Dynamic Systems and Applications 15 (2006) 269-286

STOCHASTIC HYBRID PARABOLIC SYSTEMS UNDER JUMP MARKOVIAN PERTURBATIONS-I: CONVERGENCE

AND STABILITY VIA LYAPUNOV FUNCTIONALS

M. J. ANABTAWI AND S. SATHANANTHAN

Department of Mathematics and Statistics, American University of Sharjah, P.O. Box 26666, Sharjah, UAE Department of Mathematics and Center of Excellence in ISEM, Tennessee State University, 330 10th Ave. N Suite 265G Nashville, TN 37203-3401, USA

ABSTRACT. In this paper, we investigate convergence and stability concepts in the sense of p-th mean and probability of the solution process of stochastic parabolic systems under Markovian structural perturbations. Sufficient conditions are obtained for various kinds of convergence and stability. This is achieved by developing a powerful comparison theorem in the context of vector Lyapunov-like functionals and systems of differential inequalities. AMS (MOS) Subject Classification. 93E15, 93D20, 35K40.

1. INTRODUCTION

The class of jump linear systems (JLS) was introduced by Krasovskii and Lidskii [8] in the early sixties. Since then we have seen an increasing interest in this class of systems. It has been applied to model various dynamical systems, such as manufacturing systems, socio-economic systems, etc. For more information regarding the application of such systems, we refer the reader to Mariton [8], Sethi and Zhang [9] and the references therein.

The jump nonlinear system is a hybrid system with state vector that has two components u(t) and η(t). Here u(t) is the state and η(t) is the mode. During the operation, the system can jump from one mode to another in a random way, which makes the class of systems a stochastic one. The switching between the modes is governed by a Markov process with discrete and finite state space. When the system mode is fixed it evolves like a deterministic nonlinear system. This kind of system can be used to describe abrupt phenomena, such as component and interconnection failures. There is a vast literature available in this field.

The mathematical modelling of several real world problems leads to systems of differential and partial differential equations that involve some kind of randomness due to ignorance and or uncertainties. The Markovian structural perturbations in parabolic type of partial differential equations has been investigated (see Anabtawi and Sathananthan [1]). This work is an attempt to study the Markovian structural perturbations in parabolic type of stochastic partial differential equations. We have attempted to develop comparison theorems in the context of differential inequalities and Lyapunov-like functionals to study the stability and convergence in the pth moment and in probability for such system. In Section 2, we formulate the problem. In Section 3, we develop the comparison results that will be utilized to study the qualitative properties of the system. In Sections 4 and 5, we investigate the convergence and stability analysis of the system. In Section 6, we provide an example that shows the significance of our results.

Received April 30, 2004           1056-2176 $15.00 \begin{matrix}c \\ ◯ \end{matrix}Dynamic Publishers, Inc.

2. PROBLEM FORMULATION

Let H be a bounded domain in R^{m}, Q_{t_{0}} = (t_{0}, ∞) × H for t_{0} > 0, Γ_{t_{0}} = (t_{0}, ∞) × ∂H. Consider the stochastic initial boundary value problem under Jump Markovian structural perturbations.

(2.1)  \begin{cases}du = f(t, x, u, u_{x}, u_{xx}, η(t))dt + B(η(t))udω(t), \\ u(t_{0}, x) = u_{0}(x),  for x ∈ \overline{H}, \\ Bu(t, x) = ϕ(t, x),  on Γ_{t_{0}}, \end{cases}

where B is the boundary operator defined by

(2.2)  B_{i}u_{i}(t, x) = p_{i}(t, x)u_{i} + q_{i}(t, x)\frac{∂u_{i}}{∂ν} ,  1 ≤ i ≤ n,

where p_{i}, q_{i} ∈ C[Γ_{t_{0}}, R] are such that p_{i}(t, x) ≥ 0, q_{i}(t, x) ≥ 0, and p_{i}(t, x) + q_{i}(t, x) > 0, 1 ≤ i ≤ n; \frac{∂u}{∂ν} is the outernormal derivative; η(t) is a Markov chain with a finite number of states, which will be denoted by the set S = {E_{1}, E_{2}, …, E_{s}} ≡ {1, 2, …, s}; u(t, x, η(t)) is an n-dimensional random vector process defined on a complete probability space (Ω, F, P); ϕ ∈ C[Γ_{t_{0}}, R^{n}]. B(η(t)) ∈ C[S, R^{n×n}] and ω(t) is a 1-dimensional normalized Wiener that is F_{t} measurable for t ≥ t_{0} and the increment ω_{i}(t + h) − ω_{i}(t) is independent of every event in F_{t}; F_{t} is an increasing family of sub σalgebras of F_{t}; u_{0}, η(t) and ω_{t} are independent for each t ≥ t_{0}, f ∈ C[\overline{Q}_{t_{0}} ×R^{n} ×R^{nm} ×R^{nm^{2}} ×S, R^{n}], B(η(t)), η(t) and ω(t) are F_{t}-measurable for all t ≥ t_{0}.

Let the matrix Π = (π_{ij})_{s×s} be the chain generator of the Markov chain η(t) ∈ S. The entries π_{ij}, i, j = 1, …, s of Π are interpreted as transition rates

P \{η(t + Δ) = j|η(t) = i\} = \begin{cases}π_{ij}Δ + O(Δ), & i ≠ j \\ 1 − π_{ii}Δ + O(Δ), & i = j. \end{cases}

We assume that f, B, and the domain are smooth enough so that the IBVP (2.1) has a solution process on Q_{t_{0}}.

Lemma 2.1: [11] Let \overline{H} be a bounded domain in R^{n} and ϕ belongs to a collection of twice differentiable function defined on \overline{H} into R^{n} such that

∥ϕ∥_{2}^{2} = (∫\overline{H} |ϕ|^{2}dx),

then

∥ϕ∥_{2}^{2} ≤ M[∑_{i=1}^{m} ∫\overline{H}(\frac{∂ϕ}{∂x_{i}} )^{2}dx]

where the constant M depends on the size of the given domain H.

