On the φ₀-stability of functional differential equations

Summary

Consider the following functional differential systems:

$$x' = f_1 \left( {t,x,x_1 } \right),x_{t_0 } = \theta _0 ;$$

((S1))

$$x' = C\left( {t,x} \right)x + \int_r^0 {d\sigma \left( {t,x,s} \right)x\left( {t + s} \right)} x_{t_0 } = \theta _0 , - r \leqslant s \leqslant 0;$$

((S2))

$$x' = f_2 \left( {t,x,x_t } \right) + h\left( {t,x,x_t } \right)x_{t_0 } = \theta _{0,} $$

((S3))

and the following comparison functional differential systems:

$$u' = g\left( {t,u,u_t } \right),u_{t_0 } = \sigma _0 ,$$

((S4))

$$u' = Au + Du_t ,u_{t_0 } = \sigma _0 ;$$

((S5))

$$u' = E\left( u \right)w_1 \left( u \right) + F\left( {u_t } \right)w_2 \left( {u_t } \right),u_{t_0 } = \sigma _0 ,$$

((S6))

wherex, f ₁, f₂, h ∈ R^N, g, w₁, w₂ ∈ Rⁿ are continuous functions of their respective arguments. Furthermore,x _t is the past history of the system,A = (a_ij), D = (d_ij) are constant matrices, andE andF are coefficient matrix functions.

Theorem S1. Assume that (i) V ∈ C[J × S_ϱ, K], V(t, x) is locally Lipschitzian in x and for (t, θ(0)) ∈ J × S_ϱ, D⁺ V(t, θ(0), θ) ≤_k g(t, V(t, θ(0)), V_t), V_t = V(t + s, θ(s)) (ii) g ∈ C[R₊ × Rⁿ × K, Rⁿ], g(t, u, u_t) is quasimonotone in u_t relative to K for each (t, u) ∈ R₊ × Rⁿ. (iii) f(t, 0, 0) = 0, g(t, 0, 0) = 0 and for some φ₀ ∈ K ^*₀ , b(∥x(t₀, θ₀)(t)∥) ≤ (φ₀, V(t, x(t₀, θ₀)(t))) ≤ a(t, ∥x(t₀, θ₀)(t)∥). Then the trivial solution x = 0 of (S1) is (a) stable (b) uniformly stable (c) asymptotically stable and (d) uniformly asymptotically stable if the trivial solution u = 0 of (S4) is respectively (e) φ₀-stable (f) uniformly ø₀-stable (g) asymptotically φ₀-stable and (h) uniformly asymptotically ø₀-stable.

Theorem S2. Let c_ij(t, x) ≤ a_ij, dσ_ij(t, x, s) ≤ d_ij, i, j = 1,..., n, C∥x(t₀, θ₀)(t)∥^d ≤ (φ₀, V(t, x(t₀, θ₀)(t))), V(t, u) is a cone-valued Lyapunov function for (S2). Then the steady state solution x = 0 of (S2) is exponentially stable if and only if there exist a nonnegative nonsingular n × n matrix B such that all the off-diagonal elements of B⁻¹ AB> are nonnegative.

Theorem S3. Let the trivial solution of x′ = f₁(t, x, x_l) be uniformly asymptotically stable and let h(t, x, x_t) satisfy

$$L\left( t \right)\left\| {h\left( {t,x,x_1 } \right)} \right\|_{\hat G} \leqslant \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {e_{ij} \left( V \right)\psi _2 \left( {V_t } \right) + c\left\| {x\left( {t_0 ,\theta _0 } \right)\left( t \right)} \right\|_{\hat G} } } $$

. Then the steady state solution x = 0 of (S3) is uniformly asymptotically stable if and only if there exist a nonnegative nonsingular n × n matrix B such that all the off diagonal elements of B⁻¹ J_E(u)B are nonnegative, where J_E(u) is the Jacobian of E at u.