Uniform practical stability in terms of two measures with effect of delay at the time of impulses

1 Introduction

The impulsive differential equations represent framework for mathematical modeling of many real life situations in the field of engineering, biology, chemistry, physics, control systems, population dyanamics and many more fields [1]. The stability analysis of impulsive differential equations have been extensively explored by mathematicians in last few decades [2–4, 7–9, 11, 13, 14]. The significant progress has also been made in the qualitative behaviour of impulsive functional differential equations and have very wide scope in the society. In [5] the more general stability i.e. practical stability of impulsive functional differential equations with two measures is obtained, but it is also possible that the state variables at the time of impulses are related to time delay. In present paper,we establish some criteria for uniform practical stability in terms of two measures of impulsive functional differential equations with effect of delay at the time of impulses by using piecewise continuous Lyapunov functions and Razumikhin techniques.

This paper is organized as follows. In section 2, we present some notations and definitions. In section 3, some sufficient conditions for uniform practical stability in terms of two measures of systems of impulsive functional differential equations are discussed, an example is also given to illustrate the importance of gotten result. At last concluding remarks are given in section 4.

2 Preliminaries

Consider the following Impulsive functional differential equations

where x ∈ Rⁿ, f : R⁺ × D → Rⁿ, D is an open set in PC([−𝜏, 0], Rⁿ where 𝜏 > 0 and PC([−𝜏, 0], Rⁿ) = {ϕ: [−𝜏, 0] → Rⁿ, ϕ(t) is continuous everywhere except at finite number of points at which it is right continuous and left limit exists}, I_k, J_k ∈ C(Rⁿ, Rⁿ) for k ∈ Z+ t₁ < t₂ < t₃ ... < t_k < t_k+1 ..., where t_k → ∞, as k → ∞, For each t ≥ t₀, x_t ∈ D is defined by x_t(r) = x(t + r), −𝜏 ≤ r ≤ 0.

For φ in PC([−𝜏, 0], Rⁿ), the norm of φ is defined by |φ| = sup{||φ||: −𝜏 ≤ r ≤ 0}, where ||.|| is a norm in Rⁿ.

Throughout in this paper, we introduce the following conditions.

1. For t ∈ [t₀ −𝜏, t₀], the solution x(t, t₀, Ψ) coincides with the function Ψ(t − t₀).

2. f (t, Ψ) is Lipchitzian in Ψ in each compact set in PC([−𝜏, 0], Rⁿ).

3. Functions I_k, J_k ∈ C(Rⁿ, Rⁿ), k = 1, 2, 3, .... are such that the inequality ||x + I_k(x) + J_k(x)|| < H holds if ||x|| ≤ H and I_k, J_k ≠ 0, where H = constant > 0.

4. f (t, 0) ≡ 0, I_k(0) = 0, J_k(0) = 0

Under the conditions (i) − (iv), there is a unique solutions of the problem (1) through (t₀, Ψ).

We denote the solution of impulsive functional differential equation (1) by x(t, t₀, Ψ) and maximal interval of the type [t₀ − 𝜏, β] in which above solution is defined.

Let Rτ+ = [-τ,∞), we define following:

3 Main Results

In this section, we discussed a theorem that provides sufficient conditions for the uniform practical stability in terms of two different measurements of impulsive functional differential systems (1).

Let K = {a ∈ C(R⁺, R⁺) : strictly increasing and a(0) = 0}, K₁ = {ψ ∈ C(R⁺, R⁺) : increasing and Ψ(s) < s for s > 0}.

Proof. Let x(t, t₀, ψ) be a unique solution of (1).

If (t₀, x_t0 ) ∈ R⁺ × PC([−𝜏, 0], Rⁿ) and h¯1(t0,xt0)<u, then by condition (ii) and (vi) h2(t0,x(t0)≤ψ(h¯1(t0,xt0)<ψ(u)<v.

We Claim that

For any t ∈ (t₀ − 𝜏, t₀], there exists r ∈ (−𝜏, 0], such that t = t₀ + r, then from definition 3 and condition (iii), we know that for t ∈ (t₀ − 𝜏, t₀]

h1(t,x(t))=h1(t0+r,x(t0+r))=h1(t0+r,xt0(r))≤h¯1(t0,x(t0))<u,

since 𝜉 ∈ K₁, fromcondition (iii), we have for t ∈ (t₀ − 𝜏, t₀]

Now, we prove that

If (10) does not hold, then there exists a s^* ∈ [t₀, t₁) such that

Let s_=inf{t:V(t,x(t))>ξ−1(a(u)),t∈[t0,t1)}, then V(s̄, x(s̄)) = 𝜉⁻¹(a(u)). Now V(t₀, x(t₀)) < a(u), s̄ > t₀, and for s̄ < t ≤ s^*, V(t, x(t)) > 𝜉⁻¹(a(u)). From (9) and definition of s̄, we also have for t₀ − 𝜏 ≤ t ≤ s̄, V(t, x(t)) ≤ 𝜉⁻¹(a(u)). Since a(u) < 𝜉⁻¹(a(u)), V(t₀, x(t₀)) < a(u), V(s̄, x(s̄)) = 𝜉⁻¹(a(u)), and V(t, x(t)) is continuous in [t₀, t₁), it follows that there exists a s₁ ∈ [t₀, s̄), such that V(s₁, x(s₁)) = a(u) and for s₁ ≤ t < s̄, V(t, x(t)) ≥ a(u).