The following theorem is an extension of a well-known theorem. This theorem will be used subsequently in our work

Theorem 2.1: [2,7] Assume that

(i) g^{k} ∈ C[J × R^{p}, R^{p}] and g^{k}(t, z) is quasi-monotone non-decreasing in z ∈ R^{p} for each t ∈ J and 1 ≤ k ≤ s; (ii) w ∈ C[J × (R^{p})^{s}, (R^{p})^{s}],

(2.3)  w(t, y) = (w^{1}(t, y)^{T} , w^{2}(t, y)^{T} , …, w^{s}(t, y)^{T})^{T} ,

w^{k} ∈ C[J × (R^{p})^{s}, R^{p}] and it has the following structural representation

(2.4)  w^{k}(t, y) = (g_{1}^{k}(t, y^{k}) + ∑_{l=k,l≠k}^{s} π_{kl}y_{1}^{l} − π_{kk}y_{1}^{k}, …, g_{i}^{k}(t, y^{k})

+ ∑_{l=l,l≠k}^{s} π_{kl}y_{i}^{l} − π_{kk}y_{i}^{k}, …, g_{p}^{k}(t, y^{k})

+ ∑_{l=1,l≠k}^{s} π_{kl}y_{p}^{l} − π_{kk}y_{p}^{k}),

and y^{l}, y^{k} ∈ R^{p}, l, k = 1, …, s. Then the following initial-value problem

(2.5)  y′ = w(t, y),  y(t_{0}) = y_{0},

has a maximal solution.

3. VECTOR LYAPUNOV-LIKE FUNCTIONALS

In this section, we develop a very general comparison theorem in the context of Lyapunov-like functional which will be utilized for studying the convergence and stability properties of stochastic hybrid parabolic systems under Markovian structural perturbations. Before we present our main theorem, for any V ∈ C[J × C^{2}[\overline{H}, R^{n}] × S, R_{+}^{N}], we define

(3.1) D^{+}V (t, u, k) = lim sup_{h→0^{+}} \frac{1}{h}E[V (t + h, u(t + h, x), η(t)), η(t + h))
− V (t, u, η(t))|η(t) = k],

where C^{2}[\overline{H}, R^{n}] is a collection of twice differentiable functions defined on \overline{H} into R^{n}.

Now, we present the developed comparison theorem in the context of vector Lyapunov-like functionals.

Theorem 3.1: Assume that V ∈ C[J × C^{2}[\overline{H}, R^{n}] × S, R_{+}^{N}], and V (t, u, k) is locally Lipschitzian in u for fixed t and uniformly in k. Further assume that

(i) f ∈ C[\overline{Q}_{t_{0}} × R^{n} × R^{nm} × R^{nm^{2}} × S, R^{n}], g^{k} ∈ C[J × R^{m}, R^{N}], g^{k} is concave and quasi-monotone nondecreasing in y, for fixed t ∈ J, such that

(3.2)  LV (t, u, k) ≤ g^{k}(t, V (t, u, k)),

where

(3.3)  LV (t, u, k) = \frac{∂V (t, u, k)}{∂t} + \frac{∂V (t, u, k)}{∂u}f(t, x, u, u_{x}, u_{xx}, k)

+ \frac{1}{2}tr((Bu)(Bu)^{T} )\frac{∂^{2}V (t, u, k)}{∂u^{2}},

(ii) r(t) = r(t, t_{0}, y_{0}) is the maximal solution of the differential system (2.5); (iv) for the solution process u(t, x) = u(t, x, t_{0}, u_{0}, k), E[V (t, u(t, x), η(t))] exists for t ≥ t_{0}, and

(3.4)  \{ \begin{matrix}v(t_{0}, x) ≤ y_{0}, x ∈ \overline{H} \\ Bu(t, x) ≤ Br(t, t_{0}, y_{0}), x ∈ Γ_{t_{0}}, \end{matrix}

These assumptions imply

(3.5)  v(t, x) ≤ r(t, t_{0}, y_{0}),  t ≥ t_{0},

where

(3.6)  v(t, x) = (v^{1}(t)^{T} , …, v^{k}(t)^{T} , …, v^{s}(t)^{T})^{T} ,

and

(3.7) v^{k}(t, x) = E[V (t, u(t, x), k)|u(t_{0}, x) = u_{0}, η(t) = k],  k ∈ S.

Proof: Let u(t, x) be any solution process (2.1) satisfying (3.4). For small h > 0, we set

m^{k}(t + h) = E[V (t + h, u(t + h, x), η(t + h))|u(t) = u, η(t) = k].

We note that

(3.8) m^{k}(t) = E[V (t, u(t, x), η(t))|u(t) = u, η(t) = k] = V (t, u(t), k).

Using this notation, taking the difference, m^{k}(t + h) − m^{k}(t), and using some algebraic manipulations, then taking the limit superior as h → 0^{+}, and using Lebesgue dominated convergence theorem, the properties of the right continuous stationary Markov chain, the continuity of V, and by the locally Lipschitzian nature of V, we have

(3.9)  D^{+}m^{k}(t) ≤ LV (t, u, k) + ∑_{l=1}^{s} π_{kl}[V (t, u, l) − V (t, u, k)].

From (3.2), and (3.9), we obtain

(3.10)  D^{+}m^{k}(t) ≤ g^{k}(t, m^{k}(t)) + ∑_{l=1}^{s} π_{kl}m^{l}(t) − π_{kk}m^{k}(t).

By letting

(3.11)  w^{k}(t, m^{k}(t)) = g^{k}(t, m^{k}(t)) + ∑_{l=1}^{s} π_{kl}m^{l}(t) − π_{kk}m^{k}(t).

