Báo cáo hóa học: "Research Article Total Stability Properties Based on Fixed Point Theory for a Class of Hybrid Dynamic Systems" pdf

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009, Article ID 826438, 19 pages doi:10.1155/2009/826438 Research Article Total Stability Properties Based on Fixed Point Theory for a Class of Hybrid Dynamic Systems M De la Sen Department of Electricity and Electronics, Institute of Research and Development of Processes, Faculty of Science and Technology, University of the Basque Country, Campus de Leioa (Bizkaia), 644 Apertando de Bilbao, 48080 Bilbao, Spain Correspondence should be addressed to M De la Sen, manuel.delasen@ehu.es Received 26 March 2009; Accepted 20 May 2009 Recommended by Juan J Nieto Robust stability results for nominally linear hybrid systems are obtained from total stability theorems for purely continuous-time and discrete-time systems by using the powerful tool of fixed point theory The class of hybrid systems dealt consists, in general, of coupled continuous-time and digital systems subject to state perturbations whose nominal i.e., unperturbed parts are linear and, in general, time-varying The obtained sufficient conditions on robust stability under a wide class of harmless perturbations are dependent on the values of the parameters defining the overbounding functions of those perturbations The weakness of the coupling dynamics in terms of norm among the analog and digital substates of the whole dynamic system guarantees the total stability provided that the corresponding uncoupled nominal subsystems are both exponentially stable Fixed point stability theory is used for the proofs of stability A generalization of that result is given for the case that sampling is not uniform The boundedness of the state-trajectory solution at sampling instants guarantees the global boundedness of the solutions for all time The existence of a fixed point for the sampled state-trajectory solution at sampling instants guarantees the existence of a fixed point of an extended auxiliary discrete system and the existence of a global asymptotic attractor of the solutions which is either a fixed point or a limit n globally stable asymptotic oscillation Copyright q 2009 M De la Sen This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Introduction Stability of both continuous-time and discrete-time singularly perturbed dynamic systems has received much attention 1–5 Also, stability analysis of discrete-time singularly perturbed systems with calculations of parameter bounds has been reported in 2, An assumption used in previous work to carry out the stability analysis of singularly perturbed systems is relaxed in where an upper-bound on the singular perturbation parameters is included to derive such an analysis On the other hand, the so-called hybrid models Fixed Point Theory and Applications are a very important tool for analysis in the modern computers and control technologies since they describe usual situations where continuous-time and either discrete-time and/or digital systems are coupled 6, A usual example, very common in practice, is the case when a digital controller operates over a continuous-time plant to stabilize it or to improve its performance The systems described in 6, have a more general structure since the controlled plant can also possess an hybrid nature since all the continuous-time and digital state-variables can be mutually coupled and to possess internal delays 8–10 An important class of hybrid systems of wide presence in technological applications like telecommunication, teleoperation, or control of continuous systems by discrete controllers to which the above structure belongs to is that consisting of coupled