An LMI Approach to Stability for Linear Time-Varying System with Nonlinear Perturbation on Time Scales

and Applied Analysis 3 The following are some useful relationships regarding the delta derivative, see 2 . Theorem 2.6 see 2 . Assume that f : T → R and let t ∈ T. i If f is differentiable at t, then f is continuous at t. ii If f is continuous at t and t is right scattered, then f is differentiable at t with fΔ t f σ t − f t σ t − t . 2.1 iii If f is differentiable at t and t is right dense, then fΔ t lim s→ t f t − f s t − s . 2.2 iv If f is differentiable at t, then f σ t f t μ t fΔ t . 2.3 Theorem 2.7 see 2 . Assume that f, g : T → R and let t ∈ T. i The sum f, g : T → R are differentiable at t with ( f g )Δ t ( f )Δ t ( g )Δ t . 2.4 ii For any constant α, αf : T → R is differentiable at t with ( αf )Δ t αfΔ t . 2.5 iii The product fg : T → R is differentiable at t with ( fg )Δ t fΔ t g t f σ t gΔ t f t gΔ t fΔ t g σ t . 2.6 Definition 2.8. The function f : T → R is said to be rd-continuous denoted by f ∈ Crd T,R if the following conditions hold. 4 Abstract and Applied Analysis i f is continuous at every right-dense point t ∈ T. ii lims→ t−f s exists and is finite at every ld-point t ∈ T. Definition 2.9. Let f ∈ Crd T,R . Then g : T → R is called the antiderivative of f on T if it is differentiable on T and satisfies gΔ t f t for t ∈ T. In this case, we define ∫ t a f s Δs g t − g a , a ≤ t ∈ T. 2.7 Consider the linear time-varying system with nonlinear perturbation on time scales T of the form xΔ t A t x t f t, x t , t ∈ T, 2.8 where x t ∈ R, A : T → Rn×n is an n × n matrix-valued function and f : T × R → R is rd-continuous in the first argument with f t, 0 0. The uncertain perturbation is known to satisfy a bound of the form ∥f t, x t ∥ ≤ γ‖x t ‖, 2.9 or equivalently, the perturbation is conically bounded. The solution of 2.8 through t0, x t0 satisfies the variation of constants formula x t ΦA t, t0 x t0 ∫ t t0 ΦA t, σ s f s, x s Δs, t ≥ t0. 2.10 When f t, x t 0, 2.8 becomes the linear time-varying system xΔ t A t x t , x t0 x0, t0 ∈ T. 2.11 For the case when f t, x t B t x t , B t ∈ Rn×n, 2.8 becomes the linear time-varying system xΔ t A t B t x t , x t0 x0, t0 ∈ T. 2.12 The norm of n × n matrix A is defined as ‖A‖ max ‖x‖ 1 ‖Ax‖. 2.13 The Euclidean norm of n × 1 vector x t is defined by ‖x t ‖ √ xT t x t . 2.14 Abstract and Applied Analysis 5 Definition 2.10. A function φ : 0, r → 0, ∞ is of class K if it is well-defined, continuous, and strictly increasing on 0, r with φ 0 0. Definition 2.11. Assume g : T → R. Define and denote g ∈ Crd T;R as right-dense continuous rd-continuous if g is continuous at every right-dense point t ∈ T and lims→ t−g s exists, and is finite, at every left-dense point t ∈ T. Now define the so-called set of regressive functions, R, by R p : T → R | p ∈ Crd T;R , 1 p t μ t / 0, t ∈ T } , 2.15and Applied Analysis 5 Definition 2.10. A function φ : 0, r → 0, ∞ is of class K if it is well-defined, continuous, and strictly increasing on 0, r with φ 0 0. Definition 2.11. Assume g : T → R. Define and denote g ∈ Crd T;R as right-dense continuous rd-continuous if g is continuous at every right-dense point t ∈ T and lims→ t−g s exists, and is finite, at every left-dense point t ∈ T. Now define the so-called set of regressive functions, R, by R p : T → R | p ∈ Crd T;R , 1 p t μ t / 0, t ∈ T } , 2.15 and define the set of positively regressive functions by R p ∈ R | 1 p t μ t > 0, t ∈ T. 2.16 Definition 2.12. The zero solution of system 2.8 is called uniformly stable if there exists a finite constant γ > 0 such that ‖x t, x0, t0 ‖ ≤ γ‖x0‖, 2.17 for all t ∈ T, t ≥ t0. Definition 2.13. The zero solution of system 2.8 is called uniformly exponentially stable if there exist finite constants γ, λ > 0 with −λ ∈ R such that ‖x t, x0, t0 ‖ ≤ γ‖x0‖e−λ t, t0 , 2.18 for all t ∈ T, t ≥ t0. Definition 2.14. The zero solution of system 2.8 is called ψ-uniformly stable if there exists a finite