Then

(3.12)  D^{+}m^{k}(t) ≤ w^{k}(t, m^{k}(t)),

and

D^{+}m(t) ≤ w(t, m(t)).

Applying the comparison Lemma [7], we arrive at

(3.13)  m(t) ≤ r(t, t_{0}, y_{0}),

whenever m(t_{0}) ≤ y_{0}, where

(3.14)  m(t) = (m^{1}(t)^{T} , …, m^{k}(t)^{T} , …, m^{s}(t)^{T})^{T} .

Taking the conditional expectation of both sides with respect to u_{0}, yields the desired conclusion. This completes the proof of the theorem.

The next result provides the construction of the comparison system in a systematic way.

Theorem 3.2: Assume that V_{i} ∈ C[J × C^{2}[\overline{H}, R^{n}] × S, R_{+}], and V_{i}(t, u, k) is locally lipschitzian in u for fixed t and uniformly in k. Further assume that

(i) for each 1 ≤ k ≤ s, 1 ≤ i ≤ N, and (t, u, k)

(3.15)  g_{i}^{k}(t, V_{i}(t, u, k)) ≤ ∑_{r=1}^{N} γ_{ir}^{k}(t)V_{i}(t, u, k) + γ_{i}^{k}(t),

γ_{ir}^{k}, γ_{i}^{k} ∈ C[J, R], γ_{ir}^{k} > 0 for i ≠ r; (ii) for the solution process u(t, x) of (2.1), E[V_{i}(t, u(t, x), η(t))] exit for t ≥ t_{0}. Then

v(t) ≤ ψ(t, t_{0})y_{0} + ∫_{t_{0}}^{t} ψ(t, s)h(s)ds,

whenever v(t_{0}) ≤ y_{0}, where v(t) is as defined in (3.6) and (3.7), ψ(t, t_{0}) is the fundamental matrix solution of the block system of linear differential equations

(3.16)  y′ = −C_{b}(t)y,  y(t_{0}) = y_{0},

the kth component of the block vector −C_{b}(t) is given by

(3.17)  −C^{k}(t)y^{k} + ∑_{l=1,l≠k}^{s} π_{kl}I_{N}y^{l} − π_{kk}I_{N}y^{k},

I_{N} = N × N-identity matrix,

(3.18)  C^{k}(t) = (C_{ir}^{k}(t))_{N×N},

(3.19)  C_{ir}^{k}(t) = \begin{cases}−γ_{ii}^{k}(t), & if r = i, \\ −γ_{ir}^{k}(t), & if r ≠ i, \end{cases}

and

(3.20)  h(t) = (h^{1}(t)^{T} , h^{2}(t)^{T} , …, h^{s}(t)^{T})^{T} ,

with

(3.21)  h^{k}(t) = (γ_{1}^{k}(t), γ_{2}^{k}(t), …, γ_{p}^{k}(t))^{T} .

Proof: Applying D^{+} to V_{i}(t, u, k), and by (3.2) and (3.3), we obtain

(3.22) D^{+}V_{i}(t, u, k) ≤ g_{i}^{k}(t, V_{i}(t, u, k))

+ ∑_{l=1,l≠k}^{s} π_{kl}V_{i}(t, u, l) − π_{kk}V_{i}(t, u, k), k = 1, …, s,

from (3.17), (3.18), (3.19), (3.20), and (3.22), we have

(3.23)  D^{+}V (t, u, k) ≤ − C^{k}(t)V (t, u, k) + h^{k}(t)

+ ∑_{l=1,l≠k}^{s} π_{kl}V (t, u, l) − π_{kk}V (t, u, k),

where h^{k}(t) and C^{k}(t) = (C_{ir}^{k}) are defined in (3.21) and (3.19), respectively. Now using the definitions and notations in (3.16), (3.20), and the inequality (3.23), we obtain

(3.24)  D^{+}m(t) ≤ −C_{b}(t)m(t) + h(t).

The block comparison system is described by

(3.25)  y′ = −C_{b}(t)y + h(t),  y(t_{0}) = y_{0},

and its solution through (t_{0}, y_{0}) is given by

(3.26)  y(t) = ψ(t, t_{0})y_{0} + ∫_{t_{0}}^{t} ψ(t, s)h(s)ds,

where ψ(t, t_{0}) is the fundamental matrix solution of (3.16). From the uniqueness of the solution of (3.24) the above solution is the maximal solution, r(t, t_{0}, y_{0}). This satisfies the assumption (iii) of the Theorem 3.1. Therefore all the assumptions of Theorem 3.1 are satisfied. The application of the theorem yields

v(t) ≤ ψ(t, t_{0})y_{0} + ∫_{t_{0}}^{t} ψ(t, s)h(s)ds,

whenever v(t_{0}) ≤ y_{0}. This completes the proof of the Theorem.

In the following result we reduce the block comparison systems that consist of subsystems of dimension higher than 1 described in Theorem 3.1 and Theorem 3.2, respectively, to block systems with subsystems of dimension one.

Lemma 3.1: Assume that hypotheses (i), and (iv) of Theorem 3.1 are satisfied. Further assume that g_{i}^{k} satisfies

(3.27)  ∑_{i=1}^{N} d_{i}g_{i}^{k}(t, V_{i}(t, u, k)) ≤ γ^{k}(t)ν(t, u, k) + β^{k}(t),

where

(3.28)  ν(t, u, k) = ∑_{i=1}^{N} d_{i}V_{i}(t, u, k),

d_{i} > 0 for 1 ≤ i ≤ N, γ^{k}, β^{k} ∈ C[J, R] and k ∈ S. Then

(3.29)  D^{+}ν(t, u) ≤ −C(t)ν(t, u) + β(t),

where C and β are defined as follows

(3.30)  C(t) = (c_{kl}(t))_{s×s},

where

(3.31)  c_{kl}(t) = \begin{cases}−π_{kl}, & if l ≠ k, \\ π_{kk} − γ^{k}(t), & if l = k, \end{cases}

(3.32)  β(t) = (β_{1}^{1}(t) + β_{2}^{1}(t), β_{1}^{2}(t) + β_{2}^{2}(t), …, β_{1}^{s}(t) + β_{2}^{s}(t))^{T} ,

and

(3.33)  ν(t, u) = (ν(t, u, 1), …, ν(t, u, k), …, ν(t, u, s))^{T} .

