Exponential Stability for a Class of Singularly Perturbed Itô Differential

Equations

V. Dragan and T. Morozan

Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO-70700, Bucharest, Romania vdragan@stoilow.imar.ro tmorozan@stoilow.imar.ro

Abstract

The problem of exponential stability in mean square of the zero solution for a class of singularly perturbed system of Itô differential equations is investigated.

Estimates of the block components of the fundamental random matrix are provided.

Keywords: Itô differential equations, singular perturbations, exponential stability in mean square.

AMS classification: 34D15, 34F05 ^{1}

1 Problem formulation

Consider the system of Itô differential equations:

dx_{1}(t) = [A_{11}x_{1}(t) + A_{12}x_{2}(t)]dt + ∑_{k=1}^{N} A_{11}^{k}x_{1}(t)dw_{k}(t)  (1.1)

εdx_{2}(t) = [A_{21}x_{1}(t) + A_{22}x_{2}(t)]dt + ∑_{k=1}^{N} A_{21}^{k}x_{1}(t)dw_{k}(t)

1This paper is in the final form and no version of it will be submitted for publication elsewhere.

where x_{i} ∈ R^{n_{i}}, i = 1, 2, A_{ij}, A_{ij}^{k} are constant matrices with apropriate dimensions, ε > 0 is a small parameter; w(t) = (w_{1}(t) w_{2}(t)...w_{N}(t))^{∗} is a standard Wiener process on a given probability space (Ω, F, P).

For each ε > 0 a solution of the system (1.1) is a random process x(t, ε) =

( \begin{matrix}x_{1}(t, ε) \\ x_{2}(t, ε) \end{matrix} ) which is continuous with probability 1, verifies (1.1) and for all

t, x(t, ε) is adapted to family of σ-algebras generated by the process w(t).

Definition 1.1 We say that the zero solution of the system (1.1) is mean square exponentially stable if there exist β ≥ 1, α > 0 such that for any solution x(t, ε) of the system (1.1) with x(0, ε) = x_{0} independent of w(t), t ≥ 0, we have:

E|x(t, ε)|^{2} ≤ βe^{−αt}E|x(0)|^{2}

for all t ≥ 0 E being the expectation.

Remark

a) The constants α, β from the above definition may be dependent on the small parameter ε > 0.

b) It can be seen [11] that in the definition of exponential stability we may consider only the solutions with initial conditions x(0) ∈ R^{n}(n = n_{1} + n_{2}).

The goal of this paper is to provide conditions assuring exponential stability of the zero solution of the system (1.1). Such conditions are expressed in terms of exponential stability of the zero solution of some subsytems of lower dimension not depending upon small parameter ε.

If A_{22} is an invertible matrix we may associate the following system of Itô differential equations of lower dimension:

dx_{1}(t) = A_{r}x_{1}(t)dt + ∑_{k=1}^{N} A_{r}^{k}x_{1}(t)dw_{k}(t)  (1.2)

where

A_{r} = A_{11} − A_{12}A_{22}^{−1}A_{21}
A_{r}^{k} = A_{11}^{k} − A_{12}A_{22}^{−1}A_{21}^{k}, k = 1, 2, ..., N.

We shall investigate the relationship between the exponential stability of the zero solution of the “reduced system” (1.2) and the exponential stability of the zero solution of the system (1.1) for ε > 0 small enough.

We extend to the class of stochastic systems (1.1) the well known result of Klimusev-Krasovski [9] from the ordinary differential equations.

The same problem was investigated in [6] in the case when the fast equation is

εdx_{2}(t) = [A_{21}x_{1}(t) + A_{22}x_{2}(t)]dt + \sqrt{ε} ∑_{k=1}^{N}[A_{21}^{k}x_{1}(t) + A_{22}^{k}x_{2}(t)]dw_{k}(t).

The convergence in mean square of the slow component x_{1}(t, ε) to a solution of the reduced system is proved in [4] when the system (1.1) is nonlinear.

2 Some preliminary results

Recall several results which are used in the next section.

A. For each ε > 0 we denote Φ(t, t_{0}, ε), t ≥ t_{0} the fundamental random matrix solution of the system (1.1) (see [5]), the columns of Φ(t, t_{0}, ε) are solutions of the system (1.1) and Φ(t_{0}, t_{0}, ε) = I_{n_{1}+n_{2}}.

