LARGE-SCALE STOCHASTIC HEREDITARY SYSTEMS UNDER MARKOVIAN STRUCTURAL PERTURBATIONS . PART II . QUALITATIVE ANALYSIS OF ISOLATED SUBSYSTEMS

In this part of the work, the convergence and stability analysis of isolated subsystems of stochastic hereditary systems under random structural perturbations is investigated. The variational comparison theorems developed in Part I are used to achieve this goal. Under algebraic conditions on the rate coefficients, time-delay, and an intensity matrix associated with the Markov chain, convergence and stability results are obtained. Furthermore, 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. This investigation provides a basis for drawing the conclusions about the overall large-scale system.


Introduction
The qualitative analysis plays a very important role in the study of a variety of dynamic processes in the engineering, medical, social, and physical sciences.In particular, the stability and convergence analysis is one of the most useful study for the benefit of mankind.The real world problems are highly nonlinear, nonstationary, and complex.Historically [12,14,17,[19][20][21][22][23], the Lyapunov's second method has played an important role in the study of qualititative properties of dynamic systems governed by systems of differential equations.In this part of the work, by employing the comparison results developed in [11, Part I], the stability and convergence results are established for isolated subsystems of a large-scale stochastic hereditary system.This provides a basis for the study of hierarchic systems.In Section 2, for easy reference, basic concepts of convergence and stability are presented.In Section 3, sufficient conditions in the context of the comparison theorems that insure convergence and stability of isolated stochastic hereditary subsystem are presented.Finally, to illustrate the scope and significance of the study, a simple example is 2 Large-scale stochastic systems.Part II.Isolated subsystems presented in Section 4. The presented work extend and generalize the earlier work [1, 3-6, 8, 9, 12-15, 17-23] in a systematic and unified way.

Qualitative properties
In this section, we utilize the comparison theorems developed in [11,Part I] to investigate certain qualitative properties of isolated/decoupled subsystems [11]: where, a i = (a iT 0 ,a iT 1 ,...,a iT j ,...,a iT m ) T and a i j ∈ C[J × C ni × R,R ni ] for j ∈ I(0,m); for each i ∈ I (1, p).In particular, sufficient conditions are given to insure the convergence and stability of (DHS) depending on different modes of convergence in the probabilistic analysis [2,[14][15][16][17][21][22][23].The presented sufficient conditions relate the rate coefficients, the magnitude of the past-history, and the elements of the intensity matrix associated with the Markov chain in a systematic and natural way.
For easy reference, we present several types of convergence and stability concepts depending on different modes of convergence [2,14,17,21,23] in probabilistic analysis.Without loss of generality, it is assumed that the convergence is to zero, and the stability is with respect to the trivial solution process of (DHS).Definition 2.1 [14,16].A solution process x i (t) = x(t 0 ,ϕ i 0 )(t) of stochastic system (DHS) is said to be as follows: (i) convergent in the γth mean to zero if, for every > 0, δ > 0, and t 0 ∈ R + , there exists T = T( ,δ,t 0 ) > 0 such that (ii) converge in probability to zero if, for every > 0, t 0 ∈ R, and 0 < ζ < 1, there exists a δ > 0 and a T = T( ,δ,t 0 ) > 0 such that (iii) almost sure convergent to zero if G. S. Ladde 3 Definition 2.2 [3,5,14].The trivial solution process x ≡ 0 of stochastic hereditary system (DHS) is said to be as follows: (i) stable in the γth mean if, for each > 0, t 0 ∈ R, and γ ≥ 1, there exists a positive function δ = δ(t 0 , ) such that the inequality (ii) asymptotically stable in the γth mean if it is stable in the γth mean and if, for any > 0 and t 0 ∈ R + , there exists a positive function δ 1 = δ 1 (t 0 ) and a T = T( ,δ,t 0 ) > 0 such that the inequality (iii) stable in probability if, for each > 0, 1 > ζ > 0, δ > 0, and t 0 ∈ R + , there exists a positive function δ = δ(t 0 , ,ζ) such that the inequality (iv) asymptotically stable in probability if it is stable in probability and if, for any > 0, 1 > ζ > 0, ζ > 0, and t 0 ∈ R + , there exists a positive function δ 1 = δ 1 (t 0 ) and a T = T( ,δ,t 0 ) > 0 such that (v) almost surely stable if, for each > 0 and t 0 ∈ R 0 , there exists a positive function δ = δ(t 0 , ) such that the inequality with probability one implies with probability one, (vi) almost surely asymptotically stable if it is almost surely stable with probability one and if, for any > 0 and t 0 ∈ R + , there exists a positive function δ 1 = δ 1 (t 0 ) and a T = T( ,δ,t 0 ) > 0 such that the inequality with probability one implies with probability one.
In order to prove the convergence and stability results for (DHS), we need to define the convergence and stability concepts corresponding to Definitions 2.1 and 2.2 for the solution process r i (t 0 ,σ i 0 )(t) of comparison system (BCS)/(LCS) [11]: existing for t ≥ t 0 for (t 0 ,σ i 0 ) ∈ R + × (Ꮿ qi + ) q and each i ∈ I(1, p).Again, in this definition, it is also assumed that the solution process of (BCS) convergence is to zero.Definition 2.3.The maximal solution process r i (t 0 ,σ i 0 )(t) of (BCS) [11] through (t 0 ,σ i 0 ) is said to converge to zero if for every > 0 and δ > 0, and t 0 ∈ J, there exists a positive number T = T( ,δ,t 0 ) > 0 such that for t ≥ T + t 0 .
Similarly, the stability concepts with regard to comparison system (BCS) and other stability concepts with regard to both (DHS) and (BCS) can be reformulated, analogously.For details, see [5,14,17,22].

