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Jr-Shin Li

Publications and source records attributed to Jr-Shin Li.

52 records · Page 3Linked to original sources
arXiv2012

Control and Synchronization of Neuron Ensembles

Synchronization of oscillations is a phenomenon prevalent in natural, social, and engineering systems. Controlling synchronization of oscillating systems is motivated by a wide range of applications from neurological treatment of Parkinson's disease to the design of neurocomputers. In this article, we study the control of an ensemble of uncoupled neuron oscillators described by phase models. We examine controllability of such a neuron ensemble for various phase models and, furthermore, study the related optimal control problems. In particular, by employing Pontryagin's maximum principle, we analytically derive optimal controls for spiking single- and two-neuron systems, and analyze the applicability of the latter to an ensemble system. Finally, we present a robust computational method for optimal control of spiking neurons based on pseudospectral approximations. The methodology developed here is universal to the control of general nonlinear phase oscillators.

math.OC↗
arXiv2011

Anti-phase synchronization of phase-reduced oscillators using open-loop control

In this letter, we present an elegant method to build and maintain an anti-phase configuration of two nonlinear oscillators with different natural frequencies and dynamics described by the sinusoidal phase-reduced model. The anti-phase synchronization is achieved using a common input that couples the oscillators and consists of a sequence of square pulses of appropriate amplitude and duration. This example provides a proof of principle that open-loop control can be used to create desired synchronization patterns for nonlinear oscillators, when feedback is expensive or impossible to obtain.

nlin.AO↗
arXiv2011

Optimal Asymptotic Entrainment of Phase-Reduced Oscillators

We derive optimal periodic controls for entrainment of a self-driven oscillator to a desired frequency. The alternative objectives of minimizing power and maximizing frequency range of entrainment are considered. A state space representation of the oscillator is reduced to a linearized phase model, and the optimal periodic control is computed from the phase response curve using formal averaging and the calculus of variations. Computational methods are used to calculate the periodic orbit and the phase response curve, and a numerical method for approximating the optimal controls is introduced. Our method is applied to asymptotically control the period of spiking neural oscillators modeled using the Hodgkin-Huxley equations. This example illustrates the optimality of entrainment controls derived using phase models when applied to the original state space system.

nlin.CD↗
arXiv2011

Synthesis of Optimal Ensemble Controls for Linear Systems using the Singular Value Decomposition

An emerging and challenging area in mathematical control theory called Ensemble Control encompasses a class of problems that involves the guidance of an uncountably infinite collection of structurally identical dynamical systems, which are indexed by a parameter set, by applying the same open-loop control. The subject originates from the study of complex spin dynamics in Nuclear Magnetic Resonance (NMR) spectroscopy and imaging (MRI). A fundamental question concerns ensemble controllability, which determines the existence of controls that transfer the system between desired initial and target states. For ensembles of finite-dimensional time-varying linear systems, the necessary and sufficient controllability conditions and analytical optimal control laws have been shown to depend on the singular system of the operator characterizing the system dynamics. Because analytical solutions are available only in the simplest cases, there is a need to develop numerical methods for synthesizing these controls. We introduce a direct, accurate, and computationally efficient algorithm based on the singular value decomposition (SVD) that approximates ensemble controls of minimum norm for such systems. This method enables the application of ensemble control to engineering problems involving complex, time-varying, and high-dimensional linear dynamic systems.

math.OC↗
arXiv2011

Charge-Balanced Minimum-Power Controls for Spiking Neuron Oscillators

In this paper, we study the optimal control of phase models for spiking neuron oscillators. We focus on the design of minimum-power current stimuli that elicit spikes in neurons at desired times. We furthermore take the charge-balanced constraint into account because in practice undesirable side effects may occur due to the accumulation of electric charge resulting from external stimuli. Charge-balanced minimum-power controls are derived for a general phase model using the maximum principle, where the cases with unbounded and bounded control amplitude are examined. The latter is of practical importance since phase models are more accurate for weak forcing. The developed optimal control strategies are then applied to both mathematically ideal and experimentally observed phase models to demonstrate their applicability, including the phase model for the widely studied Hodgkin-Huxley equations.

math.OC↗
arXiv2011

Optimal trajectories for efficient atomic transport without final excitation

We design optimal harmonic-trap trajectories to transport cold atoms without final excitation, combining an inverse engineering techniqe based on Lewis-Riesenfeld invariants with optimal control theory. Since actual traps are not really harmonic, we keep the relative displacement between the center of mass and the trap center bounded. Under this constraint, optimal protocols are found according to different physical criteria. The minimum time solution has a "bang-bang" form, and the minimum displacement solution is of "bang-off-bang" form. The optimal trajectories for minimizing the transient energy are also discussed.