If x(t) = x(t, t_{0}, x_{0}, ε) is the solution of the system (1.1) which verifies x(t_{0}, t_{0}, x_{0}, ε) = x_{0} then we have x(t, ε) = Φ(t, t_{0}, ε)x_{0}, x_{0} ∈ R^{n_{1}+n_{2}}.

B. Consider the linear system of Itô differential equations

dx(t) = A_{0}x(t)dt + ∑_{k=1}^{N} A_{k}x(t)dw_{k}(t)  (2.1)

we have

Lemma 2.1 [7] The following are equivalent:

(i) the zero solution of the system (2.1) is exponentially stable in mean square.

(ii) there exists a positive matrix X which solves the linear equation of Liapunov type

AX + XA^{∗} + ∑_{k=1}^{N} A^{k}X(A^{k})^{∗} + I = 0.  (2.2)

Moreover if (i) holds then the unique positive solution of the equation (2.2) is given by

X = E ∫_{0}^{∞} Φ(t, 0)Φ^{∗}(t, 0)dt  (2.3)

where Φ(t, 0) is the fundamental random matrix of the system (2.1).

C. If I ⊂ R is an interval, we denote L_{w}^{2}(I, R^{n}) the subspace of the measurable processes f : I × Ω → R^{n} which are adapted with respect to the family of the σ-algebras generated by the Wiener process w(t) and

E ∫_{I} |f(t)|^{2}dt < ∞.

Resoning as in the prove of proposition 1 in [2] we obtain:

Lemma 2.2 Assume that the zero solution of the system (2.1) is exponentially stable in mean square. If x(t) is a solution of the affine system

dx(t) = [A_{0}x(t) + f_{0}(t)]dt + ∑_{k=1}^{N}[A_{k}x(t) + f_{k}(t)]dw_{k}(t),

t ≥ 0, x(t_{0}) = x_{0} then

E|x(t)|^{2} ≤ β(e^{−α(t−t_{0})}|x_{0}|^{2} + ∑_{k=0}^{N} ∫_{t_{0}}^{t} e^{−α(t−s)}E|f_{k}(s)|^{2}ds),

for all t ∈ [t_{0}, t_{1}], f_{k} ∈ L_{w}^{2}([t_{0}, t_{1}), R^{n}), α, β are positive constants not depending upon f_{k}, x_{0}.

3 The main result

Theorem 3.1 Assume

(i) the zero solution of the reduced system (1.2) is exponentially stable in mean square;

(ii) the eigenvalues of the matrix A_{22} are located in the half plane Reλ < 0.

Then there exists ε_{0} > 0 such that for arbitrary ε ∈ (0, ε_{0}) the zero solution of the system (1.1) is exponentially stable in mean square. Moreover, if

( \begin{matrix}Φ_{11}(t, t_{0}, ε) & Φ_{12}(t, t_{0}, ε) \\ Φ_{21}(t, t_{0}, ε) & Φ_{22}(t, t_{0}, ε) \end{matrix} )

is a partition of the fundamental “random” matrix Φ(t, t_{0}, ε) of the system (1.1) we have the estimates

E|Φ_{11}(t, t_{0}, ε)|^{2} ≤ β_{1}e^{−α_{1}(t−t_{0})}
E|Φ_{12}(t, t_{0}, ε)|^{2} ≤ β_{1}εe^{−α_{1}(t−t_{0})}  (3.1)
E|Φ_{21}(t, t_{0}, ε)|^{2} ≤ \frac{β_{2}}{ε} (e^{−α_{1}(t−t_{0})} − e^{−α_{2} \frac{t−t_{0}}{ε}}) + β_{3}e^{−α_{1}(t−t_{0})}
E|Φ_{22}(t, t_{0}, ε)|^{2} ≤ β_{3}e^{−α_{1}(t−t_{0})}

for all t ≥ t_{0}, β_{k}, α_{k} positive constants not depending upon t, t_{0}, ε.

