Stability and robustness of attractors of nonlinear infinite-dimensional systems with respect to disturbances

非线性无限维系统吸引子对扰动的稳定性和鲁棒性

• 批准号: 405685496
• 负责人: Professor Dr. Sergey Dashkovskiy
• 金额: --
• 依托单位: The State Fund for Fundamental Researches
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2019
• 资助国家: 德国
• 起止时间: 2018-12-31 至 2021-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/405685496?language=en
• 关键词: Stability; robustness; attractors; nonlinear; infinite

The aim of this project is to study stability and robustness of global attractors of infinite-dimensional nonlinear systems subject to impulsive actions and external perturbations that can be distributed in the domain on which the system is considered or at the boundary of this domain. Due to impulsive actions the continuous dependence of solutions with respect to the initial conditions fails to hold. This makes the application of classical methods of global attractors theory impossible and requires for new tools and methods to study properties of global attractors. For rather wide classes of systems in infinite-dimensional state spaces we will propose sufficient conditions for the existence, stability and robustness of global attractors. These conditions will be used to study asymptotic behavior of impulsive systems generated by evolution inclusions of parabolic type with multivalued right-hand side, parabolic systems of reaction-diffusion type with non-smooth interaction functions and weakly nonlinear hyperbolic equations. Two types of impulsive actions will be considered. One of them leads to an instantaneous state transition from one given subset of the states space to another one, which is also fixed and given. This happens, for example, when the energy of a system jumps from one fixed level to another one. The other one is such that the instantaneous change of the state depends on the state value before the jump only. Such behavior appears, for example, when elastic shock happens due to a collision of a solid with a rigid body. A suitable stability concept for global attractors of such systems will be elaborated for this purpose.After this we will investigate the influence of external perturbations on global attractors. In particular the deviation of the perturbed systems flow from the unperturbed one will be studied in dependence on the size of the disturbance. We are going to establish input-to-state stability properties of global attractors for nonlinear parabolic equations and inclusions.Moreover we will consider interconnected infinite-dimensional systems having stable global attractors and ask whether the interconnection possess a global attractor as well and whether it is still input-to-state stable or not. Small-gain type conditions will be developed to answer these questions. The interactions between small-gain and dwell-time conditions that are typically needed in case of impulsive systems will be also investigated for interconnections subject to impulsive effects.

该项目的目的是研究无限二维非线性系统的全球吸引子的稳定性和稳健性，这些系统受到冲动行动和外部扰动的影响，这些系统可以分布在该域或该域的边界的域中。由于冲动行动，解决初始条件的持续依赖性无法保持。这使得不可能地应用全球吸引者理论的经典方法，并且需要使用新的工具和方法来研究全球吸引子的属性。对于无限维状态空间中相当广泛的系统，我们将为全球吸引子的存在，稳定性和鲁棒性提出足够的条件。这些条件将用于研究由抛物线类型的进化夹杂物产生的冲动系统的渐近行为，具有多价右侧，具有非平滑相互作用函数的反应扩散类型的抛物线系统和弱非线性双曲线方程。将考虑两种冲动行动。其中一个导致瞬时状态从一个给定的子集过渡到另一个空间，也是固定和给定的。例如，当系统的能量从一个固定级别跳到另一个级别时，就会发生这种情况。另一个是使状态的瞬时变化仅在跳跃之前取决于状态值。例如，当由于固体与刚体的碰撞而发生弹性冲击时，这种行为出现了。为此，将为此类系统的全球吸引者提供一个合适的稳定概念。为此，我们将研究外部扰动对全球吸引子的影响。特别是，将研究扰动系统与不受干扰的系统的偏差，以依赖干扰的大小。我们将建立全球吸引子在非线性抛物线方程和夹杂物中的投入到国家稳定性。此外，我们将考虑具有稳定的全球吸引子的互连的无限维系统，并询问该互连是否也具有全球吸引子，以及它是否仍然是输入对状态稳定的稳定。将开发小增益类型的条件来回答这些问题。在冲动系统的情况下，通常需要进行的小增加和住宿时间条件之间的相互作用也将进行脉冲效应的互连。