On the Dynamics of a Differential Inclusion Built upon a Nonconvex Constrained Minimization Problem

In this paper, we study the dynamics of a differential inclusion built upon a nonsmooth, not necessarily convex, constrained minimization problem in finite-dimensional spaces. In particular, we are interested in the investigation of the asymptotic behavior of the trajectories of the dynamical system represented by the differential inclusion. Under suitable assumptions on the constraint set and the two involved functions (one defining the constraint set, the other representing the functional to be minimized), it is proved that all the trajectories converge to the set of the constrained critical points. We present also a large class of constraint sets satisfying our assumptions. As a simple consequence, in the case of a smooth convex minimization problem, we have that any trajectory converges to the set of minimizers.Research partially supported by the research project CNR-GNAMPA Mathematical Methods for Control Theory and supported in part by the European Communitys Human Potential Program under Contract HPRN-CT-2002-00281, Evolution Equations.     

An Averaging Method for Singularly Perturbed Systems of Semilinear Differential Inclusions with C0-Semigroups

We consider a system of two semilinear differential inclusions with infinitesimal generators of C ₀-semigroups. The nonlinear terms are of high frequency with respect to time and periodic with a specified period. Moreover, they are condensing in the state variables (x,y) with respect to a suitable measure of noncompactness. The goal of the paper is to give sufficient conditions to guarantee, for >0 sufficiently small, the existence of periodic solutions and to study their behaviour as 0. The main tool to achieve this is the topological degree theory for uppersemicontinuous, condensing vector fields.     

Directionally Continuous Selections and Nonconvex Evolution Inclusions

In Bressan and Staicu, Set-Valued Anal. 2 (1994), 415–437, qualitative properties for solutions of the evolution inclusion Ax+F(t,x), with A generator of a contractive semigroup and F a lower semicontinuous closed valued multifunction defined on compact subset of R×cl(D(A)), were deduced as corollaries of similar properties for solutions of the evolution inclusion Ax+G(t,x), where G is upper semicontinuous compact convex valued, obtained from a suitable directionally continuous selection from F.In this paper we prove that the compactness assumption on the domain of F can be avoided.     

Implicit resolvent dynamical systems for quasi variational inclusions

In this paper, we suggest and analyze a class of implicit resolvent dynamical systems for quasi variational inclusions by using the resolvent operator technique. We show that the trajectory of the solution of the implicit dynamical system converges globally exponentially to the unique solution of the quasi variational inclusions. Our results can be considered as a significant extension of the previously known results.     

Variational Inclusions for Fuzzy Mappings

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On Stability and Boundedness for Lipschitzian Differential Inclusions: The Converse of Lyapunov's Theorems

For Lipschitzian differential inclusions, we prove that the existence of suitable Lyapunov functions is necessary for uniform stability and uniform boundedness of solutions.     

双周期不等长刚性线Ⅲ型应力奇异因子的精确解

研究了含双周期刚性线夹杂复合材料的反平面剪切问题,其基本胞元含有四条不等长刚性线。运用复变函数方法,获得该问题严格的闭合解,得到了复合材料体内应力场的分布表达式,并在此基础上给出了刚性线尖端III型应力奇异因子的精确公式。数值算例显示了刚性线分布特征对材料细观应力场的影响,特别是短刚性线对主刚性线尖端应力场的屏蔽与放大作用。由本文解答的特殊情形,可以直接导出近年来其他学者获得的结果和一系列有意义的新结果。     

Inferring mechanism from time-series data: Delay-differential equations

When there is qualitative information about the underlying processes and structure of a dynamical system, it may be possible to infer very accurate quantitative information about these processes using only an output time series from the system. We illustrate how this can be accomplished for time series data from a delay-differential equation with a single fixed delay. Our approach exploits modern techniques for non-parametric function estimation, is robust to fairly high levels of dynamic noise and measurement error, and can be extended straightforwardly to more general delay-differential systems and multivariate systems.     

Stability analysis of numerical methods for systems of neutral delay-differential equations

Stability analysis of some representative numerical methods for systems of neutral delay-differential equations (NDDEs) is considered. After the establishment of a sufficient condition of asymptotic stability for linear NDDEs, the stability regions of linear multistep, explicit Runge-Kutta and implicitA-stable Runge-Kutta methods are discussed when they are applied to asymptotically stable linear NDDEs. Some mentioning about the extension of the results for the multiple delay case is given.     

Optimality Conditions and Synthesis for the Minimum Time Problem

We study the minimum time optimal control problem for a nonlinear system in R ⁿ with a general target. Necessary and sufficient optimality conditions are obtained. In particular, we describe a class of costates that are included in the superdifferential of the minimum time function, even in the case when this function is only lower semicontinuous. Two set-valued maps are constructed to provide time optimal synthesis.