Proof: Taking the D^{+}−derivative of (3.23) and using the definition of D^{+}V (t, u, k) and (3.27), and (3.28), we obtain

(3.34) D^{+}ν(t, u, k) ≤ (γ^{k}(t) − π_{kk})ν(t, u, k) + ∑_{l=1,l≠k}^{s} π_{kl}ν(t, u, k) + β^{k}(t),

for each k ∈ S. This together with definitions of C(t) and β(t) in (3.30) and (3.32), respectively, yields

D^{+}ν(t, u) ≤ −C(t)ν(t, u) + β(t).

This completes the proof of the lemma.

The following result exhibits the feasibility of condition (3.27).

Corollary 3.1: Assume the hypotheses of Theorem 3.2 are satisfied. Then

(3.35)  D^{+}ν(t, u) ≤ −C(t)ν(t, u) + β(t),

where ν(t, u) and β(t) as defined before, C(t) is defined as follows.

(3.36)  C(t) = (c_{kl}(t))_{s×s},

(3.37)  c_{kl}(t) = \begin{cases}γ^{k}(t), & if k = l, \\ −π_{kl}, & if k ≠ l, \end{cases}

(3.38)  γ^{k}(t) = min_{1≤r≤N} {C_{rr}^{k}(t) + π_{kk} − d_{r}^{−1} ∑_{i=1}^{N} d_{i}|C_{ir}^{k}(t)|} > 0.

Remark 3.1: Under the assumptions of Lemma 3.1 and Corollary 3.1 in the context of the following comparison system described by

(3.39)  y′ = −C(t)y + β(t),  y(t_{0}) = y_{0},

the conclusions of Theorems 3.1 and 3.2 remain true. We note that C(t) and β(t) are defined in Corollary 3.1, and w(t, y) is defined by w(t, y) = −C(t)y + β(t), and satisfies all the assumptions of Theorems 3.1 and 3.2 with N = 1. An application of Theorem 3.1 or Theorem 3.2, gives us

(3.40)  E[ν(t, u(t))|u(t_{0}) = u_{0}, η(t) = k] ≤ r(t, t_{0}, y_{0}),

whenever

ν(t, t_{0}, u_{0}) ≤ y_{0}.

From the uniqueness of solution of (3.39), the maximal solution of (3.39), is given by

(3.41)  r(t, t_{0}, y_{0}) = ψ(t, t_{0})y_{0} + ∫_{t_{0}}^{t} ψ(t, s)β(s)ds,

ψ(t, t_{0}) is the fundamental matrix solution of

y′ = −C(t)y, y(t_{0}) = y_{0}.

The following lemma provides an explicit estimate for the solution of (2.1).

Lemma 3.2: [2, 7] Let the assumptions of Theorem 3.2 be satisfied. Then

(3.42)  ∥r(t, t_{0}, y_{0}))∥ ≤ ∥y_{0}∥ exp[− ∫_{t_{0}}^{t} α(s)ds]
+ ∫_{t_{0}}^{t} exp[− ∫_{s}^{t} α(u)du]∥β(s)∥ds,

and moreover

∥ν(t, x, u(t, x))∥ ≤ ∥r(t, t_{0}, y_{0})∥,

for t ≥ t_{0} where

(3.43)  α(t) = min_{1≤k≤s}{c_{kk}(t) − a_{k}^{−1} ∑_{l=1,l≠k}^{s} a_{l}|c_{kl}(t)|},

a_{l} are some positive number for l = 1, 2, …, s.

4. CONVERGENCE ANALYSIS

In this section we utilize the block comparison theorems developed in the previous section to investigate different modes of convergence of the solution process of (2.1). Without loss of generality, we will show the convergence of the solution process to zero. Let u(t, x, η(t)) be any solution process of the system (2.1). Let us denote

∥u(t, ω)∥_{q} = (∫\overline{H} ∥u(t, x, ω)∥^{q}dx)^{\frac{1}{q}} , for q ≥ 1, \overline{H} ⊆ R^{m}.

Here we present definitions of convergence of the solution process of (2.1).

Definition 4.1: The solution of (2.1) is said to \underline{(cm) converge in the (q, p)th-moment}, if every ϵ > 0, δ > 0 and t_{0} ∈ J, there exists a positive number T = T (ϵ, δ, t_{0}) > 0 such that

∥u(t)∥_{p} < ϵ, for t ≥ T + t_{0},

whenever

∥u_{0}∥_{p} ≤ δ,
∥u(t)∥_{p} = (E[∥u(t, ω)∥_{q}]^{p})^{\frac{1}{p}};

\underline{(cp) the qth-moment converges in probability} if for every ϵ > 0, ξ > 0, δ > 0, and t_{0} ∈ J, there exists a positive number T = T (t_{o}, ϵ, δ, ξ) such that

P [ω : ∥u(t, ω)∥_{q} ≥ ϵ] < ξ, fort ≥ T + t_{0},

whenever

P [ω : ∥u_{0}(ω)∥_{q} > δ] < ξ.

Now, we are ready to present theorems that assures the (q, p)th-moment convergence of the solution process of (2.1).

