Declarative vs Rule-based Control for Flocking Dynamics

The popularity of rule-based flocking models, such as Reynolds' classic flocking model, raises the question of whether more declarative flocking models are possible. This question is motivated by the observation that declarative models are generally simpler and easier to design, understand, and analyze than operational models. We introduce a very simple control law for flocking based on a cost function capturing cohesion (agents want to stay together) and separation (agents do not want to get too close). We refer to it as {\textit declarative flocking} (DF). We use model-predictive control (MPC) to define controllers for DF in centralized and distributed settings. A thorough performance comparison of our declarative flocking with Reynolds' model, and with more recent flocking models that use MPC with a cost function based on lattice structures, demonstrate that DF-MPC yields the best cohesion and least fragmentation, and maintains a surprisingly good level of geometric regularity while still producing natural flock shapes similar to those produced by Reynolds' model. We also show that DF-MPC has high resilience to sensor noise.Comment: 7 Page

임의성이 있는 쿠커-스메일 모형에 대한 정량적 해석에 관하여

학위논문(박사)--서울대학교 대학원 :자연과학대학 수리과학부,2020. 2. 하승열.In this thesis, we introduce random elements into the Cucker-Smale(C-S) model and provide quantitative analyses for those uncertainties. In real applications of the Cucker-Smale dynamics, we can expect that the C-S model contains some intrinsic uncertainties in itself and misses some extrinsic factors that might affect the dynamics of particles. Thus, to provide a better description for the dynamics of a C-S ensemble, one needs to incorporate such uncertain factors to the model and evaluate their effects on the dynamics or stability of the C-S system. To fulfill this, we first consider the macroscopic version of the Cucker-Smale model. Namely, we introduce random inputs from communication weights and initial data into the hydrodynamic Cucker-Smale (HCS) model to yield the random HCS model. Furthermore, we address extrinsic uncertainties in the microscopic and mesoscopic level, respectively. For a microscopic model, we introduce a randomly switching network structure to the Cucker-Smale model and investigate sufficient conditions for the emergence of flocking. As a mesoscopic model, we consider the kinetic Cucker-Smale equation perturbed by multiplicative white noise and study the well-posedness and asymptotic dynamics of solutions.본 학위 논문에서는, 쿠커-스메일 모형에 임의적 요소를 도입하여 그러한 불확실성에 대한 정량적 해석을 제시한다. 쿠커-스메일 모형의 다이나믹스를 실제로 응용함에 있어 우리는 쿠커-스메일 모형 자체가 몇몇 내적 불확실성을 포함하고 있으며 입자들의 다이나믹스에 영항을 줄 수 있는 몇 가지 외부적 요인을 놓치고 있음을 예상할 수 있다. 그러므로 쿠커-스메일 총체의 다이나믹스를 더 잘 서술하기 위해, 이러한 불확실성이 있는 요소를 모형에 도입하여 그것들이 쿠커-스메일 계의 다이나믹스나 안정성에 주는 영향을 평가할 필요가 있다. 이를 달성하기 위해, 우리는 우선 쿠커-스메일 모형의 거시적인 형태를 고려한다. 즉, 우리는 통신 가중치 함수와 초기값에서 오는 임의적 입력치를 유체역학 쿠커-스메일 모형에 포함시켜 임의적 유체역학 쿠커-스메일 모형을 유도한다. 더 나아가 미시적 그리고 중간보기적 단계에서 외적 불확실성에 대해 다룬다. 미시적 모형에 대해서, 쿠커-스메일 모형에 임의로 변하는 네트워크 구조를 도입하여 플로킹의 창발에 대한 충분조건을 알아본다. 중간보기적 단계의 모형으로서, 우리는 곱셈 백색 소음으로 동요된 쿠커-스메일 운동방정식을 고려하고 해의 존재성 및 유일성과 점근적 다이나믹스를 공부한다.1 Introduction 1 2 Preliminaries 9 2.1 Notation 9 2.2 Previous results 10 3 A local sensitivity analysis for the hydrodynamic Cucker-Smale model with random inputs 15 3.1 Pathwise well-posedness of z-variations 16 3.1.1 First-order z-variations 18 3.1.2 Higher-order z-variations 26 3.2 The local sensitivity analysis for stability estimates 32 3.2.1 Higher-order L2-stability 32 3.2.2 L2-stability estimates for z-variations 37 3.3 A local sensitivity analysis for flocking estimate 41 4 On the stochastic flocking of the Cucker-Smale flock with randomly switching topologies 48 4.1 Preliminaries 49 4.1.1 Pathwise dissipative structure 49 4.1.2 A directed graph 52 4.1.3 A scrambling matrix 53 4.1.4 A state transition matrix 54 4.1.5 Previous results 55 4.2 A description of main result 57 4.2.1 Standing assumptions 57 4.2.2 Main result 58 4.3 Emergent behavior of the randomly switching system 61 4.3.1 A matrix formulation 61 4.3.2 Pathwise flocking under a priori assumptions 62 4.3.3 Emergence of stochastic flocking 70 5 Collective stochastic dynamics of the Cucker-Smale ensemble under uncertain communication 74 5.1 Preliminaries 75 5.1.1 Derivation of the SPDE 75 5.1.2 Presentation of main results 78 5.1.3 Elementary lemmas 80 5.2 A priori estimates for classical solutions 82 5.2.1 Quantitative estimates for classical solutions 86 5.3 Global well-posedness and asymptotic dynamics of strong solutions 92 5.3.1 Construction of approximate solutions 94 5.3.2 Estimates on approximate solutions 95 5.3.3 Proof of Theorem 5.1.3 103 6 Conclusion and future works 110 Appendix A Detailed proof of Chapter 3 112 A.1 Proof of Lemma 3.1.2 112 A.2 Proof of Lemma 3.1.5 115 A.3 Proof of Lemma 3.2.4 119 A.4 Proof of Theorem 3.3.2 121 Appendix B Detailed proof of Chapter 5 124 B.1 A proof of Theorem 5.2.1 124 B.2 A proof of Proposition 5.3.3 129 Bibliography 133Docto