Proof: Consider the nonlinear equation

A_{21} + A_{22}T_{21} − εT_{21}(A_{11} + A_{12}T_{21}) = 0.  (3.2)

By a standard implicit functions argument, we deduce that there exists ε_{1} > 0 and an anlytic function T_{21}(ε) defined for |ε| < ε_{1} which is a solution for equation (3.2) with T_{21}(0) = −A_{22}^{−1}A_{21}. Moreover T_{21}(ε) = −A_{22}^{−1}A_{21} + εT̂_{21}(ε) with |T̂_{21}(ε)| ≤ ĉ < ∞, (∀)|ε| < ε_{1}.

Let T_{12}(ε) be the unique solution of the equation

ε(A_{11} + A_{12}T_{21}(ε))T_{12} − T_{12}(A_{22} − εT_{21}(ε)A_{12}) + A_{12} = 0.  (3.3)

We have T_{12}(ε) = A_{12}A_{22}^{−1} + εT̂_{12}(ε) where |T̂_{12}(ε)| ≤ ĉ < ∞ for all |ε < ε_{1}.

If ( \begin{matrix}x_{1}(t, ε) \\ x_{2}(t, ε) \end{matrix} ) is a solution of the system (1.1) we define

( \begin{matrix}ξ_{1}(t, ε) \\ ξ_{2}(t, ε) \end{matrix} ) = ( \begin{matrix}I_{n_{1}} & −εT_{12}(ε) \\ 0 & I_{n_{2}} \end{matrix} ) ( \begin{matrix}I_{n_{1}} & 0 \\ −T_{21}(ε) & I_{n_{2}} \end{matrix} ) ( \begin{matrix}x_{1}(t, ε) \\ x_{2}(t, ε) \end{matrix} ) . (3.4)

Using the Itô formula ([5]) we deduce that ( \begin{matrix}ξ_{1}(t, ε) \\ ξ_{2}(t, ε) \end{matrix} ) is a solution of the

following system:

dξ_{1}(t) = A_{1}(ε)ξ_{1}(t)dt + ∑_{k=1}^{N}(Â_{11}^{k}(ε)ξ_{1}(t) + εÂ_{12}^{k}(ε)ξ_{2}(t))dw_{k}(t)

εdξ_{2}(t) = A_{2}(ε)ξ_{2}(t)dt + ∑_{k=1}^{N}(Â_{21}^{k}(ε)ξ_{1}(t) + εÂ_{22}^{k}(ε)ξ_{2}(t))dw_{k}(t) (3.5)

where

A_{1}(ε) = A_{11} + A_{12}T_{21}(ε) = A_{r} + O(ε)
A_{2}(ε) = A_{22} − εT_{21}(ε)A_{12}
Â_{21}^{k}(ε) = A_{21}^{k} − εT_{21}(ε)A_{11}^{k}  (3.6)
A_{11}^{k}(ε) = A_{r}^{k} + O(ε), Â_{12}^{k}(ε) = Â_{11}^{k}(ε)T_{12}(ε)
Â_{22}^{k}(ε) = Â_{21}^{k}(ε)T_{12}(ε).

Since the zero solution of the reduced system (1.2) is exponentially stable in mean square it follows that the linear equation

A_{r}X + XA_{r}^{∗} + ∑_{k=1}^{N} A_{r}^{∗}X(A_{r}^{k})^{∗} + I_{n_{1}} = 0  (3.7)

has a unique symmetric solution X_{r} > 0.

Also, since A_{22} is stable, it follows that the Liapunov equation

A_{22}X_{2} + X_{2}A_{22}^{∗} + I_{n_{2}} = 0

has a unique solution X_{2} > 0 (see (2.3)).

By a standard argument based on implicit function theorem we deduce that the linear equations

A_{1}(ε)X_{1} + X_{1}A_{1}^{∗}(ε) + ∑_{k=1}^{N} Â_{11}^{k}(ε)X_{1}(A_{11}^{k}(ε))^{∗} + I_{n_{1}} = 0

A_{2}(ε)X_{2} + X_{2}A_{2}^{∗}(ε) + ε ∑_{k=1}^{N} Â_{22}^{k}(ε)X_{2}(Â_{22}^{k}(ε))^{∗} + I_{n_{2}} = 0

have positive solutions X_{1}(ε), X_{2}(ε) respectively defined for all ε ∈ (0, ε_{2}). Moreover, there exist μ_{1}, μ_{2}, ν_{1}, ν_{2} not depending upon ε such that

0 < μ_{1}I_{n_{1}} ≤ X_{1}(ε) ≤ μ_{2}I_{n_{1}};
0 < ν_{1}I_{n_{2}} ≤ X_{2}(ε) ≤ ν_{2}I_{n_{2}}.

