Exponential Stability for Singularly Perturbed Systems with State Delays

V. Dragan* and A. Ionita^{†}

* Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO-70700, Bucharest, Romania

vdragan@stoilow.imar.ro

† Institute of Theoretical and Experimental Analisys of Aeronautical Structures, P.O.Box 67-5, Bucharest, Romania

ionita@aero.incas.ro

Abstract

In this paper the problem of stability of the zero solution of singularly perturbed system of linear differential equation with state delays is investigated.

We show that if the zero solution of reduced subsystem and the one of the fast subsystem are exponentially stable, then the zero solution of the given singularly perturbed system of differential equations is also exponentially stable.

Estimates of the block components of the fundamental matrix solution are derived. These estimates are used to obtain asymptotic expansions on unbounded interval for the solutions of this class of singularly perturbed systems.

Keywords: singular perturbations, differential equations with delays, exponential stability, asymptotic expansions.

AMS classification: 34D15, 34K20. ^{1}

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

1 Problem formulation

Consider the singularly perturbed system of functional differential equations

ẋ(t) = L_{11}x_{t} + L_{12}y_{t}
εẏ(t) = L_{21}x_{t} + L_{22}y_{t}  (1.1)

where x ∈ R^{n_{1}}, y ∈ R^{n_{2}}, ε > 0 is a small parameter;

L_{j1}x_{t} = ∑_{i=0}^{p} A_{j1}^{i}x(t − τ_{i}) + ∫_{−τ_{p}}^{0} D_{j1}(s)x(t + s)ds

L_{j2}y_{t} = ∑_{i=0}^{m} A_{j2}^{i}y(t − εμ_{i}) + ∫_{−μ_{m}}^{0} D_{j2}(s)y(t + εs)ds

j = 1, 2, A_{jk}^{i} are constant matrices with apropriate dimensions D_{jk}(·) are integral matrix valued functions and 0 = τ_{0} < τ_{1} < ... < τ_{p}, 0 = μ_{0} < μ_{1} < ... < μ_{m}.

Setting L_{ε} = ( \begin{matrix}L_{11} & L_{12} \\ \frac{1}{ε}L_{21} & \frac{1}{ε}L_{22} \end{matrix} ); z = ( \begin{matrix}x \\ y \end{matrix} ) the system (1.1) may be written

in a compact form as:

ż(t) = L_{ε}z_{t}.

It is known (see [3], [4]) that the exponential stability of the zero solution of the system (1.1) is equivalent with the fact that the roots of the equation

det(λI − L_{ε}(e^{λ·}I)) = 0

are located in the half plane Re(λ) < 0.

Usually this condition is difficult to be check when the system is of high dimension.

The goal of this paper is to provide some sufficient conditions assuring the exponential stability for the system of type (1.1).

Such conditions are expressed in term of exponential stability of the zero solution of some subsystems of lower dimensions not depending upon small parameter ε.

Taking ε = 0 in (1.1) we obtain:

ẋ(t) = L_{11}x_{t} + L̃_{12}y(t)
0 = L_{21}x_{t} + L̃_{22}y(t)  (1.2)

where

L̃_{j2} = ∑_{i=0}^{m} A_{j2}^{i} + ∫_{−μ_{m}}^{0} D_{j2}(s)ds, j = 1, 2.

Assuming that the matrix L̃_{22} is invertible, we may associate the following “reduced subsystem” :

ẋ(t) = L_{r}x_{t}  (1.3)

with

L_{r}x_{t} = ∑_{i=0}^{p}[A_{r}^{i}x(t − τ_{i}) + ∫_{−τ_{p}}^{0} D_{r}(s)x(t + s)ds]  (1.4)

A_{r}^{i} = A_{11}^{i} − L̃_{12}L̃_{22}^{−1}A_{21}^{i}, i = 0, 1, ..., p, D_{r}(s) = D_{11}(s) − L̃_{12}L̃_{22}^{−1}D_{21}(s).

Sometimes (1.3) will be called “slow subsystem” associated to (1.1).

Also, we associate the so called “fast subsystem” or “boundary layer subsystem”:

y′(σ) = L_{f}y_{σ}  (1.5)

with

L_{f}y_{σ} = ∑_{i=0}^{m} A_{22}^{i}y(σ − μ_{i}) + ∫_{−μ_{m}}^{0} D_{22}(s)y(σ + s)ds

where σ = \frac{t}{ε}.