continuous-time and digital or discrete-time subsystems On the other hand, fixed point theory and related techniques are also of increasing interest for solving a wide class of mathematical problems where convergence of a trajectory or sequence to some set is essential see, e.g., 11–16 Some of the specific topics covered are, as follows The properties of the so-called n time reasonably expansive mapping are investigated in 11 in complete metric spaces X, d as those fulfilling the property that d x, T n x ≥ h d x, T x for some real constant h > The conditions for the existence of fixed points in such mappings are investigated Strong convergence of the well-known Halpern’s iteration and variants is investigated in 12 and some of the references there in Fixed point techniques have been recently used in 14 for the investigation of global stability of a wide class of time-delay dynamic systems which are modeled by functional equations Generalized contractive mappings have been investigated in 15 and references there in, weakly contractive and nonexpansive mappings are investigated in 16 and references there in The existence of fixed points of Liptchitzian semigroups has been investigated, for instance, in 13 In this paper, stability results are obtained for a wide class of hybrid dynamic systems whose nominal i.e., unperturbed parts are linear and, in general, time-varying while the state perturbations are allowed to be, in general, nonlinear, time-varying, and of a dynamic nature Those dynamic systems consist of two coupled parts, one being of a continuoustime nature being modeled by an ordinary differential equation and the other one being of a digital nature and is modeled by a difference equation Both equations are mutually coupled and, respectively, described alternatively by a set of first-order continuous-time differential equations equal to the order of the continuous-time substate and a set of first-order difference equations equal to the order of the digital substate The results about robust stability i.e., stability under tolerance to a certain amount of perturbations are obtained by first obtaining sufficient type stability conditions related to total stability for an extended discrete system which describes the overall state trajectory at sampling instants via the discretization of the continuous-time substate i.e., the state variables which describe the continuous-time component Subsequently, a result about total stability of the continuous-time substate is carried out to ensure the system’s stability during the intersample intervals Some links with the results given in about singularly perturbed systems are also given for a special hybrid system within the given class Finally, some of the results are extended for the case when the Fixed Point Theory and Applications sequence of sampling periods is allowed to be time-varying The main technique employed for deriving the stability results is based on the use of fixed point theory Notation Z and R are the sets of integer and real numbers, Z : {z ∈ Z : z > 0}, Z0 : {z ∈ Z : z ≥ 0}, R : {r ∈ R : r > 0}, R0 : {r ∈ R : r ≥ 0} Also, λmax M and det M denote, respectively, the maximum eigenvalue and denotes the direct Kronecker determinant of the square matrix M M ij The symbol product of matrices Particular norms for functions, sequences, or matrices are denoted by the appropriate subscript In the expressions being valid for any norms, those subscripts are omitted Problem Statement 2.1 Hybrid Dynamic System Σ Consider the following, in general, time-varying dynamic hybrid system System Σ xc t ˙ xd k Ac t xc t Acd t xd k δc t , 2.1 Ad k xd k Adc k xc k δd k , δc t fcc t, xc t fcd t, xd k gcc t, xc t δd k fdc k, xc k fdd k, xd k 