Theorem 4.1: Assume that

(i) all the hypotheses of Theorem 3.1 hold; (ii) for each 1 ≤ i ≤ p, V_{i}(t, u, η(t)) has the unique zero at u ≡ 0; (iii) for (t, u) ∈ J × R^{n} and k ∈ S, V_{i}(t, u, k) satisfies

(4.1)  b(∥u∥_{q}^{p}) ≤ ∑_{k=1}^{s} ∑_{i=1}^{N} V_{i}(t, u, k),

where b ∈ VK[R_{+}, R_{+}] such that b(0) = 0 and b(r) is strictly increasing in r, and b(r) → ∞ as r → ∞; (iv) there exists a matrix function T (t) such that T ∈ C[R_{+}, R^{N}], T (t) = (T_{ij}(t))_{N×N}, T_{ij}(t) ≥ 0, and for y, z ∈ R^{N}, y ≥ z, g^{k}(t, y) − g^{k}(t, z) ≥

−T (t)(y − z), for each 1 ≤ k ≤ s; (v) The maximal solution r(t, t_{0}, E[y_{0}]) of (2.5) through (t_{0}, E[y_{0}]) converges to the zero vector as t → ∞.

Then the solution process u(t, x, η(t)) of (2.1) converges in the (q, p)th moment to the zero vector.

Proof: From the conclusion of Theorem 3.1 and Lemma 3.2 [8], for each k, we have

(4.2)  E[V (t, u(t, x), k)] ≤ r^{k}(t, t_{0}, E[y_{0}]) for each k ∈ S.

Let ϵ > 0, δ and t_{0} ∈ J be given, from (iii) we have b(ϵ^{p}) > 0, b(δ^{p}) > 0, for p ≥ 1. Since the maximal solution r(t) = r(t, t_{0}, E[y_{0}]) of (2.5) through (t_{0}, E[y_{0}]) converges to zero, then, there exists T = T (b(ϵ^{p}), b(δ^{p}), t_{0}) such that

(4.3)  ∑_{k=1}^{s} ∑_{i=1}^{N} r_{i}^{k}(t, t_{0}, E[y_{0}]) < b(ϵ^{p}), for all t ≥ T + t_{0},

whenever

∑_{k=1}^{s} ∑_{i=1}^{N} E[y_{i0}^{k}] ≤ b(δ^{p}).

We choose y_{i0}(ω) such that v_{i}(t_{0}) ≤ y_{i0}(ω). This together with (4.1), implies that

(4.4) E[b(∥u_{0}∥_{q}^{p})] ≤ E[∑_{k=1}^{s} ∑_{i=1}^{N} V_{i}(t_{0}, u_{0}, k)] ≤ ∑_{k=1}^{s} ∑_{i=1}^{N} E[y_{i0}^{k}] ≤ b(δ^{p}).

Using the convexity of b and inequality (4.4), we have

∥u_{0}∥_{p} ≤ δ.

Now we claim that ∥u_{0}∥_{p} ≤ δ, implies ∥u(t)∥_{p} < ϵ, t ≥ T + t_{0}. Suppose this is false, and let there exist a sequence {t_{j}} such that t_{j} ≥ t_{0} + T, t_{j} → ∞ as j → ∞, and ∥u(t_{j})∥_{p} = ϵ. From (4.1), (4.2), (4.3), and the convexity of b and Jensen’s inequality, we arrive at the contradiction

(4.5) b(ϵ^{p}) ≤ ∑_{k=1}^{s} ∑_{i=1}^{N} E[V_{i}(t_{j}, u(t_{j}, x), k)] ≤ ∑_{k=1}^{s} ∑_{i=1}^{N} r_{i}^{k}(t_{j}, t_{0}, E[y_{0}]) < b(ϵ^{p}).

Therefore our claim is true. This completes the proof of the theorem. To exhibit the feasibility of Theorem 4.1, we present the following result. Theorem 4.2: Assume that

(i) all the hypotheses of Corollary 3.1 hold; (ii) for each 1 ≤ i ≤ N, the function V_{i}(t, u, η(t)) has the unique zero at u ≡ 0; (iii) for (t, u) ∈ J × R^{n} and k ∈ S, V_{i}(t, u, k) satisfies

b(∥u∥_{q}^{p}) < ∑_{k=1}^{s} ∑_{i=1}^{N} d_{i}V_{i}(t, u, k),

where b ∈ VK[R_{+}, R_{+}]; (iv) the matrix C defined in (3.36) satisfies the following conditions

c_{lk} ≤ 0 for l ≠ k, and c_{kk} > 0, for 1 ≤ l, k ≤ s,

and there exist positive numbers a_{l}, 1 ≤ l, k ≤ s, such that

c_{kk}(t) > a_{k}^{−1} ∑_{l=1}^{s} a_{l}|c_{lk}|, 1 ≤ k ≤ s;

(v) the functions α and β defined in (3.43) and (3.32) respectively, satisfies

(4.6)  lim_{t→∞} ∫_{t_{0}}^{t} α(s)ds = ∞,

and

(4.7)  lim_{t→∞} ∫_{t_{0}}^{t} exp[− ∫_{s}^{t} α(u)du]∥β(s)∥ds = 0.

Then the solution process u(t, x) of (2.1) converges in the (q, p)th moment to the zero vector.

Proof: From the hypotheses of Theorem 3.2 and an application of Corollary 3.1, we have

(4.8)  D^{+}ν(t, u) ≤ −C(t) ν(t, u) + β(t),

where ν(t, u), C(t), and β(t) as defined before. This together with Remark 3.1, yields

E[ν(t, u)] ≤ r(t, t_{0}, E[y_{0}]).

And from (3.42) we have

(4.9)  ∥r(t, t_{0}, E[y_{0}])∥ ≤ ∥E[y_{0}] | exp[∫_{t_{0}}^{t} −α(s)ds]
+ ∫_{t_{0}}^{t} exp[− ∫_{s}^{t} α(u)du]∥β(s)∥ds.

