Asymptotic ensemble stabilizability of the Bloch equation

In this paper we are concerned with the stabilizability to an equilibrium point of an ensemble of non interacting half-spins. We assume that the spins are immersed in a static magnetic field, with dispersion in the Larmor frequency, and are controlled by a time varying transverse field. Our goal is to steer the whole ensemble to the uniform "down" position. Two cases are addressed: for a finite ensemble of spins, we provide a control function (in feedback form) that asymptotically stabilizes the ensemble in the "down" position, generically with respect to the initial condition. For an ensemble containing a countable number of spins, we construct a sequence of control functions such that the sequence of the corresponding solutions pointwise converges, asymptotically in time, to the target state, generically with respect to the initial conditions. The control functions proposed are uniformly bounded and continuous

Controllability Canonical Forms of Linear Ensemble Systems

Ensemble control, an emerging research field focusing on the study of large populations of dynamical systems, has demonstrated great potential in numerous scientific and practical applications. Striking examples include pulse design for exciting spin ensembles in quantum physics, neurostimulation for relieving neurological disorder symptoms, and path planning for steering robot swarms. However, the control targets in such applications are generally large-scale complex and severely underactuated ensemble systems, research into which stretches the capability of techniques in classical control and dynamical systems theory to the very limit. This paper then devotes to advancing our knowledge about controllability of linear ensemble systems by integrating tools in modern algebra into the technique of separating points developed in our recent work. In particular, we give an algebraic interpretation of the dynamics of linear systems in terms of actions of polynomials on vector spaces, and this leads to the development of the functional canonical form of matrix-valued functions, which can also be viewed as the generalization of the rational canonical form of matrices in linear algebra. Then, leveraging the technique of separating points, we achieve a necessary and sufficient characterization of uniform ensemble controllability for time-invariant linear ensemble systems as the ensemble controllability canonical form, in which the system and control matrices are in the functional canonical and block diagonal form, respectively. This work successfully launches a new research scheme by adopting and tailoring finite-dimensional methods to tackle control problems involving infinite-dimensional ensemble systems, and lays a solid foundation for a more inclusive ensemble control theory targeting a much broader spectrum of control and learning problems in both scientific research and practice

Controllability Issues of Linear Ensemble Systems

We address an open problem in ensemble control: Whether there exist controllable linear ensemble systems over high dimensional parameterization spaces? We provide a negative answer: Any real-analytic linear ensemble system is not $\mathrm{L}^p$-controllable, for $2\le p \le \infty$, if the dimension of its parameterization space is greater than one