2.2 gcd t, xd k , gdc k, xc k gdd k, xd k , 2.3 2.4 for all time t ∈ kT, k T and discrete time integer index k ≥ for sampling period T where xc t and xd k are, respectively, the nc continuous-time or analog substate and nd discrete-time or digital substate The continuous-time and discrete-time variables are denoted by t and k , respectively The discretized analog substate at sampling instants is xc k The matrix functions Ac t , Acd t , Ad k , denoted as a digital signal, that is, xc kT and Adc k are of orders being compatible with the corresponding vectors in - Also, δc t and δd k are disturbances being, in general, nonlinear and time-varying subject to the following set of constraints on the real vector functions f · and g · Constraints C C1 It holds that Ad k , Adc k , fdc k, xc k , fdd k, xd k , gdc k, xc t , gdd k, xd k denote matrix and vector sequences of k of bounded entries The entries of Ac t , Acd t , fcc t, xc t , fcd t, xd k , gcc t, xc t , and gcd t, xd k are locally integrable functions of t for each fixed x in the closed ball centred at zero B 0, r : {x ∈ Rn : x ≤ r} ⊂ Rn , Max xc , xd ≤ r and all integer k ≥ and all t ≥ C2 Also, The following hold: fcc t, fdc t, 0 ∈ Rnc ; fdc k, fdd k, 0 ∈ Rnd ; 2.5 Fixed Point Theory and Applications C3 f fcc t, xc1 − fcc t, xc2 ≤ βcc xc1 − xc2 ; fcd t, xc1 − fcd t, xc2 ≤ βcd xc1 − xc2 , fdc k, xd1 − fdc k, xd2 ≤ βdc xd1 − xd2 ; fdd k, xd1 − fdd k, xd2 ≤ βdd xd1 − xd2 ; 2.6 f f 2.7 f C4 g gcc t, xc1 ≤ βcc r; gcd t, xd1 ≤ βcd r, gdc t, xci ≤ βdc r; gdd t, xdi ≤ βdd r; 2.8 g g 2.9 g for all xci ≤ r, xdi ≤ r and all integer k ≥ and all t ≥ 0, with h being any of the vector real functions or sequences of 2.3 and 2.4 and βh· h f or h g being known nonnegative real constants It turns out from Picard-Lindeloff theorem that the hybrid dynamic system 2.1 – 2.5 has a unique solution under the set of constraints C The solution is almost everywhere continuous and differentiable but it has bounded discontinuities in general at the sampling instants tk kT ; for all k ∈ Z0 The problem dealt with in this brief is the investigation of the robust stability of Σ (i.e., that of 2.1 and 2.2 with dynamic state disturbances 2.3 and 2.4 ) subject to the set of constraints C For this purpose, the state-trajectory of Σ at sampling instants is calculated in the following subsection 2.2 Extended Discrete System Σd Direct calculation of the solution of Σ at sampling instants i.e., t the following discrete extended system: Σd : x k with x k A k δ k T T T xc k , xd k Φc k Ad k δcT k T , δd k A k x k subject to x Γc k Adc k ; δ k , all integer k ≥ 0, T T xc , xd T Γc k Φc k kT ; for all k ∈ Z0 T , with xc T, kT τ Acd kT yields 2.10 xc , and τ dτ, 2.11 T T T T δc kT τ ΦT c k T, kT τ T dτ, δd k , 2.12 Fixed Point Theory and Applications Φc kT Ψc k T, kT and Φc t and Γc t being defined at sampling instants as Φc k Γc kT are the kth intersample state transition and control matrices of the continuous Γc k ˙ Ac t Ψc t, ; Ψc 0, Inc for all t ∈ kT, k T subsystem, respectively, i.e., Ψc t, and all integer k ≥ Main Rsults · The robust stability of Σsubject to the constraints C under the knowledge of the constants β · is now investigated The results on robust stability are useful for both local and global stability in the sense that stability is ensured for initial conditions of 2.1 – 2.4 being constrained to the balls xc ≤ r, xd ≤ r where the radius r is arbitrary but compatible with the validity of the constraints C on Σ 3.1 Exponential Stability of the Nominal Extended System Σ∗ d The nominal Σ is defined by zeroing δc t and δd k in - This results into the nominal T T T version Σ∗ of Σd in 2.10 – 2.12 satisfying x∗ k A k x∗ k with x∗ xc , xd d The following assumption