Applying Lemma 2.1 we conclude that the zero solution of the system

dξ_{1}(t) = A_{1}(ε)ξ_{1}(t)dt + ∑_{k=1}^{N} Â_{11}^{k}(ε)ξ_{1}(t)dw_{k}(t)  (3.8)

and the one of the system

εdξ_{2}(t) = A_{2}(ε)ξ_{2}(t)dt + ε ∑_{k=1}^{N} Â_{22}^{k}(ε)ξ_{2}(t)dw_{k}(t)  (3.9)

respectively, are exponentially stable in mean square.

Applying Lemma 2.2 to the first equation in the system (3.5) we deduce that there exist β̃_{1} > 0, α̃_{1} > 0 such that

E|ξ_{1}(t, ε)|^{2} ≤ β̃_{1}(e^{−α̃_{1}(t−t_{0})}E|ξ_{1}(t_{0})|^{2} + ε^{2} ∫_{t_{0}}^{t} e^{−α̃_{1}(t−σ)}E|ξ_{2}(s, ε)|^{2}ds. (3.10)

Also, applying Lemma 2.2 to the second equation in the system (3.5) we deduce that there exist β̃_{2} > 0, α̃_{2} > 0 such that

E|ξ_{2}(t, ε)|^{2} ≤ β̃_{2}[e^{−α̃_{2} \frac{(t−t_{0})}{ε}} E|ξ_{2}(t_{0}, ε)|^{2} + \frac{1}{ε^{2}} ∫_{t_{0}}^{t} e^{−α̃_{2} \frac{(t−σ)}{ε}} E|ξ_{1}(σ, ε)|^{2}dσ](3.11)

for all t ≥ t_{0}.

Substituting (3.11) in (3.10) and changing the order of integration we get

E|ξ_{1}(t, ε)|^{2} ≤ β̃_{3}e^{−α̃_{1}(t−t_{0})}[E|ξ_{1}(t_{0}, ε)|^{2} + εE|ξ_{2}(t_{0}, ε)|^{2}]
+β̃_{4} ∫_{t_{0}}^{t}[εe^{−α̃_{1}(t−s)} + e^{−α̃_{2} \frac{(t−s)}{ε}}]E|ξ_{1}(s, ε)|^{2}ds.

By a standard argument in singular perturbation theory (see [1, 12]) we deduce that there exists 0 < ε_{0} ≤ ε_{2} such that for arbitrary ε ∈ (0, ε_{0}) we have

E|ξ_{1}(t, ε)|^{2} ≤ β̃_{5}e^{−α_{1}(t−t_{0})}[E|ξ_{1}(t_{0}, ε)|^{2} + εE|ξ_{2}(t_{0}, ε)|^{2}]  (3.12)

for all t ≥ t_{0}, where β̃_{5} > 0, α_{1} ∈ (0, α̃_{1}).

Using (3.12) in (3.11) we obtain

E|ξ_{2}(t, ε)| ≤ β̃_{2}e^{−α̃_{2} \frac{(t−t_{0})}{ε}} [E|ξ_{2}(t_{0}, ε)|^{2}]  (3.13)

+\frac{β̃_{6}}{ε} [e^{−α_{1}(t−t_{0})} − e^{−α̃_{2} \frac{(t−t_{0})}{ε}}][E|ξ_{1}(t_{0}, ε)|^{2} + εE|ξ_{2}(t_{0}, ε)|^{2}].

