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

本课题的目的是研究受脉冲作用和外部扰动影响的无限维非线性系统的全局吸引子的稳定性和鲁棒性，这些吸引子可以分布在系统所考虑的区域或该区域的边界上。由于脉冲作用，解对初始条件的连续依赖不能成立。这使得全局吸引子理论的经典方法无法应用，需要新的工具和方法来研究全局吸引子的性质。对于无限维状态空间中相当广泛的系统，我们将给出全局吸引子存在性、稳定性和鲁棒性的充分条件。这些条件将用于研究由具有多值右侧的抛物型演化包涵、具有非光滑相互作用函数的反应-扩散型抛物型系统和弱非线性双曲型方程所产生的脉冲系统的渐近行为。我们将考虑两种类型的冲动行为。其中之一导致从状态空间的一个给定子集到另一个给定子集的瞬时状态转移，这也是固定的和给定的。例如，当一个系统的能量从一个固定能级跃迁到另一个固定能级时，就会发生这种情况。另一种是状态的瞬时变化仅取决于跳转前的状态值。例如，当固体与刚体碰撞而产生弹性冲击时，就会出现这种行为。为此目的，将阐述这类系统的全局吸引子的一个适当的稳定性概念。在此之后，我们将研究外部扰动对全局吸引子的影响。特别地，受扰动系统与未受扰动系统的流动偏差将根据扰动的大小进行研究。我们将建立非线性抛物方程和包含的全局吸引子的输入-状态稳定性。此外，我们将考虑具有稳定全局吸引子的互联无限维系统，并询问互联是否也具有全局吸引子，以及它是否仍然是输入到状态稳定的。将开发小增益型条件来回答这些问题。对于受脉冲效应影响的互连，还将研究脉冲系统中通常需要的小增益和停留时间条件之间的相互作用。

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.