A Class of Impulsive Pulse-Width Sampler Systems and Its Steady-State Control in Infinite Dimensional Spaces

and Applied Analysis 3 input signal v t v1 t , v2 t , . . . , vq t T of the pulse-width sampler satisfy the following dynamic relation: ui t ⎧ ⎨ ⎩ signαni , nT0 ≤ t < n |αni | T0, i 1, 2, . . . , q; 0, n |αni | T0 ≤ t < n 1 T0, n 0, 1, . . . , 1.4 αni ⎧ ⎨ ⎩ vi nT0 , |vi nT0 | ≤ 1, i 1, 2, . . . , q; signvi nT0 , |vi nT0 | ≥ 1, n 0, 1, . . . , 1.5 where T0 is called the sampling period of the pulse-width sampler which is the same as the period of f and τk, k 1, 2, . . . . We end this introduction by giving some definitions. Definition 1.1. The closed-loop system 1.1 , 1.3 – 1.5 is called linear pulse-width sampler control system with impulses. The closed-loop system 1.2 , 1.3 – 1.5 is called semilinear pulsewidth sampler control system with impulses. Definition 1.2. In the closed-loop system 1.1 , 1.3 – 1.5 or system 1.2 , 1.3 – 1.5 , the q dimensional vector αn αn1 , αn2 , . . . , αnq T is called the duration ratio of the pulse-width sampler in the nth sampling period, n 0, 1, . . . . We defined a closed cube Ω in R as Ω { α ( α1, α2, . . . , αq )T ∈ R | |αi| ≤ 1, i 1, 2, . . . , q } , 1.6 then we have αn ∈ Ω, for n 0, 1, . . . . Definition 1.3. In the closed-loop system 1.1 , 1.3 – 1.5 or system 1.2 , 1.3 – 1.5 , if there exists a q dimensional vector α ( αn1 , αn2 , . . . , αnq )T ∈ Ω, 1.7 and a corresponding periodicity rectangular-wave control signal u t u t, α defined by ui t ui t, α ⎧ ⎨ ⎩ sign αi, nT0 ≤ t < n |αni | T0, i 1, 2, . . . , q; 0, n |αni | T0 ≤ t < n 1 T0, n 0, 1, . . . . 1.8 such that the closed-loop system 1.1 , 1.3 – 1.5 or system 1.2 , 1.3 – 1.5 , has a corresponding T0-periodic trajectory x · x ·, α : x t T0, α x t, α , t ≥ 0, then the control signal 1.8 is called the steady-state control with respect to the disturbance f . The T0periodic trajectory x · is called steady-state corresponding to steady-state control u · and the constant vector α ∈ Ω of steady-state control 1.8 is called to be a steady-state duration ratio. 4 Abstract and Applied Analysis Definition 1.4. In the closed-loop system 1.1 , 1.3 – 1.5 or system 1.2 , 1.3 – 1.5 , if there exists some α ∈ Ω such that lim n→∞ αn α, where αn ( αn1 , αn2 , . . . , αnq )T , α ( α1, α2, . . . , αq )T , 1.9 then system 1.1 , 1.3 – 1.5 or system 1.2 , 1.3 – 1.5 , corresponding to the disturbance f is called to be stead-state stable. Further, system 1.1 , 1.3 – 1.5 or system 1.2 , 1.3 – 1.5 , corresponding to the perturbation f is called stead-state stabilizability if we can choose a suitable T0 > 0 andK2 such that system 1.1 , 1.3 – 1.5 or system 1.2 , 1.3 – 1.5 , is stead-state stable. 2. Mathematical Preliminaries Let L X,X denote the space of linear operators from X to X, Lb X,X denote the space of bounded linear operators from X to X, Lb R, X denote the space of bounded linear operators from R to X, and Lb X,R denote the space of bounded linear operators from X to R. It is obvious that Lb X,X , Lb R, X , and Lb X,R is the Banach space with the usual supremum norm. Define D̃ {τ1, . . . , τδ} ⊂ 0, T0 , where 0 < τ1 < τ2 < · · · < τδ < T0. We introduce PC 0, T0 ;X ≡ {x : 0, T0 → X | x is continuous at t ∈ 0, T0 \ D̃, x is continuous from left and has right hand limits at t ∈ D̃}, and PC1 0, T0 ;X ≡ {x ∈ PC 0, T0 ;X | ẋ ∈ PC 0, T0 ;X }. Set ‖x‖PC max { sup t∈ 0,T0 ‖x t 0 ‖, sup t∈ 0,T0 ‖x t − 0 ‖ } , ‖x‖PC1 ‖x‖PC ‖ẋ‖PC. 2.1 It can be seen that endowed with the norm ‖ · ‖PC ‖ · ‖PC1 , PC 0, T0 ;X PC1 0, T0 ;X is a Banach space. We introduce the following assumption H1 . i H1.1 A is the infinitesimal generator of a C0-semigroup {T t , t ≥ 0} on X with domain D A . ii H1.2 There exists δ such that τk δ τk T0. iii H1.3 For each k ∈ Z 0 , Bk ∈ Lb X,X and Bk δ Bk. We first recall the homogeneous linear impulsive periodic system ẋ t Ax t , t / τk, Δx t Bkx t , t τk, 2.2 Abstract and Applied Analysis 5 and the associated Cauchy problem ẋ t Ax t , t ∈ 0, T0 \ D̃, Δx τk Bkx τk , k 1, 2, . . . , δ, x 0 x. 2.3and Applied Analysis 5 and the associated Cauchy problem ẋ t Ax t , t ∈ 0, T0 \ D̃, Δx τk Bkx τk , k 1, 2, . . . , δ, x 0 x. 2.3 If x ∈ D A and D A is an invariant subspace of Bk, using 18, Theorem 5.2.2, page 144 , step by step, one can verify that the Cauchy problem 2.3 has a unique classical solution x ∈ PC1 0, T0 ;X represented by x t S t, 0 x, where S ·, · : Δ { t, θ ∈ 0, T0 × 0, T0 | 0 ≤ θ ≤ t ≤ T0} −→ Lb X,X 2.4