In this paper we show that the exponential stability of the zero solution of the slow subsystem (1.3) and the one of the fast subsystem (1.5) implies the exponential stability of the zero solution of the system (1.1) for arbitrary ε > 0 small enough.

Our result extends to the class of systems of functional differential equations with state delays of type (1.1) the well known result of Klimusev-Krasovski [6].

Systems of differential equations of type (1.1) with m = 0, p = 1 and D_{jk}(s) = 0 were considered in [5], where asymptotic structure of solutions was studied.

In [2] the system (1.1) with p = 0 was considered and the separation of time scales was proposed.

2 The main result

We make the following assumptions:

H_{1}) The roots of the equation

det[λI_{n_{2}} − ∑_{i=0}^{m} A_{22}^{i}e^{−λμ_{i}} − ∫_{−μ_{m}}^{0} D_{22}(s)e^{λs}ds] = 0  (2.1)

are located in the half plane Re(λ) < −γ_{f} < 0 for some positive constant γ_{f}. H_{2}) The roots of the equation

det [λI_{n_{1}} − ∑_{i=0}^{p} A_{r}^{i}e^{−λτ_{i}} − ∫_{−τ_{p}}^{0} D_{r}(s)D_{r}(s)e^{λs}ds] = 0  (2.2)

are located in the half plane Re(λ) ≤ −γ_{r} < 0 for some positive constant γ_{r}. Remark:

a) Under the assumption H_{1}) it follows that λ = 0 is not a root of the equation (2.1) and hence the matrix L̃_{22} is invertible. Thus the matrices A_{r}^{i}, i = 0, 1, ..., p and D_{r}(s) are well defined.

b) If 0 < α_{f} < γ_{f} there exists β_{f} ≥ 1 such that the solutions of the system (1.5) satisfy:

|y(σ)| ≤ β_{f}e^{−α_{f} σ}||y_{0}||  (2.3)

for all σ ≥ 0; ||y_{0}|| = sup_{s∈[−μ_{m},0]} |y(s)|.

c) If 0 < α_{r} < γ_{r} there exists β_{r} ≥ 1 such that for any solutions of the system (1.3) we have

|x(t)| ≤ β_{r}e^{−α_{r}t}||x_{0}||, (∀) t ≥ 0.  (2.4)

The main result of this paper is: Theorem 2.1 Under the assumptions H_{1}), H_{2}) there exists ε_{0} > 0 such that for arbitrary ε ∈ (0, ε_{0}) the zero solution of the system (1.1) is exponentially stable. Moreover, if ( \begin{matrix}Φ_{11}(t, ε) & Φ_{12}(t, ε) \\ Φ_{21}(t, ε) & Φ_{22}(t, ε) \end{matrix} ) is the partition of the fundamental matrix solution, we have

|Φ_{11}(t, ε)| ≤ β_{1}e^{−α_{1}t}
|Φ_{12}(t, ε)| ≤ εβ_{1}e^{−α_{1}(t)}
|Φ_{21}(t, ε)| ≤ β_{2}e^{−α_{1}t}
|Φ_{22}(t, ε)| ≤ β_{2}(e^{−α_{2} \frac{t}{ε}} + εe^{−α_{1}t}), (∀)t ≥ 0, ε ∈ (0, ε_{0})

α_{j}, β_{j}, j = 1, 2 are positive constants which depend by α_{f}, α_{r}, β_{f}, β_{r}, respectively.

Proof:

Let Φ_{r}(t) be the fundamental matrix solution of the reduced system (1.3) and Φ_{f}(σ) be the fundamental matrix solution of the fast system (1.5).

It is easy to see that t → Φ_{f}(\frac{t}{ε} is the fundamental matrix solution of the system

εẏ(t) = L_{22}y_{t}, t ≥ 0.

Let ( \begin{matrix}x(t, ε) \\ y(t, ε) \end{matrix} ) be a solution of the system (1.1) with the initial conditions:

x(s, ε) = 0, −τ_{p} ≤ s < 0
y(s, ε) = 0, −εμ_{m} ≤ s < 0.

Using the variation of constants formula we obtain:

y(t, ε) = Φ_{f}(\frac{t}{ε})y(0) + \frac{1}{ε} ∫_{0}^{t} Φ_{f}(\frac{t − s}{ε})L_{21}x_{s}ds  (2.5)

x(t, ε) = Φ_{r}(t)x(0) + ∫_{0}^{t} Φ_{r}(t − s)L_{12}y_{s}ds + ∫_{0}^{t} Φ_{r}(t − s)L̃_{12}L̃_{22}^{−1}L_{21}x_{s}ds.(2.6)

Further we write:

∫_{0}^{t} Φ_{r}(t − s)L_{12}y_{s}ds = ∑_{i=0}^{m} ∫_{0}^{t} Φ_{r}(t − s)A_{12}^{i}y(s − εμ_{i}, ε)ds
+ ∫_{0}^{t} Φ_{r}(t − s) ∫_{−μ_{m}}^{0} D_{12}(θ)y(s + εθ, ε)dθds
= ∑_{i=0}^{m} ∫_{εμ_{i}}^{t} Φ_{r}(t − s)A_{12}^{i}y(s − εμ_{i}, ε)ds
+ ∫_{−μ_{m}}^{0} ∫_{θ}^{t} Φ_{r}(t − s)D_{12}(θ)y(s + εθ, ε)dsdθ (2.7)
= ∑_{i=0}^{m} ∫_{0}^{t} Φ_{r}(t − s − εμ_{i})A_{12}^{i}y(s, ε)ds
+ ∫_{0}^{t} ∫_{−μ_{m}}^{0} Φ_{r}(t − s + θ)D_{12}(θ)dθy(s, ε)ds.

Substituting (2.5) in (2.7) we get:

∑_{i=0}^{m} ∫_{0}^{t} Φ_{r}(t − s − εμ_{i})A_{12}^{i}y(s, ε)ds

= ∑_{i=0}^{m} ∫_{0}^{t} Φ_{r}(t − s − εμ_{i})A_{12}^{i}Φ_{f}(\frac{s}{ε})y(0)ds

+\frac{1}{ε} ∑_{i=0}^{m} ∫_{0}^{t} Φ_{r}(t − s − εμ_{i})A_{12}^{i} ∫_{0}^{s} Φ_{f}(\frac{s − σ}{ε})L_{21}y_{σ}dσds.  (2.8)

Applying Fubini theorem we have:

\frac{1}{ε} ∫_{0}^{t} Φ_{r}(t − s − εμ_{i})A_{12}^{i} ∫_{0}^{s} Φ_{f}(\frac{s − σ}{ε})L_{21}x_{σ}dσds

= \frac{1}{ε} ∫_{0}^{t} ∫_{s}^{t} Φ_{r}(t − σ − εμ_{i})A_{12}^{i}Φ_{f}(\frac{σ − s}{ε})dσL_{21}x_{s}ds.  (2.9)

On the other hand:

\frac{1}{ε} ∫_{s}^{t} Φ_{r}(t − σ − εμ_{i})A_{12}^{i}Φ_{f}(\frac{σ − s}{ε})dσ

= ∫0^{\frac{t−s}{ε}} Φ_{r}(t − s − εσ − εμ_{i})A_{12}^{i}Φ_{f}(σ)dσ

= Φ_{r}(t − s)A_{12}^{i} ∫_{0}^{∞} Φ_{f}(σ) + Ψ_{1}(t, s, ε)

where

Ψ_{1}(t, s, ε) = ∫_{0}^{∞}[Φ_{r}(t − s − εσ − εμ_{i}) − Φ_{r}(t − s)]A_{12}^{i}Φ_{f}(σ)dσ
− ∫ ∞_{\frac{t−s}{ε}} Φ_{r}(t − s − εσ − εμ_{i})A_{12}^{i}Φ_{f}(σ)dσ.

Based on (2.3) and (2.4) we deduce:

|Ψ_{1}(t, s, ε)| ≤ β̃_{1}[e^{−α_{2} \frac{t−s}{ε}} + εe^{−α_{1}(t−s)}]

α_{j} > 0, β̃_{1} > 0, j = 1, 2. It is easy to check that

∫_{0}^{∞} Φ_{f}(σ)dσ = −L̃_{22}^{−1}.