Reversing (3.4) and taking into account (3.12), (3.13) we get:

E|x_{1}(t, ε)|^{2} ≤ β_{1}e^{−α_{1}(t−t_{0})}[E|x_{1}(t_{0}, ε)|^{2} + εE|x_{2}(t_{0}, ε)|^{2}]
E|x_{2}(t, ε)|^{2} ≤ β_{2}e^{−α_{2} \frac{(t−t_{0})}{ε}} E|x_{2}(t_{0}, ε)|  (3.14)

+\frac{β̃_{2}}{ε} (e^{−α_{1}(t−t_{0})} − e^{−α_{2} \frac{(t−t_{0})}{ε}})(E|x_{1}(t_{0}, ε)|^{2} + εE|x_{2}(t_{0}, ε)|^{2})

which means the exponential stability of the zero solution of the system (1.1).

The estimations (3.1) follows from (3.14) replacing ( \begin{matrix}x_{1}(t_{0}, ε) \\ x_{2}(t_{0}, ε) \end{matrix} ) by ( \begin{matrix}I_{n_{1}} \\ 0 \end{matrix} )

and ( \begin{matrix}0 \\ I_{n_{2}} \end{matrix} ) respectively.

Remark. The estimates of the block components of the fundamental matrix solution Φ(t, t_{0}, ε) of the system (1.1) obtained in the above theorem differ essentially from the ones obtained in the deterministic framework (see [1]).

This is due to the intensity of the white noise from the second equation of the system (1.1). Let us consider the system of differential equations:

dx_{1}(t) = [A_{11}x_{1}(t)+A_{12}x_{2}(t)]dt+∑_{k=1}^{N}[A_{11}^{k}x_{1}(t)+εA_{12}^{k}x_{1}(t)]dw_{k}(t) (3.15)
εdx_{2}(t) = [A_{21}x_{1}(t)+A_{22}x_{2}(t)]dt+\sqrt{ε} ∑_{k=1}^{N}[A_{21}^{k}x_{1}(t)+A_{22}^{k}x_{2}(t)]dw_{k}(t).

In this case the reduced system is :

dx_{1}(t) = [A_{11} − A_{12}A_{22}^{−1}A_{21}]x_{1}(t)dt + ∑_{k=1}^{N} A_{11}^{k}x_{1}(t)dw_{k}(t).  (3.16)

We associate to system (3.15) the following system:

dx_{2}(τ) = A_{22}x_{2}(τ)dτ + ∑_{k=1}^{N} A_{22}^{k}x_{2}(τ)dw_{k}(τ)  (3.17)

which is called “the boundary layer subsystem”. If in the system (3.15) we perform the coordinates transformation (3.4) we get:

dξ_{1}(t) = A_{1}(ε)ξ_{1}(t)dt + ∑_{k=1}^{N}(Â_{11}^{k}(ε)ξ_{1}(t) + εÂ_{12}^{k}(εξ_{2}(t))dw_{k}(t) (3.18)
εdξ_{2}(t) = A_{2}(ε)ξ_{2}(t)dt + \sqrt{ε} ∑_{k=1}^{N}(Â_{21}^{k}(ε)ξ_{1}(t) + Â_{22}^{k}(ε)ξ_{2}(t))dw_{k}(t).

With the same arguments as in the proof of Theorem 3.1 we obtain: Theorem 3.1’ Assume

a) A_{22} is a invertible matrix.

b) The zero solution of the reduced system (3.16) and the one of the boundary layer system (3.17), respectively, is exponentially stable in mean square. Under these assumptions the block components of the fundamental random matrix solution of the system (3.15) verify:

E|Φ_{11}(t, t_{0}, ε)|^{2} ≤ β_{1}e^{−α_{1}(t−t_{0})}
E|Φ_{12}(t, t_{0}, ε)|^{2} ≤ β_{1}εe^{−α_{1}(t−t_{0})}
E|Φ_{21}(t, t_{0}, ε)|^{2} ≤ β_{2}e^{−α_{1}(t−t_{0})}  (3.19)
E|Φ_{22}(t, t_{0}, ε)|^{2} ≤ β_{2}(e^{−α_{2} \frac{(t−t_{0})}{ε}} + εe^{−α_{1}(t−t_{0})})

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 7, p. 8 for all t ≥ t_{0}, ε > 0 small enough; α_{k}, β_{k} positive constants not depending upon t, t_{0}, ε.