constant γ > 0 such that for any t0 and x t0 , the corresponding solution satisfies ∥ψ t x t, x0, t0 ∥ ≤ γ∥ψ t0 x0 ∥, 2.19 for all t ∈ T, t ≥ t0. Definition 2.15. System 2.8 is called an h-system if there exist a positive function h : T → R, a constant c ≥ 1 and δ > 0 such that ‖x t, x0, t0 ‖ ≤ c‖x0‖h t h t0 −1, t ≥ t0, 2.20 if ‖x0‖ < δ here h t −1 1/h t . If h is bounded, then 2.8 is said to be h-stable. Definition 2.16. A continuous function P : T → R with P 0 0 is called positive definite negative definite on T if there exists a function φ ∈ K such that φ t ≤ P t φ t ≤ −P t for all t ∈ T. 6 Abstract and Applied Analysis Definition 2.17. A continuous function P : T → Rwith P 0 0 is called positive semidefinite negative semi-definite on T if P t ≥ 0 P t ≤ 0 for all t ∈ T. Definition 2.18. A continuous function P : T × R → R with P t, 0 0 is called positive definite negative definite onT×Rn if there exists a function φ ∈ K such that φ ‖x‖ ≤ P t, x φ ‖x‖ ≤ −P t, x for all t ∈ T and x ∈ R. Definition 2.19. A continuous function P : T × R → R with P t, 0 0 is called positive semi-definite negative semi-definite on T × R if 0 ≤ P t, x 0 ≥ P t, x for all t ∈ T and x ∈ R. Lemma 2.20 7 , Completing the square . assume that S ∈ Mn×n is a symmetric positive definite matrix. Then for every Q ∈ Mn×n, we obtain 2xQy − ySy ≤ xTQS−1QTx, ∀x, y ∈ R. 2.21 3. Main Results In this section, we first introduce Lyapunov stability theory of various types stability for linear time varying system with nonlinear perturbation on time scales. Then, we use this Lyapunov stability theory to obtain sufficient conditions for various types of stabilities of this system. 3.1. Lyapunov Stability Theory Theorem 3.1. If there exist a continuously differentiable positive definite function V t, x t ∈ Crd T × R,R , and a, b ∈ R such that i VΔ t, x t ≤ 0, ii a‖x t ‖ ≤ V t, x t ≤ b‖x t ‖, then the zero solution of system 2.8 is ψ-uniformly stable if there exists ψ t ∈ Crd T,R satisfying ψΔ t ≤ 0. Proof. For t0 ∈ T, we let x t0 x0. Then, by i , we have ∫ t t0 VΔ s, x s Δs V t, x t − V t0, x t0 ≤ 0, ∫ t t0 ψΔ s Δs ψ t − ψ t0 ≤ 0. 3.1 We obtain V t, x t ≤ V t0, x t0 and ψ t ≤ ψ t0 for all t ∈ T, t ≥ t0. By ii , we get the estimation as follows: a ∥ψ t ∥2‖x t ‖ ≤ ∥ψ t ∥2V t, x t ≤ ∥ψ t0 ∥2V t0, x t0 ≤ b ∥ψ t0 ∥2‖x t0 ‖. 3.2 We conclude that ‖ψ t x t ‖ ≤ γ‖ψ t0 x t0 ‖where γ √ b/a > 0. Therefore, the zero solution of system 2.8 is ψ-uniformly stable. The proof of the theorem is complete. Abstract and Applied Analysis 7 Corollary 3.2. If there exist a continuously differentiable positive definite function V t, x t ∈ Crd T × R,R and a, b ∈ R such that i VΔ t, x t ≤ 0, ii a‖x t ‖ ≤ V t, x t ≤ b‖x t ‖, then the zero solution of system 2.8 is uniformly stable. Theorem 3.3. If there exist a continuously differentiable positive definite function V t, x t ∈ Crd T × R,R and a, b, ∈ R with − /b ∈ R satisfying i VΔ t, x t ≤ − ‖x t ‖, ii a‖x t ‖ ≤ V t, x t ≤ b‖x t ‖, then the zero solution of system 2.8 is uniformly exponentially stable. Proof. For t0 ∈ T, we let x t0 x0. We obtain, by i and ii , that for all t ≥ t0, VΔ t, x t ≤ − ‖x t ‖ ≤ − b V t, x t . 3.3and Applied Analysis 7 Corollary 3.2. If there exist a continuously differentiable positive definite function V t, x t ∈ Crd T × R,R and a, b ∈ R such that i VΔ t, x t ≤ 0, ii a‖x t ‖ ≤ V t, x t ≤ b‖x t ‖, then the zero solution of system 2.8 is uniformly stable. Theorem 3.3. If there exist a continuously differentiable positive definite function V t, x t ∈ Crd T × R,R and a, b, ∈ R with − /b ∈ R satisfying i VΔ t, x t ≤ − ‖x t ‖, ii a‖x t ‖ ≤ V t, x t ≤ b‖x t ‖, then the zero solution of system 2.8 is uniformly exponentially stable. Proof. For t0 ∈ T, we let x t0 x0. We obtain, by i and ii , that for all t ≥ t0, VΔ t, x t ≤ − ‖x t ‖ ≤ − b V t, x t . 3.3 Since − /b ∈ R , it follows from Gronwall’s inequality for time scales 2 and ii that a‖x t ‖ ≤ V t, x t ≤ V t0, x t0 e− /b t, t0 ≤ b‖x t0 ‖e− /b t, t0 . 3.4