is given Assumption 3.1 The nominal uncoupled continuous-time and digital subsystems xc t ˙∗ ∗ ∗ ∗ Ad k xd k are both exponentially stable, that is, there exist normAc t xc t and xd k dependent real constants Kc ≥ and Kd ≥ such that Ψc t2 , t1 ≤ Kc e−ac t2 −t1 and Ψd k2 , k1 ≤ Kd ak2 −k1 for some real constants ac > and ad ∈ 0, where Ψc ·, · and Ψd ·, · are the stated ˙ transition matrices of the uncoupled continuous-time and digital subsystems in Σ (i.e.,Ψc t, j k2 −1 Ac t Ψc t, ; Ψc 0, Inc with Ψc k2 T, k1 T Φ j between two sampling instants j k1 j k2 −1 Ad k1 and Ψd k2 , k1 j integers k2 ≥ k1 ≥ j with Ψd 0, Ind for all t ≥ 0, any real t2 ≥ t1 ≥ and any The following stability result holds for the nominal extended system i.e., δ ≡ in 2.10 Proposition 3.2 Define ⎛ ρk Max⎝ Max 1≤i≤nc nc T j ⎞ Ψc ij k T, kT ⎛ τ Acd τ dτ ⎠, Max ⎝ 1≤i≤nd nd j ⎞ ij Adc k ⎠ 3.1 Thus, the nominal extended discrete system is exponentially stable if Assumption 3.1 holds and ρk < − Max e−ac T , ad for all integer k ≥ Proof Decompose A k A0 k A k in 2.11 with A0 k Block Diag Φc k , Ad k , Ψc k T, kT and Ad k Ψd k T, k being the one sampling period kth Φc k A k x∗ k is exponentially stable if there exist real transition matrices Thus, x∗ k constants K ≥ being norm-dependent and a ∈ 0, such that its state transition matrix k2 −1 Ψ k2 , k1 A j satisfies Ψ k2 , k1 ≤ Kak2 −k1 , K for the 12 -matrix norm given j k1 by the maximum modulus within the whole set of eigenvalues Also, ρk A k 2, from Fixed Point Theory and Applications A k − A0 k , lies in the union nc1 Ri of the discs Ri the definition of ρk and A k i nc nd {z : |z| ≤ j |A ij k |} from Gerschgorin’s circle theorem, 17 Therefore, A k ≤ ρk ≤ a < for all integer k ≥ if Assumption 3.1 and 3.1 hold Thus, the Max e−ac T , ad nominal extended system is exponentially stable and the result has been proved 3.2 Stability of the Discrete Disturbed Extended System Σd The following result gives sufficient conditions for stability of the extended discrete system 2.10 within a closed ball of the extended state x · Proposition 3.3 Assume that Proposition 3.2 holds under Assumption 3.1 (i.e., the nominal extended system 2.10 is exponentially stable) under the stronger condition A k ≤ a < − K d /K < where the real constants K and a are related to the state transition matrix of Σd 2.10 and defined in the proof of Proposition 3.2, and Kc a−1 βc c Kd βh for h f f βhc βhd 3.2 Kd βd g g βhc 3.3 βhd c, d; and K d < Thus, the state vector is uniformly bounded according to x k ≤ ak K Kd for all integer k ≥ provided that Max xc , xd − ak 1−a r≤r 3.4 < r/2K ≤ r/2 Proof First , note from direct calculus from 2.6 – 2.9 that the disturbance signal δ k in 2.10 satisfies δ k ≤ δc k δd k ≤ K d r, 3.5 provided that Max xc k , xd k < r/2 for all integer k ≥ since K > and a < − K d /K imply a K d < Consider the set of sequences {y k , k ≥ 0} equipped with the ∞ norm for sequences y ∞ Max0≤k≤∞ y k Thus, the operator Td defined by Td y k A k y k δ k is a contraction on the closed subset Rd of bounded nd -vector sequences {y k , k ≥ : y ∞ ≤ r} By the contraction mapping theorem 17, 18 , there is a unique solution y k Td y k fixed point with sequences in Rd , and x k Ψ k, x k−1 i Ψ k, i δ i ≤ Kak x k−1 K Kd r i , 3.6 Fixed Point Theory and Applications which leads directly to 3.4 since a < − K d /K < implies k−1 − ak ; 1−a i ak K Kd − ak ≤1 1−a ak − a ≤ 3.7 3.3 Stability of the Continuous-Time Substate and State Boundedness inbetween Consecutive Sampling Instants Now, the solution to 2.1 subject to 2.2 and 2.3 is analyzed by taking into account that x k ≤ r provided that Proposition 3.3 holds A total stability argument is used as main tool for the proof of stability of the continuous-time