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Given any finite family of real d-by-d nonsingular matrices {S1,…,Sl}, by extending the well-known Li–Yorke chaos of a deterministic nonlinear dynamical system to a discrete-time linear inclusion or hybrid or switched system:
xn∈{Skxn?1;1≤k≤l},x0∈Rdandn≥1,
we study the chaotic dynamics of the state trajectory (xn(x0, σ))n ≥ 1 with initial state x0∈Rd, governed by a switching law σ:N→{1,…,l}. Two sufficient conditions are given so that for a “large” set of switching laws σ, there exhibits the scrambled dynamics as follows: for all x0,y0∈Rd,x0≠y0,
lim infn→+∞∥xn(x0,σ)?xn(y0,σ)∥=0andlim supn→+∞∥xn(x0,σ)?xn(y0,σ)∥=∞.
This implies that there coexist positive, zero and negative Lyapunov exponents and that the trajectories (xn(x0, σ))n ≥ 1 are extremely sensitive to the initial states x0∈Rd. We also show that a periodically stable linear inclusion system, which may be product unbounded, does not exhibit any such chaotic behavior. An explicit simple example shows the discontinuity of Lyapunov exponents with respect to the switching laws.  相似文献   

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We consider the function Lyapunov equation f*(A)X+Xf(A)=C, where A and C are given matrices, f(z) is a function holomorphic on a neighborhood of the spectrum σ(A) of A. For a solution X of that equation, norm estimates are established. By these estimates we investigate perturbations of a matrix A whose spectrum satisfies the condition inf?σ(f(A))>0. In the case f(z)=zν with a positive integer ν we obtain conditions that provide localization of the spectrum of a perturbed matrix in a given angle.  相似文献   

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The present work proposes a relaxed gradient based iterative (RGI) algorithm to find the solutions of coupled Sylvester matrix equations AX+YB=C,DX+YE=F. It is proved that the proposed iterative method can obtain the solutions of the coupled Sylvester matrix equations for any initial matrices X0 and Y0. Next the RGI algorithm is extended to the generalized coupled Sylvester matrix equations of the form Ai1X1Bi1+Ai2X2Bi2+?+AipXpBip=Ci,(i=1,2,…,p). Then, we compare their convergence rate and find RGI is faster than GI, which has maximum convergence rate, under an appropriative positive number ω and the same convergence factor µ1 and µ2. Finally, a numerical example is included to demonstrate that the introduced iterative algorithm is more efficient than the gradient based iterative (GI) algorithm of (Ding and Chen 2006) in speed, elapsed time and iterative steps.  相似文献   

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Let {Πτ(m, n): m?≥?n?≥?0} be the family of periodic discrete transition matrices generated by bounded valued square matrices Λτ(n), where τ:[0,1,2,?)→Ω is an arbitrary switching signal. We prove that the family {Πτ(m, n): m?≥?n?≥?0} of bounded linear operator is uniformly exponentially stable if and only if the sequence n?∑k=0neiαkΠτ(n,k)w(k):Z+→R is bounded.  相似文献   

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This paper discusses the problem of synchronization for delayed neural networks using sampled-data control. We introduce a new Lyapunov functional, called complete sampling-interval-dependent discontinuous Lyapunov functional, which can adequately capture sampling information on both intervals from r(t?τ¯) to r(tk?τ¯) and from r(t?τ¯) to r(tk+1?τ¯). Based on this Lyapunov functional and an improved integral inequality, less conservative conditions are derived to ensure the stability of the synchronization error system, leading to the fact that the drive neural network is synchronized with the response neural network. The desired sampled-data controller is designed in terms of solutions to linear matrix inequalities. A numerical example is provided to demonstrate that the proposed approaches are effective and superior to some existing ones in the literature.  相似文献   

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This paper studies the stability of linear continuous-time systems with time-varying delay by employing new Lyapunov–Krasovskii functionals. Based on the new Lyapunov–Krasovskii functionals, more relaxed stability criteria are obtained. Firstly, in order to coordinate with the use of the third-order Bessel-Legendre inequality, a proper quadratic functional is constructed. Secondly, two couples of integral terms {∫t?htsx(s)ds,∫stx(s)ds} and {∫t?hMsx(s)ds,∫st?htx(s)ds} are involved in the integral functionals ∫t?htt(·)ds and ∫t?hMt?ht(·)ds, respectively, so that the coupling information between them can be fully utilized. Finally, two commonly-used numerical examples are given to demonstrate the effectiveness of the proposed method.  相似文献   

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Consider the continuous-time matrix Riccati operator Ricc(Q)=AQ+QA′?QSQ+R. In this work, we consider the robustness of this operator to direct perturbations of the matrices (A, R, S) and, in particular, the flow robustness of the corresponding Riccati differential equation. For a given class of perturbation, we show that the corresponding differential equation is well defined in the sense it is bounded above and below, it has a well-defined fixed point, and it converges to this fixed point exponentially fast. Moreover, the flow of the perturbed Riccati flow is close to the nominal Riccati flow when the perturbation is small; i.e. we prove a continuity-type condition in the size of the perturbation.  相似文献   

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