Finally we obtain

∫_{0}^{t} Φ_{r}(t − s)L_{12}y_{s}ds = − ∫_{0}^{t} Φ_{r}(t − s)L̃_{12}L̃_{22}^{−1}L_{21}x_{s}ds  (2.10)
+ ∫_{0}^{t} Ψ_{0}(t, s, ε)dsy(0) + ∫_{0}^{t} Ψ_{2}(t, s, ε)L_{21}x_{s}ds

where

|Ψ_{0}(t, s, ε)| ≤ β̃_{0}e^{−α_{1}(t−s)}e^{−α_{2} \frac{s}{ε}}
|Ψ_{2}(t, s, ε)| ≤ β̃_{2}(e^{α_{2} \frac{(t−s)}{ε}} + εe^{−α_{1}(t−s)}).

Substituting (2.10) into (2.6) we get

x(t, ε) = Φ_{r}(t)x(0) + ∫_{0}^{t} Ψ_{0}(t, s, ε)dsy(0) + ∫_{0}^{t} Ψ_{2}(t, s, ε)L_{21}x_{s}ds.

Directly

|x(t, ε)| ≤ β̃_{3}e^{−α_{1}t}(|x(0)| + ε|y(0)|) + β̃_{4} ∫_{0}^{t}(e^{−α_{2} \frac{t−s}{ε}} + εe^{−α_{1}(t−s)})|L_{21}x_{s}|ds.

By standard techniques in singular perturbation theory [8], [1] we obtain that there exists ε_{0} > 0 such that for arbitrary 0 < ε < ε_{0} we have

|x(t, ε)| ≤ β_{1}e^{−α_{1}t}(|x(0)| + ε|y(0)|).  (2.11)

Using (2.11) in (2.5) we deduce

|y(t, ε) ≤ β_{2}e^{−α_{2} \frac{t}{ε}} |y(0)| + β_{3}e^{−α_{1}t}(|x(0)| + ε|y(0)|).  (2.12)

From (2.11) and (2.12) we conclude that the zero solution of the system (1.1) is exponentially stable. The estimates of the block components of the fundamental matrix solution follows from (2.11) and (2.12) taking x(0) = I_{n_{1}}, y(0) = 0 and x(0) = 0, y(0) = I_{n_{2}}, respectively. Thus the proof is complete.

3 Asymptotic expansions

Let us consider system (1.1) with D_{jk}(θ) = 0 for all θ < 0.

Based on the result of Theorem 2.1 we have:

Theorem 3.1 Under the assumptions H_{1} − H_{2} the block components of the fundamental matrix solutions of the system (1.1) have the following asymptotic structure:

Φ_{11}(t, ε) = Φ_{r}(t) + εΦ̂_{11}(t, ε)
Φ_{12}(t, ε) = −εΦ_{r}(t)L̃_{12}L̃_{22}^{−1} + εΦ̃_{12}(\frac{t}{ε}, ε) + ε^{2}Φ̂_{12}(t, ε)
Φ_{21}(t, ε) = −L̃_{22}^{−1}L_{21}(Φ_{r})_{t} + Φ̃_{21}(\frac{t}{ε}, ε) + εΦ̂_{21}(t, ε)
Φ_{22}(t, ε) = Φ_{f}(\frac{t}{ε}) + εΦ̂_{22}(t, ε)

|Φ̂_{jk}(t, ε)| ≤ β̂e^{−α̂t}, ∀t ≥ 0
|Φ̃_{jl}(σ, ε)| ≤ β̃e^{−α̃σ} ∀σ ≥ 0,

α̂, α̃, β̂, β̃ being positive constants not depending upon ε, t, σ.

The asymptotic formulae in Theorem 3.1 allow us to obtain the asymptotic structure of the solutions of a singularly perturbed affine system of functional differential equations:

ẋ(t) = L_{11}x_{t} + L_{12}y_{t} + f(t)
εẏ(t) = L_{21}x_{t} + L_{22}y_{t} + g(t) t ≥ 0

where f(·), g(·) are integrable functions.

References

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[4] J. Hale, S. Lunel - Introduction to functional equations, Springer-Verlag, 1993.

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