Since t₀ − 𝜏 ≤ t ≤ s̄, V(t, x(t)) ≤ 𝜉⁻¹(a(u)), for s₁ ≤ t < s̄, V(t, x(t)) ≥ a(u) and s₁ ∈ [t₀, s̄), then for t ∈ [s₁, s̄] and r ∈ [−𝜏, 0], we have V(t+r,x(t+r))≤ξ−1(a(u))≤ξ−1(V(t,x(t)))

By condition (iv), we have for t ∈ [s₁, s̄],

On integration over (s₁, s̄), and by condition (v), we get

On the other hand

which is a contradiction, so (10) holds.

By condition (iv) and (10), we have

By similar arguments, we can prove that

By condition (iv), we get

continuing in the same way as before, we get

and

Now a(u) < 𝜉⁻¹(a(u)), so by using conditions (vi) and (iii), we get

therefore,

Hence the proof theorem is complete. □

Example: Consider the following differential equation

in which x ∈ Rⁿ; c, d > 0, 0 < c + d < 1,𝜏 > 0, p(t), q(t) ∈ C[R⁺R⁺p(t)≥p, q(t) ≤ q, (1+1c+d)q−2p>0. Denote x ∈ Rⁿ by x = (x₁, x₂, ... x_n).

Let h2(t,x)=‖x‖1=Σi=1n|xi|,h1(t,x)=‖x‖∞=max1≤i≤n|xi|. From definition of h̄₁, we know that h¯1(t,xt)=sup−τ≤r≤0h1(t+r,x(t+r))=sup−τ≤r≤0‖x(t+r)‖∞=|xt|∞.

For Given (u,v) with 0<u<(1n√n)(c+d)v, if following assumptions hold:

1. tk−tk−1<−ln(c+d)/(−2p+(1+1c+d)q).

2. |x_t|∞ < u implies that for any r∈[−τ,0],‖x(t)‖1<1c+d‖x(t+r)‖1 holds.

Then the equation(12) with respect to (u, v) is (h̄₁, h₂)-uniformly practically stable.

Proof. We choose the functions in Theorem 3.2 as follows:

1. If h¯1(t,xt)<u, we have for any r ∈ [−𝜏, 0],

   h2(t,x)=‖x‖1<1c+d‖x(t+r)‖1≤nc+d‖x(t+r)‖∞≤nc+d|xt|∞=ψ(h¯1(t,xt))

   Thus condition (ii) of Theorem 3.2 is satisfied

2. since 11n2‖x‖12≤‖x‖22≤n‖x‖∞2, then b(h₂(t, x)) ≤ V(t, x) ≤ a(h₁(t, x)) holds

3. For any solution x(t) of (12)

   sup{V(t+r,x(t+r):r∈[−τ,0]}<ξ−1(V(t,x(t)))

   we have clear that

   sup{xT(t+r)x(t+r):r∈[−τ,0]}<1c+d(xT(t)x(t))).

   Therefore

   V′(t,x(t))=(−p(t)xT(t)+q(t)xT(t−τ))x(t)+xT(t)(−p(t)x(t)+q(t)x(t−τ))=−2p(t)xT(t)x(t)+2q(t)xT(t−τ)x(t)≤−2p(t)xT(t)x(t)+q(t)((xT(t−τ)x(t−τ)+xT(t)x(t))≤[−2p+(1+1c+d)q]xT(t)x(t)=g(t)w(V(t,x(t))).

   It is also holds that

   V(tk)=V(tk,x(tk−)+Ik(x(tk−)+Jk(x(tk−−τ))=V(tk,cx(tk−)+dx(tk−−τ))=c2xT(tk−)x(tk−)+2cdxT(tk−−τ)x(tk−)+d2xT(tk−−τ)x(tk−−τ)≤c2xT(tk−)x(tk−)+cd[xT(tk−−τ)x(tk−−τ)+xT(tk−)x(tk−)]+d2xT(tk−−τ)x(tk−−τ)=(c2+cd)xT(tk−)x(tk−)+(cd+d2)xT(tk−−τ)x(tk−−τ)=(c+d)[cxT(tk−)x(tk−)+dxT(tk−−τ)x(tk−−τ)]≤ξ[cV(tk−,x(tk−))+dV(tk−−τ,x(tk−−τ))]

   Thus condition (iv) in Theorem 3.2 also satisfied.

4. Now from the choice of functions g(t) and w(t), we got

   ∫tk−1tkg(s)ds=[−2p+(1+1c+d)q](tk−tk−1)<−ln(c+d)

   and

   ∫jξ−1(j)dsw(s)=∫jj/c2dss=−ln(c+d)

   Let A = − ln(c+d) > 0, then condition (v) of Theorem 3.2 satisfied

5. Since 0<c+d<1,0<u<v,u<(1/nn)(c+d)v, therefore ψ(u)=(n/c+d)u<(1/n)v≤v, and a(u)−nu2<n(1/n3)(c+d)2v2=(1/n2)(c+d)2v2≤(1/n2)c+d)v2=ξ(b(v))

   So condition (vi) satisfied.

From (1)-(5) it is clear that all the conditions of Theorem 3.2 are satisfied. So for given (u,v) with 0 < u < 0<u<(1nn)(c+d)v, the equation(12) with respect to (u, v) is (h¯1,h2)-uniformly practically stable.

4 Conclusion

In this paper, we have extended the criterions of uniform practical stability in terms of two measurements of impulsive functional differential equations to a more general system in which effect of delay at the time of impulses are also considered. An example is given to prove the effectiveness of our so obtained result. We can also see that impulses do contribute to the uniform practical stability of the system.