2020/08/07 by Jr-Shin Li, Wei Zhang, Li, Jr-Shin +1
Biochemistry, Genetics and Molecular Biology · Computer Science · Physics and Astronomy · #22E65 #34K35 #93A15 #93B05 #93C10 #Atomic and Subatomic Physics Research #Dynamical Systems (math.DS) #FOS: Mathematics #Hemoglobin structure and function #Nonlinear Dynamics and Pattern Formation #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2008.03243
openalex publication_date 2020/08/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Problems involving control of large ensmebles of structurally identical dynamical systems, called ensemble control, arise in numerous scientific areas from quantum control and robotics to brain medicine. In many of such applications, control can only be implemented at the population level, i.e., through broadcasting an input signal to all the systems in the population, and this new control paradigm challenges the classical systems theory. In recent years, considerable efforts have been made to investigate controllability properties of ensemble systems, and most works emphasized on linear and some forms of bilinear and nonlinear ensemble systems. In this paper, we study controllability of a broad class of bilinear ensemble systems defined on semisimple Lie groups, for which we define the notion of ensemble controllability through a Riemannian structure of the state space Lie group. Leveraging the Cartan decomposition of semisimple Lie algebras in representation theory, we develop a covering method that decomposes the state space Lie group into a collection of Lie subgroups generating the Lie group, which enables the determination of ensemble controllability by controllability of the subsystems evolving on these Lie subgroups. Using the covering method, we show the equivalence between ensemble and classical controllability, i.e., controllability of each individual system in the ensemble implies ensemble controllability, for bilinear ensemble systems evolving on semisimple Lie groups. This equivalence makes the examination of controllability for infinite-dimensional ensemble systems as tractable as for a finite-dimensional single system.