such a system is an Itô-type stochastic functional differential system perturbed by a finite-state Markov process. In this work, by using decomposition-aggregation method [2,3,6,11,9,[12][13][14]18], we propose to study the qualitative behavior of a solution of such a large-scale stochastic hereditary system. Moreover, by utilizing variational comparison theorems [9] for each isolated subsystem in the context of energy/Lyapunov-like functions and the nature of interactions among the subsystems of a large-scale system, a very general variational comparison theorem is formulated. These comparison theorems are used to investigate 2 Large-scale stochastic systems. Part III. Qualitative analysis various modes of convergence of a solution process of the stochastic large-scale hereditary system. Stability results for the trivial solution of the stochastic large-scale system are also obtained.
In Section 2, variational comparison results of isolated subsystems (DHS) [9] are extended to large-scale stochastic hereditary systems under Markovian structural perturbations (LHS). The comparison results of this section are utilized to investigate the convergence and stability analysis of hierarchic systems in Section 3. The byproduct of the study generates a concept of block quasimonotone nondecreasing property [1,2,[15][16][17]. Moreover, the scope and the significance of the presented results are outlined in Section 4. In particular, it contains remarks concerning the effects of the hereditary and random perturbations of the system on the convergence and stability. The presented results extend and generalize the earlier work [1][2][3][4][5][6][7][8][13][14][15][16][17][18] in a systematic and unified way.

Variational comparison theorems
Let us recall large-scale stochastic hereditary system (LHS) described in [9]. It is decomposed into smaller, simpler, and suitable p-interconnected subsystems (perturbed) of the following form: ..,(x p ) T ) T , and p i=1 n i = n for each i ∈ I(1, p); the interactions (perturbations) among the p subsystems of system (LHS) are described by c i .
In this section, analogous to the results of [9, Section 3], we develop a few auxiliary comparison results for stochastic large-scale hereditary system (LHS). For this purpose, we utilize an energy/Lyapunov-like function associated with each decoupled/isolated stochastic hereditary subsystem (DHS) corresponding to (LHS) under Markovian structural perturbations: in [10] for i ∈ I(1, p). We use the same auxiliary systems of differential equations as in (DAS) as described in [9]. As stated before, we use vector Lyapunov-like functions associated with each isolated system i ∈ I(1, p) in (DHS). By following the definition in [9], we define D + (LHS) V i (s,ϕ, z i (t,s,ϕ i (0)),η(s)), for each subsystem of large-scale system (LHS). where L C i is defined and can be represented as Before we present a main variational comparison result for large-scale system, we present two lemmas. The first lemma is analogous to [9, Lemma 3.1] with respect to large-scale system (LHS). The second lemma is an extension of the well-known maximal solution for systems of functional differential equations [15][16][17].
Prior to stating the second lemma, we need to introduce a concept of block quasimonotone nondecreasing property of a comparison function.
The function G(t,u,σ) is said to possess a block quasimonotone nondecreasing property in u for each (t,σ) if for each j ∈ I(1, q), (a) the block G j (t,u,σ) is nondecreasing in 4 Large-scale stochastic systems. Part III. Qualitative analysis u(l) for l = j, and (b) for l = j and each i ∈ I(1, p), G i j (t,u,σ) is quasimonotone nondecreasing in u i j and nondecreasing in u k j for k = i and k ∈ I(1, p).
Proof. We note that G defined in (2.4) satisfies the block quasimonotone nondecreasing property in u. The rest of the proof of the lemma can be constructed by following the argument used in the proofs of the results in [15][16][17]. The details are left to the reader. Now, we are ready to present a result that is similar to [9, Theorem 3.1].
Theorem 2.5. Assume that (a) g i j , h i j , G i j , and G satisfy the conditions of Lemma 2.4; (b) r(t) = r(t 0 ,σ 0 )(t) is the maximal solution of system of comparison differential equations (2.6) existing for t ≥ t 0 ; (c) z i (t,s,z i 0 ) solution process of auxiliary system (DAS) in [9] through (s,ϕ i (0)), t 0 ≤ s ≤ t and its second derivative (∂ 2 /∂x i ∂x i )z i (t,s,z i 0 ) is locally Lipschitzian in z i 0 for each i ∈ I(1, p); s,ϕ i (0)),η(s)) satisfies the following relation: for each i ∈ I(1, p) and j ∈ I(1, q); (e) for each i ∈ I(1, p) and j ∈ I(1, q) the interaction functions among the subsystems satisfy the inequality where, I i is as defined in (2.2) and

Qualitative analysis of large-scale systems
Now, we are ready to formulate the convergence and stability results for stochastic largescale hereditary system (LHS). First, we present very general results regarding convergence and stability properties of the solution process of (LHS).
In the following, we provide the sufficient conditions to insure the almost-sure convergence of a solution process of (LHS) to the zero vector.
Then a solution process of (LHS) converges to the zero vector (a.s.) as t → ∞.
Proof. The proof of the theorem can be constructed by following the argument of [10, Theorem 3.2], and the argument used in the proofs of the results in [1,3,8,13,14]. The details are left to the reader. Now, we present sufficient conditions that assure the almost-sure stability properties of the trivial solution process of (LHS). Theorem 3.3. Assume that hypotheses of Theorem 2.5 are satisfied. Further assume that (i) for each t ∈ R + , V i (t,x, j) satisfies the inequality where a ∈ ᐂ and b ∈ Ꮿ; (ii) F(t,0,η(t)) ≡ 0 and G(t,0,0) ≡ 0. Then, (a) the stability of the trivial solution of r(t) ≡ 0 of (2.6) implies the γth mean stability of the trivial solution process x(t) ≡ 0 of (LHS); (b) the asymptotic stability of the trivial solution of r(t) ≡ 0 of (2.6) implies the γth mean asymptotic stability of the trivial solution process x(t) ≡ 0 of (LHS).
Proof. The proof of the theorem can be formulated by following the argument used in the proof of [10,Theorem 3.3]. The details are left to the reader.
We present sufficient conditions that assure the almost-sure stability properties of the trivial solution process of (LHS).
Proof. The proof of the theorem can be constructed by employing the proof of [

Conclusions
From our stability analysis, we can draw several conclusions. We briefly state some of them. However, details will appear elsewhere. Stability conditions in [10], for example: for some α > 0 for each i ∈ I (1, q).
(3) These conditions are algebraically simple and easy to compute. (4) They are expressed in terms of rate coefficients, time-delay, and intensity matrix. (5) Condition (4.2) gives an estimate on time-delay τ, magnitude of intensity matrix, and magnitude of hereditary interactions.