On an asymptotically autonomous system with Tikhonov type regularizing term

We study the asymptotic behaviour of the trajectories of the second order equation ${\ddot{x}(t)+\gamma \dot{x}(t)+\nabla\phi(x(t))+\varepsilon(t)x(t)=g(t)}We study the asymptotic behaviour of the trajectories of the second order equation [(x)\ddot](t)+g[(x)\dot](t)+?f(x(t))+e(t)x(t)=g(t){\ddot{x}(t)+\gamma \dot{x}(t)+\nabla\phi(x(t))+\varepsilon(t)x(t)=g(t)} , where γ > 0, g ? L¹([0,+￥[;H){g \in L^1([0,+\infty[;H)}, Φ is a C ² convex function and e{\varepsilon} is a positive nonincreasing function.     

The realization of Smale solenoid type attractors in 3-manifolds

We study how to realize Smale solenoid type attractors in 3-manifolds. It is already known that we can restrict the 3-manifolds to lens spaces. We get all Smale solenoids realized in a given lens space through an inductive construction. We turn this around to address the question of how to decide whether a closed braid is a trivial knot in S³. For a diffeomorphism f of a 3-manifold M that realizes a Smale solenoid, it is natural to ask whether f⁻¹ also realizes a Smale solenoid. We relate this question to exchangeable braids, and for some special positive case, we describe the relation between the two Smale solenoids of f and f⁻¹.     

Solutions with internal jump for an autonomous elliptic system of FitzHugh–Nagumo type

Systems of elliptic partial differential equations which are coupled in a noncooperative way, such as the FitzHugh–Nagumo type studied in this paper, in general do not satisfy order preserving properties. This not only results in technical complications but also yields a richer solution structure. We prove the existence of multiple nontrivial solutions. In particular we show that there exists a solution with boundary layer type behaviour, and we will give evidence that this autonomous system for a certain range of parameters has a solution with both a boundary and an internal layer. The analysis uses results from bifurcation theory, variational methods, as well as some pointwise a priori estimates. The final section contains some numerically obtained results.     

一类三阶自治系统的积分法

研究一类三阶自治系统的积分方法,证明了在一定条件下,这类系统的积分问题归结为求解由该方程的系数所确定的Riccati方程     

耗散KDV型方程Cauchy问题的整体吸引子

该文对耗散ＫＤＶ型方程的动力学行为进行了讨论,得到了该方程在Ｈ＾２（Ｒ＾１）上存在整体吸引子。     

Stability of an autonomous dynamic system with fast Markov switching

For an autonomous dynamic system with a fast Markov switching, sufficient conditions for asymptotic stability in the presence of a Lyapunov function which assures stability of a stationary averaged system are established.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 9, pp. 1176–1181, September, 1991.     

Attractor for lattice system of dissipative Zakharov equation

We consider the asymptotic behavior of solutions of an infinite lattice dynamical system of dissipative Zakharov equation. By introducing new weight inner product and norm in the space and establishing uniform estimate on ＂Tail End＂ of solutions, we overcome some difficulties caused by the lack of Sobolev compact embedding under infinite lattice system, and prove the existence of the global attractor; then by using element decomposition and the covering property of a polyhedron in the finite-dimensional space, we obtain an upper bound for the Kolmogorov ε-entropy of the global attractor; finally, we present the upper semicontinuity of the global attractor.     

Existence of heteroclinic orbits of the Shil'nikov type in a 3D quadratic autonomous chaotic system

In this paper, the existence of heteroclinic orbits of Shil'nikov type in a three-dimensional quadratic autonomous system is proved. Four heteroclinic orbits and four critical points together constitute two cycles simultaneously. The dynamical behaviors of the system are also studied.     

