Weak coupling of Lohe tensor models

Seung Yeal Ha

Seoul National University

Abstract: In this talk, we present a systematic algebraic approach for the weak coupling of Cauchy problems to multiple Lohe tensor models. For this, we identify an admissible Cauchy problem to the Lohe tensor (LT) model with a characteristic symbol consisting of four tuples in terms of a size vector, a natural frequency tensor, a coupling strength tensor and admissible initial configuration. In this way, the collection of all admissible Cauchy problems to the LT models is equivalent to the space of characteristic symbols. On the other hand, we introduce a binary operation, namely fusion operation as a binary operation between characteristic symbols. It turns out that the fusion operation satisfies an associativity and admits the identity element in the space of characteristic symbols which naturally forms a monoid. By virtue of the fusion operation, the weakly coupled system of multi tensor models can be obtained by applying the fusion operation of multiple characteristic symbols corresponding to the Lohe tensor models. As a concrete example, we consider a weak coupling of the swarm sphere model and the Lohe matrix model, and provide sufficient framework leading to emergent dynamics to the proposed weakly coupled model. This is a joint work with Dohyun Kim (Sungshin Women’s Univ.) and Hansol Park (Simon Fraser Univ.)

时间: 2023.08.08  8:30-9:30

地点：理学楼609

Some progresses on Hilbert expansion of kinetic equations with collision

Qinghua Xiao（肖清华）

Chinese Academy of Science

Abstract: We report some progresses on Hilbert expansion of kinetic equations with collision. Through new weighted energy methods, we justify the corresponding Euler limits of Landau-type equations in both the relativistic framework and non-relativistic framework as the Knudsen number shrinks to zero. As a further development of the energy method, the recent progress on Hilbert expansion of the Vlasov-Maxwell-Boltzmann system with(without) cutoff will also be discussed. 

时间: 2023.08.08  9:30-10:20

地点：理学楼609

Flocking and pattern formation in the Cucker-Smale model

Yinglong Zhang（张英龙）

Shandong University

Abstract: Self-organized collective motions are often observed in complex biological systems, such as the flocking of birds, swarming of fish, and synchronization of fireflies. Many phenomenological agent-based models were proposed and studied due to their potential applications to unmanned aerial vehicles and client networks. In this presentation, we talk about the flocking model proposed by F. Cucker and S. Smale [Cucker-Smale, IEEE, 2007], which is called the Cucker-Smale model. We talk about the conditions of the global and local flockings, and we also talk about the pattern formation problem in this model.

时间: 2023.08.08    10:20-11:10

地点：理学楼609

Leader-following rendezvous control for generalized Cucker-Smale model on Riemannian manifolds

Xiaoyu Li（李小雨）

Nanjing University of Posts and Telecommunications

Abstract: We study a leader-following rendezvous problem for the generalized Cucker-Smale model, a double-integrator multi-agent system, on some Riemannian manifolds. By using intrinsic properties of covariant derivative, logarithm map, and parallel transport on the Riemannian manifolds, we design a feedback control law and prove that this feedback control law enables all followers to track the trajectory of the moving leader when the Riemannian manifold is compact or flat. As concrete examples, we consider the leader-following rendezvous problem on the unit sphere, Euclidean space, unit circle, and infinite cylinder and present the corresponding feedback control laws. Meanwhile, numerical examples are given for the aforementioned Riemannian manifolds to illustrate and verify the theoretical results.

时间: 2023.08.08    11:10-12:00

地点：理学楼609