subsystem ≤ acd , xc Proposition 3.4 Assume that Proposition 3.3 holds, Sup0≤t or it is not Schur stable for any ε > ii If v ε has positive zeros, let ε be the smallest such zero If A ε1 is Schur stable for any ε1 ∈ 0, ε , then ε∗ ε Otherwise, A ε is not Schur stable for all sufficiently small and positive values of ε iii The extended discrete system Σd ε is stable for all ε ∈ 0, ε∗ satisfying ∗ ≤ r/2K and A ε K d Max0≤ε≤ε∗ ΔA ε < with K d Max xc , xd defined in 3.2 10 Fixed Point Theory and Applications Numerical Example The following third-order system so-called controlled plant , whose state-space description lie within the class of hybrid system 2.1 , is considered: a1 y t ˙ a2 y t a3 y k 4y k − b0 u t y t ă b1 u t b2 u k 3u k − zk 0.2z k 1.1u k ˙ δ t −7δ t 0.3 z k δ t , 4.1 1.3y k , 8.5u t , for all t ∈ kT, k T and any integer k ∈ Z0 The signal u t is a stabilizing outputfeedback control signal generated from an hybrid controller as follows: G1 D, q ut G1 D, q D q2 − q2 D G2 D, q L D, q D2 q G2 D, q ut 1.25q2 − Dq L D, q y t, 0.25q − 1.44187D2 1.12792 D2 q2 − 0.269774q2 D L D, q D − 0.5 q 0.206426D − 2.54251, 1.10629 , 0.5 , 4.2 where q, defined by qv t v t T for any real vector function v : R0 → Rp , is the discrete one-step advance operator and D : d/dt is the time-derivative operator After substituting the control law in the plant description, the resulting closed-loop system is of the general form given while driven only by the disturbance δ t The signal δ t δc t is a perturbation which satisfies the general assumptions—constraints C of the theory of total stability There are six parameters to be estimated by the estimation schemes: a1 −1, a2 2, a3 3, b0 b2 ,and b3 The constant sampling period is T 0.4 Finally, the reference 1, b1 model is a third-order highly damped one of discrete regulation The plant output i.e., the solution of 4.1 is shown in Figure Note that both the extended discrete system and the continuous one are Lyapunov stable since the output is bounded for all time Remark 3.5, which is a combined interpretation of Propositions 3.2–3.4, holds with all the relevant functions in the control scheme which are uniformly bounded for all time, that is, “at” and “in-between” sampling instants If the disturbance δ t is zeroed, then the closed-loop system is globally asymptotically stable Generalizations to Hybrid Systems with Time-Varying Sampling Periods The more general case when the sequence of sampling periods is, in general, not constant and is discussed in the following by using fixed point theory The main mathematical result is concentrated in Theorem 5.1 which also contains many of the above results as particular case Some corollaries to Theorem 5.1 are also given Equations 2.1 – 2.5 are now assumed Fixed Point Theory and Applications 11 20 Output 15 10 10 20 30 40 50 Time ×0.1 s Figure 1: Output versus time to run for all time t ∈ tk , tk , k ∈ Z0 , where {tk }k ∈ Z0 and {Tk }k ∈ Z0 are the sequences of sampling instants and sampling periods i.e., interval lengths inbetween two consecutive k−1 0, for all k ∈ Z discrete time integer sampling instants , respectively, with tk j T j , t0 index k ∈ Z0 The following result holds Theorem 5.1 Assume that constraints C hold, {tk }k ∈ Z0 is a real monotone strictly increasing Tk : sequence of sampling instants and {Tk }k ∈ Z0 is the real sequence of sampling periods with R tk − tk ∈ εT , T ⊂ 0, ∞ for some constants εT , T ≥ εT ∈ R Then, the following properties hold: f i Assume that Max xc , xd ≤ r, Ac is a stability matrix satisfying ac > Kc βcc f g g βcd 0, {A k }k ∈Z0 is a real sequence of convergent (or Schur) matrices, βcd , βcc f and the real sequence {ac Ψc tk f τ, tk / ac − βcc βcd Kc }k ∈ Z has all