Assumptions (iv), and (v) together with (4.9) yields

lim_{t→∞} ∥r(t)∥ = 0.

Therefore all the assumptions of Theorem 4.1 are satisfied. Therefore the conclusion of the theorem remains valid. This completes the proof of the theorem.

In the following we state and prove a result concerning the qth-moment convergence in probability of solution process of (2.1).

Theorem 4.3: Assume that all the hypotheses of Theorem 4.1 hold. Then the qth-moment of the solution process u(t, x, η(t)) of (2.1) converges in probability to the zero vector.

Proof: The proof of this theorem can be formulated by following the proof of Theorem 4.1 and by using Chebyshev’s inequality.

Theorem 4.4: Assume that all the hypotheses of Theorem 4.2 hold. Then the qth-moment of the solution process u(t, x, η(t)) of (2.1) converges in probability to the zero vector.

Proof: The proof of this theorem can be formulated by following the proofs of Theorem 4.2 and Theorem 4.3. Further details are omitted.

5. STABILITY ANALYSIS

In this section, we employ the block comparison theorems developed in Section 3 to investigate different kinds of stability of the trivial solution process of (2.1).

First, we formulate the definitions of stability of the solution process of (2.1).

Definition 5.1: The trivial solution of the partial differential system (2.1) is said to be \underline{(SM^{∗}) stable in the (q, p)th-moment}, if for each ϵ > 0, t_{0} ∈ J, there exists a positive function δ = δ(t_{0}, ϵ) such that the inequality

∥u_{0}∥_{p} ≤ δ,

implies

∥u(t)∥_{p} < ϵ, t ≥ t_{0};

\underline{(ASM^{∗}) asymptotically stable in the (q, p)th-moment}, if it is stable in the (q, p)th moment and if for ϵ > 0, t_{0} ∈ J there exists positive numbers δ_{0} = δ(t_{0}) and T = T (t_{0}, ϵ) such that the inequality

∥u_{0}∥_{p} ≤ δ,

implies

∥u(t)∥_{p} < ϵ, t ≥ t_{0} + T;

\underline{(SP^{∗}) the qth-moment is stable in probability}, if for each ϵ > 0, ξ > 0, t_{0} ∈ J, there exist a positive function δ = δ(t_{0}, ϵ, ξ) that is continuous in t_{0} for each ϵ and ξ such that the inequality

P [ω : ||u_{0}(ω)||_{q} > δ] < ξ,

implies

P [ω : ||u(t, ω)||_{q} ≥ ϵ] < ξ, t ≥ t_{0};

\underline{(ASP^{∗}) the qth-moment is asymptotically stable in probability}, if it is stable in probability and if for each ϵ > 0, ξ > 0, t_{0} ∈ J, there exist positive numbers δ_{0} = δ(t_{0}) and T = T (t_{0}, ϵ, ξ) such that the inequality

P [ω : ||u_{0}(ω)||_{q} > δ] < ξ,

implies

P [ω : ||u(t, ω)||_{q} ≥ ϵ] < ξ, t ≥ t_{0} + T.

Definition 5.2: The trivial solution y = 0 of the block comparison system (2.5) is said to be \underline{(S) stable}, if for each ϵ > 0, t_{0} ∈ J, there exist a positive function δ(t_{0}, ϵ) that is continuous in t_{0} for each ϵ such that the inequality

∑_{k=1}^{s} ∑_{i=1}^{N} E[y_{i_{0}}^{k}] ≤ δ,

implies

∑_{k=1}^{s} ∑_{i=1}^{N} y_{i}^{k}(t, t_{0}, E[y_{0}] < ϵ, t ≥ t_{0}.

\underline{(AS) asymptotically stable}, if it is stable and converges to the zero vector.

Theorem 5.1: Assume that all the hypotheses of Theorem 4.1 are satisfied. Further assume that f(t, x, 0, 0, 0) = 0, w(t, 0) ≡ 0, and

b(∥u∥_{q}^{p}) ≤ ∑_{k=1}^{s} ∑_{i=1}^{N} V_{i}(t, u, k) ≤ a(t, ∥u∥_{q}^{p}),

a, b ∈ C[J × R_{+}, R_{+}], a ∈ CK for each t ∈ J and b ∈ VK.

Then

(S) implies (SM^{∗}),

and

(AS)  implies  (ASM^{∗}).

Proof: From the conclusion of Theorem 3.1 and Lemma 3.2 [8], we have

(5.1) ∑_{k=1}^{s} ∑_{i=1}^{N} E[V_{i}(t, u(t, x), k)] ≤ ∑_{k=1}^{s} ∑_{i=1}^{N} r_{i}^{k}(t, t_{0}, E[y_{0}]), for each k ∈ S.

Let ϵ > 0 and t_{0} ∈ J be given. From the nature of b, b(ϵ^{p}) > 0, for p ≥ 1. From the assumption that the trivial solution of (2.5) is stable, then there exist δ_{1} = δ_{1}(t_{0}, ϵ) > 0, that is continuous in t_{0} for each ϵ such that

(5.2)  ∑_{k=1}^{s} ∑_{i=1}^{N} r_{i}^{k}(t, t_{0}, E([y_{0}]) < b(ϵ^{p}),  for all t ≥ t_{0},

whenever

∑_{k=1}^{s} ∑_{i=1}^{N} E[yk_{i_{0}} ] ≤ δ_{1},

where r(t, t_{0}, E[y_{0}]) is the maximal solution of (2.5) through (t_{0}, E[y_{0}]). We choose y_{i0}(ω) such that v_{i}(t_{0}) ≤ y_{i0}(ω), and

∑_{k=1}^{s} ∑_{i=1}^{N} E[y_{i_{0}}^{k}] = a(t_{0}, E[∥u_{0}∥_{q}^{p}]).