Qualitative analysis of isolated subsystem
Now, depending on different modes of convergence, we are ready to present convergence and stability results.The following result deals with γth mean convergence result.

G. S. Ladde 5
Theorem 3.1.Assume that hypotheses of [11,Theorem 3.1] 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) the maximal solution process r i (t 0 ,σ i 0 )(t) of (BCS) converges to zero.Then, the solution process x i (t 0 ,ϕ i 0 )(t) of (DHS) converges in the γth mean to the zero.Proof.The proof of the result can be constructed by imitating the arguments used in the proofs of results in [1,3,14,17].The details are left to the reader.
In the following, we provide the sufficient conditions to insure the almost sure convergence of a solution process of (DHS) to the zero vector.Theorem 3.2.Assume that hypotheses of [11,Theorem 3.1] , where 0 ∈ R qi + denotes the zero vector, (ii) there exists a matrix function and is a block diagonal matrix which is defined by Υ i (t) = diag T i1 (t),T i2 (t),...,T i j (t),...,T iq (t) , where where r i (t) = r i (t 0 ,σ i 0 )(t) is the maximal solution u = w i (t,u i ,σ i ), through (t 0 ,σ i 0 ) and (iv) the mean of the maximal solution E[r i (t)] = E[r(t 0 ,σ i 0 )(t)] of (BCS) through (t 0 ,σ i 0 ) converges to the zero vector as t → ∞.Then a solution process of (DHS) converges to the zero vector almost surely as t → ∞.
6 Large-scale stochastic systems.Part II.Isolated subsystems Proof.For t 0 ≤ s ≤ t, we define a function Z : R + × (R qi From the assumption of the theorem E[Z i j (s)] exists.Moreover, we note that s t0 Z i j (v)dv exists and nondecreasing in s.Under the assumptions on g i j and r i j (t 0 ,σ i 0 )(s), the following inequalities λ jl E r il t 0 ,σ i 0 (s) ds, T i j (s)E r i j t 0 ,σ i 0 (s) + g i j s,E r i j t 0 ,σ i 0 (s) ,E r i j s λ jl E r il t 0 ,σ i 0 (s) ds are valid.From (3.8) and assumption (iii), we conclude that For each i ∈ I(1, p), we now define Applying the operators defined in [11, Theorem 3.1, in particular, (3.1), (3.2), and (3.3)] to W i in (3.9) along a solution process x i (t 0 ,ϕ i 0 )(s) of (DHS) and utilizing hypotheses of [11, Theorem 3.1, in particular, (3.5), (3.7), (3.8)] and the definition of Z i j in (3.6), we obtain where ϕ i (0) = x i (t 0 ,ϕ i 0 )(s) for t 0 ≤ s ≤ t.From the definitions of π j j and T i j , the nature G. S. Ladde 7 of V i , and (3.10), it is clear that Also, from the assumption of the existence of E[V i (t 0 ,z i (t,t 0 ,ϕ i 0 (0)),η(t 0 ))], (3.4), (3.8), and (3.9), we obtain (3.12) The nonnegativity of W i , (3.11), (3.12), and an application of well-known result [2,21] gives us that W i (t,x i (t 0 ,ϕ i 0 )(t),η(t)) is non-negative supermartingale.Therefore, from the property of nonnegativity of the supermartingale, W i (t,x i (t 0 ,ϕ i 0 )(t),η(t)) converges almost surely as t → ∞.From (3.8) and (3.9), it is clear that V i (t,x i (t 0 ,ϕ i 0 )(t),η(t)) converges almost surely as t → ∞.In order to conclude the proof of the theorem, we first need to show that V i (t,x i (t 0 ,ϕ i 0 )(t),η(t)) → 0 as t → ∞.For this purpose, we note from [11, Theorem 3.1] that for i ∈ I(1, q i ) and j ∈ I(1, q), E V i t,x i (t), j ≤ E r i j t 0 ,σ i 0 (t) , t ≥ t 0 .
(3.13) Assumption (iv) of the theorem provides us that E[r i (t 0 ,σ i 0 )(t)] → 0 as t → ∞.From this and (3.13) and the application of Fatou's lemma [2,14,21], we have From assumption (i), we have and therefore from (3.14), we obtain lim t→∞ E[V i (t,x i (t), j)] = 0.This together with the properties of expectation establishes lim t→∞ [V i (t,x i (t), j)] = 0. From this, assumption (i), and continuity of V i , the almost sure convergence of the solution process of (DHS) to zero follows, immediately.This completes the proof.Now, we present sufficient conditions that assure the γth mean and almost sure stability properties of the trivial solution process of (DHS).Theorem 3.3.Assume that hypotheses of [11,Theorem 3.1] 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) a i (t,0,η(t)) ≡ 0 and w(t,0,0) ≡ 0.Then, (a) the stability of trivial solution of r i (t) ≡ 0 of (BCS) implies the γth mean stability of the trivial solution process x i (t) ≡ 0 of (DHS), (b) the asymptotic stability of trivial solution of r i (t) ≡ 0 of (BCS) implies the γth mean asymptotic stability of the trivial solution process x i (t) ≡ 0 of (DHS).