its elements T T T : R0 × Rnc × not greater than unity Then, the state trajectory solution xc t , xd k Z0 Rnd → Rn is in the closed ball B 0, r : {x ∈ Rn : x ≤ r} ⊂ Rn centred at zero; for all t, k ∈ R0 × Z0 Thus, it is totally stable within such a ball ii Assume that Max xc , xd ≤ r Then, the state trajectory solution T T T : R0 × Rnc × Z0 × Rnd → Rn is totally stable within the closed ball xc t , xd k B 0, r : {x ∈ Rn : x ≤ r} ⊂ Rn centred at zero; for all t, k ∈ R0 × Z0 if ⎛ ⎜ ρ⎝1 f βcc ⎞ f βcd ρc T − ρ f βdc f ⎟ βdd ⎠ 1 −ρ g g g βcd ρc T βcc g βdd ≤ 1, βdc 5.1 for all k ∈ Z , where ρ is an upper-bound of the sequence { Ψ tk , }k ∈ Z0 with k−1 Ψ tk , : provided that > ρ ≥ maxk ∈ Z Ψ tk , , > ρc ≥ i A i f f f f maxk ∈ Z Ψc tk , kk−1 ≥ Kc /ac or, if ρ ≤ 1, provided that βcc βcd βdc βdd g g g g βcc βcd βdc βdd (i.e., in case of absence of perturbations) iii If ρ f βdc f f βdd < and βcc f βcd f βdc f βdd g βcc g βcd g βdc g βdd 0, then 12 Fixed Point Theory and Applications ∗ a the bounded sequence {x k }k∈Z0 has a unique fixed point x0 in the convex closed ∗ ∗ xc , xd ball B 0, r so that x k → x∗ ≤ r, Max xc , xd T T T as Z0 k → ∞ for any k → ∞ Then, the whole state trajectory b Assume that Tk → T ∈ εT , T as Z0 T T T nc : R0 × R × Z0 × Rnd → Rn has a fixed point x∗ τ solution xc t , xd k ∗ ∗ xc τ , xd T T T ∈ B 0, r for each τ ∈ 0, T iv Assume that the constraints C4 are modified as follows: g C4 : gcc t, xci ≤ βcc xci ; g gcd t, xdi ≤ βcd xdi , g gdd t, xdi ≤ βdd xdi ; i g gdc t, xci ≤ βdc xci ; 5.2 1, T T Then, a unique fixed point z∗ ∈ B 0, r exists for the whole state trajectory solution xc t , xd k nc nd n R0 × R × Z0 × R → R provided that the following constraint holds: ρ f f βdc f f βdd f ρc T βcc f f g βcd βcc g g βdc βcd g βdd < T : 5.3 g For sufficiently small βdc βdd βdc βcd and any given ρ < 1, there exists a sufficiently small upper- bound of the sequence of sampling periods T such that the above constraint holds for a given f f g g constant βcc βcd βcc βcd Proof i One gets directly from 3.10 via 2.10 that x tk T xc tk τ : Ψ c tk T τ , xd k T τ τ, tk xc tk Ψ c tk τ, tk Ψc t k τ τ, tk E1 x k Ψ c tk τ, tk s δc t k Ψ c tk τ, tk s δc t k Ψ c tk τ, tk s Acd tk τ, tk x k s ds 5.4 Ψ c tk τ s δc t k s ds s ds, where Ψ c tk τ, tk : Ψc tk τ τ, tk E1 s ds E2 , 5.5 and E1 : Diag Inc , 0nd , E2 : Diag 0nc , Ind In − E1 ; for all τ ∈ 0, Tk , for all k ∈ Z0 Since the threshold εT for the minimum sampling period between any two consecutive Fixed Point Theory and Applications 13 samples exist, the state trajectory solution of 2.1 – 2.4 is unique for each given bounded initial condition Then, for any two state-trajectory solutions x, y : R0 → Rn of 5.4 : τ − y tk x tk τ ≤ Ψ c tk τ x k −y k τ, tk Ψ c tk τ, tk s s − δcy tk δcx tk ds s ≤ Ψ c tk f βcc g Ψ c tk s xcx tk τ, tk s − xcy tk s ds τ, tk s ds τ g βcd r τ, tk × Max tk ≤s≤ tk τ f βcd βcc ≤ Ψ c tk x k −y k τ, tk Ψ c tk Kc − e−ac τ ac x k −y k f f βcc βcd xcx s − xcy s Kc − e−ac τ ac g βcc g ∀τ ∈ 0, Tk βcd r; 5.6 f If ac > Kc βcc x tk ≤ f βcd , then one deduces from 5.6 that τ − y tk 1− τ Kc f βcc ac f βcd −1 Ψc tk τ, tk x k −y k Kc g βcc ac ∀k ∈ Z0 , ≤ ac ac − f βcc f βcd Kc Ψ c tk τ, tk x k −y k Kc g βcc ac ∀k ∈ Z0 , ≤ ac ac − f βcc f βcd Kc Ψ c tk τ, tk x k −y k 5.7 ∀τ ∈ 0, Tk g βcd r 5.8 ∀τ ∈ 0, Tk Kc g βcc ac ∀k ∈ Z0 , g βcd r g βcd r 5.9 ∀τ ∈ 0, Tk On the other hand the combination of 3.6 and 2.12 when extended to the case of time varying-sampling leads to the following constraints at sampling instants since 14 Fixed Point Theory and Applications Φc k Ψc k T, kT : k−1 Ψ tk , x − y x k −y k Ψ t k , ti Ψ t k , ti δx i − δ y i i k−1 Ψ tk , x − y i Ti × T T τ − δcy ti T δcx ti ≤ Ψ tk , k−1 x −y Ψ T ti c τ Ψ t k , ti T T − τ dτ, δdx i − δdy i i Ti × Ψ c t i , ti τ − δcy ti τ δcx ti δdx i − δdy i τ dτ Ti × f x −y ≤ Ψ tk , Ψ c t i , ti βcc k−1 f βcd τ − yc ti τ xc ti Ψ t k , ti i τ dτ f k−1 f βdc βdd k−1 2r Ψ tk , ti xd i − yd i i Ψ t k , ti g βcc i g βcd Ti Ψ c t i , ti g βdc τ dτ g βdd 5.10 k c1 ¸ with Ψ tk , : ≤ r; for i A i Proceed by complete induction by assuming that x t all t ∈ 0, tk−1 for any given initial conditions and any k ∈ Z Then, x tk ≤ r from 5.10 if f Ψ tk , βcc k−1 i k−1 f βcd Ψ tk , ti Ψ t k , ti Ψ c t i , ti f βcc g βcd Ti Ψ c t i , ti τ dτ f βdc τ dτ i g Ti g βdc βdd k−1 Ψ t k , ti i g βdd ≤ 5.11 since Max Maxj ≤ k−1 ∈ Z xd k , Maxt ≤ tk−1 ∈ R xc t − ≤ r by taking y t as the identically zero solution on R0 and also Max Maxj ≤ k−1 ∈ Z xd k − yd k , Maxt ≤ tk−1 ∈ R xc t − yc t ≤ 2r for any two solutions in B 0, r , for all t ≤ tk−1 Then, 5.8 and Property i holds Fixed Point Theory and Applications 15 ii It holds if ⎛ f ⎜ ρ⎝1 ⎞ f βcc βcd ρc T f ⎟ βdd ⎠ f βdc 1−ρ 1−ρ g g g βcd ρc T βcc βdc g βdd ≤ 1, 5.12 for all k ∈ Z provided that > ρ ≥ maxk ∈ Z Ψ tk , , > ρc ≥ maxk ∈ Z Ψc tk , tk−1 ≥ f f f f g g g Kc /ac , then 5.8 holds, or if ρ ≤ provided that βcc βcd βdc βdd βcc βcd βdc g βdd 0, i.e., in case of absence of perturbations Furthermore, there exists a real constant M ∈ R , dependent on r, such that x t ≤ r M < ∞; for all t ∈ R0 This conclusion follows since the real sequence { x k }k ∈ Z0 is uniformly bounded for any initial conditions fulfilling x ≤ r and the mild continuously time differentiable state trajectory solution xc t cannot be unbounded on any open finite-time interval tk , tk ; for all k ∈ Z0 since xc tk ≤ x k ≤ r iii The inequality 5.10 adopts the following particular form at t tk by taking initial conditions at t tk : −y k x k ≤ Ψ t k , tk Tk × f x k −y k Ψ c t k , tk f βcc τ xc tk βcd τ − yc tk τ dτ 5.13 f f βdc g 2r ≤ ρ xd k − yd k βdd βcd f f βdc r ρc T Tk g βcc f βcd g βcc since, by construction, xd k − yd k f βcc g βdc τ dτ g ∀k ∈ Z0 βdd x k −y k βdd f βcc Ψ c t k , tk f βcd f βdc g βcd g βdc 5.14 g βdd ∀k ∈ Z0 ≤ x k − y k Now, if f βdd g βcc g g βcd g βdc f βdd 5.15 f i.e., all the perturbations are identically zero and ρ βdc βdd < 1, then from Schauder’s first fixed point theorem on 5.14 , the bounded sequence {x k }k∈ Z0 has a unique fixed point in the convex closed B 0, r One gets from 5.6 that x t −y t tk ,tk τ ≤ Ψ c tk τ, tk Kc − e−ac τ ac Kc − e−ac τ ac g βcc g βcd r f βcc f βcd x k −y k 5.16 ∀τ ∈ 0, Tk 16 Fixed Point Theory and Applications for all τ ∈ 0, Tk , for all k ∈ Z0 Thus, there is a unique fixed point x∗ in B 0, r , then the T T T : R0 × Rnc × Z0 × Rnd → Rn has a fixed point x∗ τ state trajectory solution xc t , xd k in B 0, r for each τ ∈ 0, T This point coincides with that of the real sequence {x k }k ∈ Z0 for τ from 5.14 Property iii has been proven iv Assume that the constraints C4 are modified as follows: g g gcc t, xci ≤ βcc xci ; g gcd t, xdi ≤ βcd xdi , g gdd t, xdi ≤ βdd xdi ; gdc t, xci ≤ βdc xci ; i 5.17 1, Then, 5.14 and 5.16 are, respectively, modified as follows: x k −y k ≤ ρ f ≤ f βdc x t −y t βdd f f ρc T βcc g βcd g βcc βcd g βdc g 5.18 x k −y k , x k −y k βdd , tk ,tk τ Ψ c tk Kc − e−ac τ ac τ, tk f βcc f βcd g βcc g βcd 5.19 for all τ ∈ 0, Tk , for all k ∈ Z0 so that state trajectory solution x : R0 → Rn possesses a unique fixed point z∗ ∈ B 0, r for any harmless perturbations guaranteeing that ρ f f βdc f βdd f ρc T βcc βcd g βcc g βcd g βdc g βdd < 1, 5.20 provided that ρ < Also, assume that ρ < and the sequence of sampling periods {Tk }k ∈ Z0 converges asymptotically to a limit T in εT , T for some sufficiently small upperf bound T < 1/ρc βcc f βcd g g βcc f f βcd , for all perturbations 5.17 constrained subject to any g g