Since a ∈ CK, we can find a δ = δ(t_{0}, ϵ) such that

(5.3)  E[∥u_{0}∥_{q}^{p}] ≤ δ^{p} implies a(t_{0}, E[∥u_{0}∥_{q}^{p}]) ≤ δ_{1}.

Therefore

∥u_{0}∥_{p} ≤ δ.

Now we claim that ∥u_{0}∥_{p} ≤ δ, implies ∥u(t)∥_{p} < ϵ, t ≥ t_{0}. Suppose this is false, then there would exist a solution process u(t, x, ω) with ∥u_{0}∥_{p} ≤ δ and t_{1} > t_{0} such that

(5.4)  ∥u(t_{1})∥_{p} = ϵ.

From the inequality in the assumptions of the theorem, (5.1), (5.4), and the convexity of b and Jensen’s inequality, we arrive at the contradiction

b(ϵ^{p}) ≤ ∑_{k=1}^{s} ∑_{i=1}^{N} E[V_{i}(t_{1}, u(t_{1}, x), k)] ≤ ∑_{k=1}^{s} ∑_{i=1}^{N} r_{i}^{k}(t_{1}, t_{0}, E[y_{0}])] < b(ϵ^{p}).

Therefore our claim is true. This proves the first statement (S) implies (SM^{∗}).

Now, to prove the second conclusion of the theorem, from (AS) of the trivial solution of (2.5), the stability of the trivial solution of (2.1) follows immediately as shown in the proof of the first conclusion. The proof of the second part in the asymptotic stability definition follows by following an argument similar to the argument in the proof of Theorem 4.1. The details are omitted.

Here we present a special case of Theorem 5.1 showing the applicability of Theorem 5.1.

Theorem 5.2: Assume that all the hypotheses of Theorem 4.2 are satisfied. Further assume that f(t, x, 0, 0, 0) = 0, β(t) ≡ 0, and

b(∥u∥_{q}^{p}) ≤ ∑_{k=1}^{s} ∑_{i=1}^{N} V_{i}(t, u^{i}, k) ≤ a(t, ∥u∥_{q}^{p}),

a ∈ C[J × R_{+}, R_{+}], a ∈ CK for each t ∈ J and b ∈ VK. Then

(S) implies (SM^{∗}),

and

(AS) implies (ASM∗).

Proof: From the hypotheses of Theorem 3.1 and an application of Corollary 3.1, we have

(5.5)  ∥r(t, t_{0}, E[y_{0}])∥ ≤ ∥E[y_{0}]∥ exp[∫_{t_{0}}^{t} −α(s)ds]
+ ∫_{t_{0}}^{t} exp[− ∫_{s}^{t} α(u)du]∥β(s)∥ds.

This together with assumptions (iv), and (v) of Theorem 4.2, yields

lim_{t→∞} ∥r(t)∥ = 0.

In addition, because of β(t) ≡ 0, it follows that the trivial solution of (3.39) is asymptotically stable. Therefore all the assumptions of Theorem 5.1 are satisfied. The conclusion of the theorem follows immediately.

Now, we state without proof theorems similar to Theorems 4.3 and 4.4 that assures the qth-moment stability in probability of the trivial solution of (2.1).

Theorem 5.3: Assume that all the hypotheses of Theorem 4.3 are satisfied. Further assume that f(t, x, 0, 0, 0) = 0, w(t, 0) ≡ 0, and

b(∥u∥_{q}) ≤ ∑_{k=1}^{s} ∑_{i=1}^{N} V_{i}(t, u, k) ≤ a(t, ∥u∥_{q}),

a ∈ C[J × R_{+}, R_{+}], a ∈ CK for each t ∈ J and b ∈ VK. Then

(S) implies (SP^{∗}),

and

(AS) implies (ASP^{∗}).

Theorem 5.4: Assume that all the hypotheses of Theorem 4.4 are satisfied. Further assume that f(t, x, 0, 0, 0) = 0, β(t) ≡ 0, and

b(∥u∥_{q}) ≤ ∑_{k=1}^{s} ∑_{i=1}^{N} V_{i}(t, u, k) ≤ a(t, ∥u∥_{q}),

a ∈ C[J × R_{+}, R_{+}], a ∈ CK for each t ∈ J and b ∈ VK. Then

(S) implies (SP^{∗}),

and

(AS) implies (ASP^{∗}).

Remark 5.1: The proofs of Theorem 5.3 and Theorem 5.4 can be formulated by following the proofs of Theorem 5.1 and Theorem 5.2 together with Chebychev’s inequality.

Remark 5.2: We note that assumption (iv) of Theorem 4.2 implies that the matrix C in (3.36) has positive diagonal elements, that is

(5.6) γ^{k}(t) = min_{1≤r≤N}{C_{rr}^{k}(t) + π_{kk} − d_{r}^{−1} ∑_{i=1,i≠r}^{N} d_{i}|C_{ir}^{k}(t)|} > 0, k ∈ S,

and it satisfies the quasidominant property

(5.7)  c_{kk}(t) > a_{k}^{−1} ∑_{l=1,l≠k}^{s} a_{l}|c_{lk}|,

where C^{k}(t) = (C_{ir}^{k}(t)) is defined in (3.18) and a_{l} > 0. From (5.5) and π_{kk} > 0 for each k ∈ S, the positivity of γ^{k}(t) is influenced by the random structural perturbations. Therefore, the (q, p)th-moment convergence and stability properties are influenced by the Markovian structural perturbations. Indeed they are the stabilizing agents.