The estimates obtained in Theorem 3.1’ are similar to the ones obtained in deterministic framework [1]. In the case of system (3.15) we are able to prove the following converse result:

Theorem 3.2 Assume that A_{22} is an invertible matrix. If the fundamental random matrix solution Φ(t, t_{0}, ε) of the system (3.15) verifies the estimates (3.19), then the zero solution of the reduced subsystem (3.16) and the one of the boundary layer subsystem (3,17) respectively is exponentially stable in mean square.

Proof. Denote X(ε) = E ∫_{0}^{∞} Φ(t, 0, ε)I(ε)Φ^{∗}(t, 0, ε)dt where

I(ε) = ( \begin{matrix}I_{n_{1}} & 0 \\ 0 & ε^{−1}I_{n_{2}} \end{matrix} ). If ( \begin{matrix}X_{11}(ε) & X_{12}(ε) \\ X_{12}^{∗}(ε) & X_{22}(ε) \end{matrix} ) is the partition of X(ε)

it is easy to see that

X_{11}(ε) = E ∫_{0}^{∞}(Φ_{11}(t, 0, ε)Φ_{11}^{∗}(t, 0, ε) + ε^{−1}Φ_{12}(t, 0, ε)Φ_{12}^{∗}(t, 0, ε))dt
X_{12}(ε) = E ∫_{0}^{∞}(Φ_{11}(t, 0, ε)Φ_{21}^{∗}(t, 0, ε) + ε^{−1}Φ_{12}(t, 0, ε)Φ_{22}^{∗}(t, 0, ε))dt
X_{22}(ε) = E ∫_{0}^{∞}(Φ_{21}(t, 0, ε)Φ_{21}^{∗}(t, 0, ε) + ε^{−1}Φ_{22}(t, 0, ε)Φ_{22}^{∗}(t, 0, ε))dt.

Using the estimates (3.19) we see hat the integral from definition of X(ε) is convegent. Moreover we get:

X_{ij}(ε) ≤ ĉ < ∞  (3.20)

for ε > 0 small enough, ĉ not depending upon ε.

Based on Itô formula [5] we conclude that X(ε) verifies the equation

A(ε)X(ε) + X(ε)A^{∗}(ε) + ∑_{k=1}^{N} A^{k}(ε)X(ε)(A^{k}(ε))^{∗} + I(ε) = 0 (3.21)

where

A(ε) = ( \begin{matrix}A_{11} & A_{12} \\ ε^{−1}A_{21} & ε^{−1}A_{22} \end{matrix} )
A^{k}(ε) = ( \begin{matrix}A_{11}^{k} & εA_{12}^{k} \\ ε^{− \frac{1}{2}} A_{21}^{k} & ε^{− \frac{1}{2}} A_{22}^{k} \end{matrix} ) .

If

T (ε) = ( \begin{matrix}I_{n_{1}} & −εT_{12}(ε) \\ 0 & I_{n_{2}} \end{matrix} ) ( \begin{matrix}I_{n_{1}} & 0 \\ −T_{21}(ε) & I_{n_{2}} \end{matrix} )

is the matrix used in (3.4) we denote

( \begin{matrix}Y_{11}(ε) & Y_{12}(ε) \\ Y_{12}^{∗}(ε) & Y_{22}(ε) \end{matrix} ) = T (ε)X(ε)T^{∗}(ε).

Based on (3.20) we have

|Y_{ij}(ε)| ≤ c̃ < ∞  (3.22)

for ε > 0 small enough, c̃ not depending upon ε. Using (3.21) we obtain that Y_{ij}(ε) solve the following system

A_{1}(ε)Y_{11}(ε) + Y_{11}(ε)A_{1}^{∗}(ε) + ∑_{i=1}^{N}[Â_{11}^{i}(ε)Y_{11}(ε)(Â_{11}^{i}(ε))^{∗}

+εÂ_{12}^{i}(ε)Y_{12}^{∗}(ε)(Â_{11}^{i}(ε))^{∗} + εÂ_{11}^{i}(ε)Y_{12}(ε)(Â_{12}^{i}(ε))^{∗}
+ε^{2}Â_{12}^{i}(ε)Y_{22}(ε)(Â_{12}^{i}(ε))^{∗}] + M_{11}(ε) = 0

εA_{1}(ε)Y_{12}(ε) + Y_{12}(ε)A_{2}^{∗}(ε) + ∑_{i=1}^{N}[ε^{\frac{1}{2}} Â_{11}^{i}(ε)Y_{11}(ε)(Â_{21}^{i}(ε))^{∗}

