Absolute Stability of Neutral Systems with Lurie Type Nonlinearity

1 Introduction

The paper considers absolute stability of a system of nonlinear differential equations of neutral type

where x=(x₁, ⋅, x_n)^T:[−τ, ∞) → ℝⁿ, A, B and D are n × n constant matrices, b is an n-dimensional column constant vector, 𝜏 > 0 is a constant delay, f:R→R , f(0)=0, is a Lurie-type nonlinear function satisfying Lipschitz condition,

and c is a column n-dimensional constant vector where the superscript T denotes the transpose of a vector (or a matrix).

2 Exponential C⁰ Stability

Below, we give conditions for C⁰ exponential stability of (1.1) . From the proof, an estimate of the convergence of solutions follows as well. To solve the problem of absolute stability within the meaning of Definition 4 for system (1.1) under restriction (1.3) , the Lyapunov-Krasovskii functional (2.1) will be used of the quadratic type with respect to both the current coordinates and their derivatives,

In (2.1), t=0, x is a solution of system (1.1), H, G₁, G₂ are positive-definite symmetric matrices, ς1>0,ς2>0 and β⩾0. The condition (1.3) and the well-known properties of quadratic forms imply the two-sided estimate of functional (2.1)

These will be used in the paper. We also use the notation

The stability result formulated below depends, among others, on the positive definiteness of an auxiliary (3n + 1)  ×  (3n + 1) matrix S. This matrix depends on matrices, vectors and constants used in (1.1) –(1.3) , (2.1) and on a parameter ν⩾0, not yet involved. In other words, the elements of S are constructed using β, ς₁, ς₂, k, ν, b, c, A, B, D, H, G₁, G₂, formally

In its block form, the matrix S is defined as

with block-matrices

where I in S₁₄ is the unit n × n matrix.

Proof. The proof splits into two parts. In part i), an estimate of the total derivative of the functional (2.1) along solutions of the system (1.1) is computed. Part ii) deals with the exponential stability of solutions to system (1.1) in the metrix C⁰.

i) Let us calculate the total derivative of the functional (2.1) along solutions of (1.1) . Assuming t ∈ [0, ∞)∖{τ s, s=0, 1, ⋅}, we have

Before writing this derivative in a form with the matrix S, we rewrite it using system (1.1) as follows

Define an auxiliary vector

Now, all the non-integral terms in the right-hand side of the last expression can be written using this vector and the matrix S. Omitting technicalities, it can be verified term by term that the identity

holds. Then, (2.5) can be written as

Due to (1.3),

As S is positive definite, we have

ii) Prove the exponential stability of the solutions to system (1.1) in the metrix C⁰. Below we analyze two cases dealing with ς ⩾ δ and ς < δ where ς:=min{ς1,ς2}>0 .

ii₁) Let ς ⩾ δ . Then, from inequality (2.2), we get

and

Inequality (2.8) yields

or

We get,

Integrating this inequality over (0, t), which is a correct operation because the set of isolated points {t=τ s, s=0, 1, ⋅} is countable, for t=0, we obtain

ii₂) Let ς < δ. Rewrite the right-hand part of the inequality (2.2) as follows

Now, using (2.8), we get

Then,

and integrating this inequality over (0, t), we obtain (as in ii₁))

From (2.9) and (2.10) we deduce that, in both cases ii₁), ii₂) considered, we have

Utilizing (2.2) and (2.11), we get (the below inequalities hold for t≥0 as well)

hence, it follows that

Now, after a simple estimation of the right-hand side,

where

Put η₀=γ/2. Inequality (2.3) is proved. The zero solution of (1.1) is C⁰ exponentially stable within the meaning of Definition 2 since (1.6) holds. □

3 Exponential C¹ Stability and Absolute Stability

To prove the exponential stability in metric C¹, we need the following lemma providing a formula transforming neutral system (1.1) to a delayed one of the Volterra type.

Proof. We will prove the lemma by induction. Let x(t) be the solution of problem (1.1), (1.4). Then, for m=1, the lemma holds because formula (3.1) coincides with initial system (1.1) . Let us recall that, if we deal with a sum of the form ∑i=10… , then, by customary definitions, we set such sum equal to zero. This concerns the first sum in (3.1).Assuming that the conclusion of the lemma holds for m=j−1 where j=2 is a fixed natural number, we will show that, then, it holds for m=j as well. For m=j−1, t ∈ ((j − 2)τ, (j − 1)τ), we get from (3.1)

If m=j and t ∈ ((j − 1)τ, jτ), then, using (1.1), we get for the term x˙(t−(j−1)τ) in (3.2):

Finally, substituting (3.3) into the right-hand side of (3.2), we derive

i.e., the formula (3.1) holds for m=j as well. □

Proof. By Theorem 1, the zero solution of the system (1.1) is exponentially stable in the metric C⁰. We show that the zero solution of (1.1) is also C¹ exponentially stable. As it follows from formula (3.1) in Lemma 1, for t ∈ ((m − 1)τ, mτ), m=1, 2, ⋅, we have

From (1.3) and (1.2) we derive ∣f(σ(t − iτ))∣ ⩽ k∣ σ(t − iτ)∣ ⩽ k∣c∣ x(t − iτ) ∣ and, therefore,

Let us estimate ∣x(t)∣ and ∣x(t − iτ) ∣ in (3.6) by formula (2.3). We obtain

Since t ∈ ((m − 1)τ, mτ), then ∣D∣^m ⩽ exp((t/τ)ln∣ D∣). We have

and, by (3.4),

From (3.7), we get

Then, for any admissible m, inequality (3.8) with the term

omitted, yields

where

Inequality (3.5) is proved. The zero solution of (1.1) is C¹ exponentially stable within the meaning of Definition 3 where η₁=γ/2 and (1.7) holds. □

The following theorem on absolute stability of system (1.1) within the meaning of Definition 4 directly follows from Theorem 1 and Theorem 2.

4 Example

We will investigate system (1.1) where n=2, τ=1,

i.e., the system

with initial conditions (1.4). Set ς₁=0.01, ς₂=0.001, k=1, ν=0.1, β=0.01 and

Part of the below numerical computations was performed by MATLAB. For the eigenvalues of matrices G₁, G₂ and H, we get λ_min(G₁)ėq 0.1697, λ_max(G₁)ėq 0.5303, λ_min(G₂)=λ_max(G₂)=0.06, λ_min(H)=0.24, λ_max(H)=1.1. The matrix S takes the form