6. EXAMPLE

Let us consider the following system

(6.1) \begin{cases}du = [ϕ(η(t))\frac{∂^{2}u}{∂x^{2}} + ψ(η(t))\frac{∂u}{∂x} + θ(η(t))u]dt + β(η(t))udω(t), \\ u(t_{0}, x) = u_{0}(x)  on \overline{H}, \\ u(t, x) = ϕ(t, x)  on Γ_{t_{0}}, \end{cases}

where ϕ, ψ, θ, and β are continuous functions, η(t) is right-continuous twostate Markov chain. We choose S = {E_{1}, E_{2}}, ω(t) is 1-dimensional Wiener process. We choose our Lyapunov functional to be

V (t, u, η(t)) = \frac{1}{2} ∫\overline{H} H(η(t))u^{2}(t, x)dx,

Looking at the following difference

V (t + h, u(t + h, x), η(t + h)) − V (t, u(t, x), η(t)) (6.2)

= \frac{1}{2} ∫ [u^{2}(t + h, x)H(η(t + h)) − u^{2}(t, x)H(η(t))]dx

= \frac{1}{2} ∫ [(u^{2}(t + h, x) − u^{2}(t, x))H(η(t + h))

+ u^{2}(t, x)(H(η(t + h)) − H(η(t)))]dx

= \frac{1}{2} ∫ [(2u∂uh + ϵ(h))H(η(t + h))

+ u^{2}(t, x)(H(η(t + h)) − H(η(t)))]dx

= \frac{1}{2} ∫ [(2uh[ϕ(η(t))\frac{∂^{2}u}{∂x^{2}} + ψ(η(t))\frac{∂u}{∂x} + θ(η(t))u

+ β(η(t))dω + \frac{1}{2}β(η(t))uβ(η(t))]H(η(t + h))

+ ϵ(h))H(η(t + h)) + u^{2}(t, x)(H(η(t + h)) − H(η(t)))]dx

Taking the conditional expectation of both sides, we obtain

E[V (t + h, u(t + h, x), η(t + h))|η(t) = k, u(t_{0}, x) = u_{0}] (6.3)

− E[V (t, u(t, x), η(t))|η(t) = k, u(t_{0}, x) = u_{0}]

= \frac{1}{2} ∫ [h[2uϕ(k)\frac{∂^{2}u}{∂x^{2}} + 2uψ(k)\frac{∂u}{∂x} + 2uθ(k)u

+ β^{2}(k)u^{2}](H(k) (1 − ∑_{l=1,l≠k}^{s} π_{lk}h)

+ ϵ(h)E[H(η(t + h))|η(t) = k]

+ u^{2} ( ∑_{l=1,l≠k}^{s} H(l)π_{lk}h + H(k)π_{kk}h + o(h) − H(k))]dx.

Dividing both sides by h and taking the limit superior as h → 0, and by using the Divergence Theorem and Schwartz inequality together with the definition of V, we arrive at

(6.4)  D^{+}V (t, u, k) ≤ − (\frac{2ϕ(k)}{M} − 2θ(k) − β^{2}(k)) V (t, u, k),

+ ∑_{l=1,l≠k}^{s} π_{lk}V (t, u, k) − π_{kk}V (t, u, k),

where M depends on the domain of integration. From (6.2), we note that

(6.5)  w^{k}(t, V (t, u, k)) = − (\frac{2ϕ(k)}{M} − 2θ(k) − \frac{1}{2}β^{2}(k)) V (t, u, k)

+ ∑_{l=1,l≠k}^{s} π_{lk}V (t, u, k) − π_{kk}V (t, u, k).

Here our comparison system is

(6.6)  y′ = w^{k}(t, y) = −Cy,  y(t_{0}) = y_{0}.
(6.7)  C = (c_{lk})_{2×2},

where

c_{lk} = \begin{cases}\frac{2ϕ(k)}{M} − 2θ(k) − β^{2}(k) + π_{kk}, & l = k \\ −π_{lk} & l ≠ k. \end{cases}

Here, it is obvious that the comparison system (6.4) has a maximal solution, r(t, t_{0}, y_{0}). Therefore all the assumptions of Theorem 3.1 are satisfied. The application of Theorem 3.1 yields

E[V (t, u, k)|u_{0}, k] ≤ r(t, t_{0}, y_{0}),

For the coefficient matrix, C, we assume that

−\frac{2ϕ(k)}{M} + 2θ(k) + β^{2}(k) ≤ π_{kk}, k = 1, 2

and

α = min_{1≤k≤2} {c_{kk} − d_{k}^{−1} ∑_{l=1,l≠k}^{2} d_{l}|c_{lk}|} > 0, for d_{k} > 0,

in this case the solution through (t_{0}, E[y_{0}]), satisfies the inequality

||r(t, t_{0}, E[y_{0}])|| ≤ E[||y_{0}||] exp[−α(t − t_{0})],

which converges to zero through (t_{0}, E[y_{0}]) as t ← ∞, provided α > 0. Now define

(6.8)  b(∥u∥_{q}^{2}) = m∥u∥_{q}^{2}, and a(t, ∥u∥_{q}^{2}) = M∥u∥_{q}^{2},

where m = min_{1≤k≤s} H(E_{k}), M = max_{1≤k≤s} H(E_{k}), H(E_{k}) > 0, and p = 2. Then

b(∥u∥_{q}^{2}) ≤ ∑_{k=1}^{2} V (t, u, k) ≤ a(t, ∥u∥_{q}^{2})

From the above discussions and the assumption that the stochastic system has the trivial solution, all the assumptions of Theorem 4.2 and Theorem 5.2 are satisfied. Therefore the mean-square convergence and stability of the qth-moment of the solution process of the system are guaranteed.

7. ACKNOWLEDGMENTS

This research is supported in part by NASA grant No. NCCW-0085, NSF grant No. HRD-9706268, HRD-0531490 and U.S. Army Research Office grant No. DAAD190110740 and the support is greatly appreciated.

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