+ε^{\frac{3}{2}} Â_{12}^{i}(ε)Y_{12}^{∗}(ε)(Â_{21}^{i}(ε))^{∗} + ε^{\frac{1}{2}} Â_{11}^{i}(ε)Y_{12}(ε)(Â_{22}^{i}(ε))^{∗}

+ε^{\frac{3}{2}} Â_{12}^{i}(ε)Y_{22}(ε)(Â_{22}^{i}(ε))^{∗} + M_{12}(ε) = 0  (3.23)

A_{2}(ε)Y_{22}(ε) + Y_{22}(ε)A_{2}^{∗}(ε) + ∑_{i=1}^{N}[Â_{21}^{i}(ε)Y_{11}(ε)(Â_{21}^{i}(ε))^{∗}

+Â_{22}^{i}(ε)Y_{12}^{∗}(ε)(Â_{21}^{i}(ε))^{∗} + Â_{21}^{i}(ε)Y_{12}(ε)(Â_{22}^{i}(ε))^{∗}
+Â_{22}^{i}(ε)Y_{22}(ε)(Â_{22}^{i}(ε))^{∗}] + M_{22}(ε) = 0

where

M_{11}(ε) = [I_{n_{1}} + εT_{12}(ε)T_{21}(ε)][I_{n_{1}} + εT_{12}(ε)T_{21}(ε)]^{∗} + εT_{12}(ε)T_{12}^{∗}(ε)
M_{12}(ε) = −[I_{n_{1}} + εT_{12}(ε)T_{21}(ε)]T_{21}^{∗}(ε) − T_{12}(ε)
M_{22}(ε) = I_{n_{2}} + εT_{21}(ε)T_{21}^{∗}(ε).

If ε_{k}, k ≥ 0 is a sequence with lim_{k→∞} ε_{k} = 0 then from (3.22) it follows that there exist a subsequence ε_{k_{l}} such that lim_{l→∞} Y_{ij}(ε_{k_{l}}) are well defined.

Set Y_{ij}^{0} = lim_{l→∞} Y_{ij}(ε_{k_{l}}).Replacing ε by ε_{k_{l}} in (3.23) and taking the limit for l → ∞ we get

A_{r}Y_{11}^{0} + Y_{11}^{0}A_{r}^{∗} + ∑_{i=1}^{N} A_{r}^{i}Y_{11}^{0}(A_{r}^{i})^{∗} + I_{n_{1}} = 0

A_{22}Y_{22}^{0} + Y_{22}^{0}A_{22}^{∗} + ∑_{i=1}^{N} A_{22}^{i}Y_{22}^{0}(A_{22}^{i})^{∗} + I_{n_{2}} + ∑_{i=1}^{N} A_{21}^{i}Y_{11}^{0}(A_{21}^{i})^{∗} = 0 (3.24)
Y_{12}^{0} = 0.

The conclusion follows using Lemma 2.1 and the first two equations of the system (3.24).

4 Application

In this section we shall present an application of the result of Theorem 3.1 designing of a stabilizing state feedback for the following class of controlled stochastic linear systems:

dx_{1}(t) = [A_{11}x_{1}(t)+A_{12}x_{2}(t)+B_{1}u(t)]dt+∑_{k=1}^{N} A_{11}^{k}x_{1}(t)dw_{k}(t) (4.1)

εdx_{2}(t) = [A_{21}x_{1}(t)+A_{22}x_{2}(t)+B_{2}u(t)]dt+∑_{k=1}^{N} A_{21}^{k}x_{1}(t)dw_{k}(t)

u ∈ R^{m} is the control, x_{i} ∈ R^{n_{i}}, i = 1, 2 are the states and A_{ij}^{k}, B_{j} are real constant matrices of appropriate dimensions.