Proof.The proof of the theorem can be formulated by imitating the proofs of theorems in [3,5,6,10].However, the detailed proofs will appear elsewhere.
Theorem 3.4.Assume that hypotheses of [11,Theorem 3.1] 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) there exists a matrix function and is a block diagonal matrix which is defined by where r i (t) = r i (t 0 ,σ i 0 )(t) is the maximal solution of u = w i (t,u i ,σ i ) through (t 0 ,σ i 0 ), (iv) a i (t,0,η(t)) ≡ 0 and w(t,0,0) ≡ 0.Then, (a) the stability of trivial solution of r i (t) ≡ 0 of (BCS) implies the almost sure stability of the trivial solution process x i (t) ≡ 0 of (DHS), G. S. Ladde 9 (b) the asymptotic stability of trivial solution of r i (t) ≡ 0 of (BCS) implies the γth mean asymptotic stability of the trivial solution process x i (t) ≡ 0 of (DHS).
Proof.The proof of the theorem can be constructed by employing the proof of Theorem 3.2.The details are left to the reader.
Corollary 3.5.Assume that hypotheses of [11,Corollary 3.1] are satisfied.Further assume that (i) the assumption (i) of Theorem 3.1 is satisfied; (ii) a i (t,x i t ,η(t)) ≡ 0, (iii) the elements of matrices A i (t), B i (t), and past-history τ satisfy the following inequalities: for some d j > 0 for every j ∈ I(1, q).Then, (a) α ∈ L 1 [R + ,R + ] implies the γth mean stability of the trivial solution process of (DHS), (b) liminf (t−t0)→∞ [(1/(t − t 0 ))( t t0 α(θ)dθ)] ≥ α > 0 implies the γth mean asymptotic stability of the trivial solution process of (DHS).Proof.To conclude the validity of the conclusions of the corollary, it is enough to prove that (1) assumptions of Theorem 3.3 are satisfied, (2) α ∈ L 1 [R + ,R + ] implies the stability of the trivial solution process of (LCS) [11]; ≥ α > 0 implies asymptotic stability of the trivial solution process of (LCS).These statements together with the assumptions of corollary would fulfill all the hypotheses of Theorem 3.3.Hence, by the application of Theorem 3.3, the conclusions of the corollary follow, immediately.In fact, we first note that w(t,u,u t ) = −A i u + L i (u t ) satisfies the relation w(t,0,0) ≡ 0. To prove the remaining statements (2) and (3), we use a scalar Lyapunov function for (LCS), ordinary differential inequality, and comparison theorem in the context of minimal class 10 Large-scale stochastic systems.Part II.Isolated subsystems of functions defined by the following: where a real-valued continuous function A is determined by From (3.21) and (3.24), we observe that G(0) < 0, and G(λ) is increasing on (0,∞).Therefore, we can find a positive number α such that G(α) ≤ 0. We can take A(t) = exp[αt].The validity of the corollary follows by following the argument used in [7].

Examples
In the following, we present simple examples that illustrate the usefulness of the presented results of [11,Part I] as well as the results of this paper.
Example 4.2.The asymptotic stability (exponential stability) in the mean of the trivial solution of (DHS) is assured if the coefficient of comparison system in [11, in particular, in (LCS)] satisfies the conditions π j j − α i j (t) − τβ i j (t) > 0, j ∈ I(1, q), π j j − a i j (t) − τβ i j (t) − d −1 j q l = j d l π l j > α > 0 (4.15) for some α > 0 for each i ∈ I(1, q).Remark 4.3.It is clear that the sufficient conditions for the validity of qualitative properties of stochastic hereditary decoupled subsystems under Markovian structural perturbations are related with coefficients of intensity matrix Π = (π l j ) q×q , time-delay, and the estimates on the rate functions.Moreover, from (4.15) one can draw several implications about the effects of Markovian structural perturbations as well as the hereditary effects on systems.