论文标题
希尔伯特空间动力学系统的收缩理论
Contraction Theory for Dynamical Systems on Hilbert Spaces
论文作者
论文摘要
欧几里得空间上动力学系统的收缩理论是良好的。对于收缩（分别半缩）系统，任何两个轨迹之间的距离（分别半距离）呈指数速度降低。对于部分收缩系统，每个轨迹都将指数级收敛到不变的子空间。 在本说明中，我们开发了有关希尔伯特空间的收缩理论。首先，我们为合并性提供了一种新颖的积分条件，对于时间不变的系统，我们确定了独特的全球稳定平衡的存在。其次，我们介绍了部分和半收缩的概念，并为时间变化和时间流逝的系统提供了各种足够的条件。最后，我们将理论应用于经典的反应扩散系统。
Contraction theory for dynamical systems on Euclidean spaces is well-established. For contractive (resp. semi-contractive) systems, the distance (resp. semi-distance) between any two trajectories decreases exponentially fast. For partially contractive systems, each trajectory converges exponentially fast to an invariant subspace. In this note, we develop contraction theory on Hilbert spaces. First, we provide a novel integral condition for contractivity, and for time-invariant systems, we establish the existence of a unique globally exponentially stable equilibrium. Second, we introduce the notions of partial and semi-contraction and we provide various sufficient conditions for time-varying and time-invariant systems. Finally, we apply the theory on a classic reaction-diffusion system.