Our aim is to construct a control law of the form:

u(t) = F_{1}x_{1}(t) + F_{2}x_{2}(t)  (4.2)

with the property that the zero solution of the corresponding closed-loop system:

dx_{1}(t) = [(A_{11}+B_{1}F_{1})x_{1}(t)+(A_{12}+B_{1}F_{2})x_{2}(t)]dt
+ ∑_{k=1}^{N} A_{11}^{k}x_{1}(t)dw_{k}(t)  (4.3)
εdx_{2}(t) = [(A_{21}+B_{2}F_{1})x_{1}(t)+(A_{22}+B_{2}F_{2})x_{2}(t)]dt

+ ∑_{k=1}^{N} A_{21}^{k}x_{1}(t)dw_{k}(t)

be mean square exponentially stable. To avoind the ill conditioning due to the presence of the small parameter ε in the system (4.1) the designing of the stabilizing state feedback (4.2) is made in two steps. In each step we shall design a stabilizing feedback gain for a controlled system of lower dimensions and independent of the small parameter ε.

To the system (4.1) with A_{22} invertible we associate the following reduced subsystem:

dx_{1}(t) = [A_{r}x_{1}(t) + B_{r}u(t)]dt + ∑_{k=1}^{N} A_{r}^{k}x_{1}(t)dw_{k}(t)  (4.4)

where A_{r} and A_{r}^{k} are the same as in (1.2) and B_{r} = B_{1} = A_{12}A_{22}^{−1}B_{2}.

The corresponding boundary layer subsystem of the system (4.1) is:

x′_{2}(σ) = A_{22}x_{2}(σ) + B_{2}u(σ)  (4.5)

σ = \frac{t}{ε}. If the pair (A_{22}, B_{2}) is stabilizable (see [10]) we choose F_{2} ∈ R^{m×n_{2}} such that the matrix A_{22} + B_{2}F_{2} have the eigenvalues located in the half plane Reλ < 0.

Now, using this choice of the feedback gain F_{2} we consider the following system:

dx_{1}(t) = [A_{r}x_{1}(t) + B_{r}u(t)]dt + ∑_{k=1}^{N} A_{r}^{k}(F_{2})x_{1}(t)dw_{k}(t)  (4.6)

where A_{r}^{k}(F_{2}) = A_{11}^{k} − [A_{12} + B_{1}F_{2}][A_{22} + B_{2}F_{2}]^{−1}A_{21}^{k}.

Assuming that the system (4.6) is stochastically stabilizable we choose the matrix F_{r} ∈ R^{m×n_{1}} such that the zero solution of the closed-loop system

\frac{dx_{1}}{dt} = [A_{r} + B_{r}F_{r}]x_{1}(t)dt + ∑_{k=1}^{N} A_{r}^{k}(F_{2})x_{1}(t)dw_{k}(t)  (4.7)

be mean square exponentially stable.

Set

F_{1} = [I_{m} + F_{2}A_{22}^{−1}B_{2}]F_{r} + F_{2}A_{22}^{−1}A_{21}.  (4.8)

The matrix F_{1} given by (4.8) and the matrix F_{2} obtained before, can be used to provide a stabilizing control law (4.2).

To show that such a control stabilizes the system (4.1) for ε > 0 small enough, we shall use the result of the Theorem 3.1 to the closed-loop system (4.3). To this end we show that the assumption of the Theorem 3.1 are fulfild.

Indeed the matrix coefficients of the boundary layer subsystem is now A_{22} + B_{2}F_{2} which is stable. On the other hand the reduced subsystem associated to the closed-loop system is

dx_{1}(t) = [A_{11}+B_{1}F_{1}−(A_{12}+B_{1}F_{2})(A_{22}+B_{2}F_{2})^{−1}(A_{21}+B_{2}F_{1})]x_{1}(t)dt
+ ∑_{k=1}^{N} A_{r}^{k}(F_{2})x_{1}(t)dw_{k}(t).  (4.9)

By algebraic manipulations, similar with the ones in deterministic framework, (see e.g. [3, 8]) we obtain that (4.9) is equivalent to (4.7) and hence the zero solution of the system (4.9) is exponentially stable in mean square.

Thus we obtain that the assumptions from Theorem 3.1 hold and hence the zero solution of the closed-loop system is mean square exponentially stable for arbitrary ε > 0 small enough.

Remark: In the case of the stochastic systems (4.1) the designing of the feedback gain F_{r} cannot be performed independently of the stabilizing feedback gain F_{2} as it happens in the deterministic framework [3, 8].

References

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