Centralized and distributed cooperative Receding Horizon control of autonomous vehicle missions

We consider a setting where multiple vehicles form a team cooperating to visit multiple target points and collect rewards associated with them. The team objective is to maximize the total reward accumulated over a given time interval. Complicating factors include uncertainties regarding the locations of target points and the effectiveness of collecting rewards, differences among vehicle capabilities, and the fact that rewards are time-varying. We present a Receding Horizon (RH) control scheme which dynamically determines vehicle trajectories by solving a sequence of optimization problems over a planning horizon and executing them over a shorter action horizon. A key property of this scheme is that the trajectories it generates are stationary, in the sense that they ultimately guide vehicles to target points, even though the controller is not designed to perform any discrete point assignments. The proposed scheme is centralized and it induces a cooperative behavior. We subsequently develop a distributed cooperative controller which does not require a vehicle to maintain perfect information on the entire team and whose computational cost is scalable and significantly lower than the centralized case, making it attractive for applications with real-time constraints. We include simulation-based comparisons between the centralized algorithm and the distributed version, which illustrate the effectiveness of the latter.     

Non autonomous evolution operators of hyperbolic type

In this paper we consider a Cauchy problem in a Banach spaceE:u′(t)=A(t)u(t)+f(t), t∈[t ₀, T], u(t₀)=u₀, whereA(·) is a family of linear operators inE which satisfy all the requirements of Kato's semigroup approach to the non autonomous hyperbolic equations except for the density of the common domains ofA(t). An application is given to a hyperbolic partial differential equation with discontinuous coefficients.     

Solutions of the autonomous Kirchhoff type equations in RN

Recently, the existence, the multiplicity and the concentration of solutions for the Kirchhoff type equations or systems have been extensively established by a number of authors. Such problems contain a nonlocal term which implies that the equation or system is no longer a pointwise identity. Thus, some researchers think that the nonlocal phenomenon will cause some mathematical difficulties. In this note, however, we will give a simple transformation so that the solutions of the autonomous Kirchhoff type equation or system are easily obtained by using the known solutions of the corresponding local equation or system. In particular, some qualitative properties of solutions for the local problems are also inherited.     

Complete damping of a linear autonomous differential-difference system by a controller of the same type

For a spectrally controllable linear autonomous system of retarded type with commensurable delays, we construct a dynamic state feedback ensuring a complete damping of the original system. The closed-loop system is of retarded type as well. The results are illustrated by an example.     

The necessary conditions of the preservation of an invariant manifold of an autonomous system near an equilibrium point

Let a smooth autonomous system of ordinary differential equations have a smooth locally invariant manifold passing through its equilibrium point. The sufficient conditions are known under which each perturbed system has at least one smooth invariant manifoldC ¹ close to the original one. In the paper we prove the necessity of these conditions.     

Finitely smooth normal form of an autonomous system with two pure imaginary roots

We consider the problem of finitely smooth normalization of a system of ordinary differential equations whose linear part has two eigenvalues, while the other eigenvalues lie outside the imaginary axis.     

Uncertainty modeling in autonomous robust spacecraft system design

In the last few years, much research has been dedicated to the development of decisions support systems for the space system engineers or even of completely automated design methods capturing the reasoning of the system experts. However, the problem of taking into account the uncertainties of the variables and models to determine an optimal and robust spacecraft design has not been tackled effectively yet. Based on the clouds formalism we propose a novel approach to process the uncertainty information provided by expert designers and incorporate it into the automated search for a robust optimal design. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global Attractor for Weakly Damped Nonlinear Schrodinger-Boussinesq Equations in an Unbounded Domain

In this paper the authors consider the Cauchy problem of nonlinear Schrödinger-Boussinesq equations in R and prove the existence of the maximal attractor.     

Linear functional differential equations of neutral type asymptotically autonomous

Summary It is studied the relationship between the solutions of the linear functional differential equations(1) (d/dx) D(x_t)=L(x_t) and its perturbed equation(2) [(d/dx) D(x_t)−G(t, x_t)]= =L(x_t)+F(t, x_t) and is proved, under certain hypotheses which will be precised bellow that, if μ is a simple characteristic root of(1), then there exist a σ > 0 and a non zero vector a such that system(2) has a solution satisfying where δ(t)=αd{F(t, ϕ_μ)+μG(t, ϕ_μ)+F(t, X₀G(t, ϕ_μ))}, ϕ_μ(θ)=c·exp (μθ), −r⩾θ⩾0 and α, d, X₀ are given constants. Entrata in Redazione il 5 gennaio 1972.