given additive constant βcc βcd βcc βcd Then, a unique fixed point z∗ τ ∈ B 0, r exists for each τ ∈ 0, T for any harmless perturbations subject to ρ f βdc f βdd g βdc g βdd < − ρc T f βcc f βcd g βcc g βcd 5.21 and Property iv has been proven Note that Theorem 5.1 is also applicable, in particular, for constant sampling periods Remark 5.2 Note that the fixed points of Theorem 5.1 iii - iv are reached asymptotically for the state-trajectory solution for each τ ∈ 0, T provided that the sequences of sampling instants and periods converge to finite limits This does not mean that there is a unique equilibrium point for such a trajectory The physical conclusion of Theorem 5.1 iii - iv is that the unique asymptotically sable attractor might be either a globally asymptotically stable equilibrium point or a stable limit oscillation of period at most T Fixed Point Theory and Applications 17 From Remark 5.2 and Theorem 5.1 iii - iv , one concludes the existence of globally stable equilibrium points or that of globally stable limit oscillations as follows in the next two particular results in view of 5.16 and 5.19 in Theorem 5.1(iii), then x∗ τ x∗ ≡ 0; for all τ ∈ 0, T Then, Corollary 5.3 If x∗ 0 x t → as t → ∞ and zero is a globally asymptotically stable equilibrium point If z∗ in Theorem 5.1(iv) then, x t → z∗ ≡ 0, as t → ∞ and zero is a globally asymptotically stable equilibrium point under the modified constraints C4 Corollary 5.4 Assume that x∗ τ1 / x∗ τ2 for τ1 , τ2 / τ1 ∈ 0, T in Theorem 5.1(iii) Then, any k → ∞ state-trajectory solution converges to a unique limit oscillation of period T ∗ ≤ T as Z0 If x∗ τ1 / x∗ τ2 for τ1 , τ2 / τ1 ∈ 0, T in Theorem 5.1(iv) Then, any state-trajectory solution k → ∞ of period not larger that converges to a unique limit oscillation x kT τ → z∗ τ as Z0 T ; for all τ ∈ 0, T , under the modified constraints C4 The condition of convergence of the sequence of sampling periods to a constant limit is not necessary to derive Theorem 5.1 iii - iv and Corollaries 5.3-5.4 The following ad-hoc generalization follows: k−1 j Corollary 5.5 Construct the continuous time argument t : tk τ Tj τ; for all t ∈ tk , tk ; k j for all τ ∈ 0, Tk ; for all k ∈ Z0 Assume that limk → ∞ Tj τ1 k / limk → ∞ k Tj j τ2 k for τ1 k , τ2 k / τ1 k ∈ tk , tk in Theorem 5.1(iii) Then, any state-trajectory solution k → ∞ converges to a unique limit oscillation of period T ∗ ≤ T as Z0 Remark 5.6 An inequality for the maximum allowable L∞ -norm of the error among any two state-trajectory solutions is now derived Note from the constraints C.2–C.4, and 2.6 – 2.9 that xd k − yd k Ad k xd k − yd k ≤ md k f βdc f Adc k xc k − yc k g x k −y k βdd βdc δdx k − δdy k g βdd r, 5.22 where ∞ > md ≥ md k : Adc k , Ad k Then, from 5.16 and 5.17 , it is possible to take into account the discontinuities at sampling instants by evaluating the argument τ on the closed interval 0, Tk as follows: x t −y t tk ,tk τ ≤ Max 0≤s≤τ Kc − e−ac τ ac Ψc tk s, tk × x t −y t tk ,tk τ Kc − e−ac τ ac f f βcc βcd g βcc g βcd md k g βdc f βdc f βdd g βdd r ∀τ ∈ 0, Tk 5.23 18 Fixed Point Theory and Applications However, 5.23 leads to x t −y t ≤ x−y ≤ ρc 0,t Kc Max Max − e−ac τ ac k ∈ Z0 τ ∈ 0,Tk 2K c g βcc ac ≤α x−y g βcd β; ∞ r f f βcc βcd 1−e x t −y t −ac τ ∞ 5.24 ∀t ∈ R0 , Max Max k ∈ Z0 τ ∈ 0,Tk where α : ρc β: 2K c g βcc ac Then, x − y ∞ Kc Max − e−ac τ ac τ ∈ 0,T f βcc f βcd , 5.25 g βcd r Max τ ∈ 0,T 1−e −ac τ , ρc : Max Max Ψc tk k ∈ Z0 0≤τ≤Tk τ, tk ≤ Max 2r, − α−1 β if α < Acknowledgments The author is very grateful to the Spanish Ministry of Education by its partial support of this work through Project DPI2006-00714 He is also grateful to the Basque Government 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