quant-ph↗
arXiv2011

Minimum-Time Quantum Transport with Bounded Trap Velocity

We formulate the problem of efficient transport of a quantum particle trapped in a harmonic potential which can move with a bounded velocity, as a minimum-time problem on a linear system with bounded input. We completely solve the corresponding optimal control problem and obtain an interesting bang-bang solution. These results are expected to find applications in quantum information processing, where quantum transport between the storage and processing units of a quantum computer is an essential step. They can also be extended to the efficient transport of Bose-Einstein condensates, where the ability to control them is crucial for their potential use as interferometric sensors.

math.OC↗
arXiv2011

The Role of Singular Control in Frictionless Atom Cooling in a Harmonic Trapping Potential

In this article we study the frictionless cooling of atoms trapped in a harmonic potential, while minimizing the transient energy of the system. We show that in the case of unbounded control, this goal is achieved by a singular control, which is also the time-minimal solution for a "dual" problem, where the energy is held fixed. In addition, we examine briefly how the solution is modified when there are bounds on the control. The results presented here have a broad range of applications, from the cooling of a Bose-Einstein condensate confined in a harmonic trap to adiabatic quantum computing and finite time thermodynamic processes.

math.OC↗
arXiv2011

Optimal Control of Inhomogeneous Ensembles

Inhomogeneity, in its many forms, appears frequently in practical physical systems. Readily apparent in quantum systems, inhomogeneity is caused by hardware imperfections, measurement inaccuracies, and environmental variations, and subsequently limits the performance and efficiency achievable in current experiments. In this paper, we provide a systematic methodology to mathematically characterize and optimally manipulate inhomogeneous ensembles with concepts taken from ensemble control. In particular, we develop a computational method to solve practical quantum pulse design problems cast as optimal ensemble control problems, based on multidimensional pseudospectral approximations. We motivate the utility of this method by designing pulses for both standard and novel applications. We also show the convergence of the pseudospectral method for optimal ensemble control. The concepts developed here are applicable beyond quantum control, such as to neuron systems, and furthermore to systems with by parameter uncertainty, which pervade all areas of science and engineering.

math.OC↗
arXiv2010

Frictionless atom cooling in harmonic traps: a time-optimal approach

In this article we formulate frictionless atom cooling in harmonic traps as a time-optimal control problem, permitting imaginary values of the trap frequency for trasient time intervals during which the trap becomes an expulsive parabolic potential. We show that the minimum time solution has "bang-bang" form, where the frequency jumps suddenly at certain instants and then remains constant, and calculate estimates of the minimum cooling time for various numbers of such jumps. A numerical optimization method based on pseudospectral approximations is used to obtain suboptimal realistic solutions without discontinuities, which may be implemented experimentally.

quant-ph↗
arXiv2010

Optimal Design of Minimum-Power Stimuli for Spiking Neurons

In this article, we study optimal control problems of spiking neurons whose dynamics are described by a phase model. We design minimum-power current stimuli (controls) that lead to targeted spiking times of neurons, where the cases with unbounded and bounded control amplitude are considered. We show that theoretically the spiking period of a neuron, modeled by phase dynamics, can be arbitrarily altered by a smooth control. However, if the control amplitude is bounded, the range of possible spiking times is constrained and determined by the bound, and feasible spiking times are optimally achieved by piecewise continuous controls. We present analytic expressions of these minimum-power stimuli for spiking neurons and illustrate the optimal solutions with numerical simulations.

math.DS↗
arXiv2010

Constrained Minimum-Energy Optimal Control of the Dissipative Bloch Equations

In this letter, we apply optimal control theory to design minimum-energy $π/2$ and $π$ pulses for the Bloch system in the presence of relaxation with constrained control amplitude. We consider a commonly encountered case in which the transverse relaxation rate is much larger than the longitudinal one so that the latter can be neglected. Using the Pontryagin's Maximum Principle, we derive optimal feedback laws which are characterized by the number of switches, depending on the control bound and the coordinates of the desired final state.

math.OC↗
arXiv2009

A Pseudospectral Method for Optimal Control of Open Quantum Systems

In this paper, we present a unified computational method based on pseudospectral approximations for the design of optimal pulse sequences in open quantum systems. The proposed method transforms the problem of optimal pulse design, which is formulated as a continuous time optimal control problem, to a finite dimensional constrained nonlinear programming problem. This resulting optimization problem can then be solved using existing numerical optimization suites. We apply the Legendre pseudospectral method to a series of optimal control problems on open quantum systems that arise in Nuclear Magnetic Resonance (NMR) spectroscopy in liquids. These problems have been well studied in previous literature and analytical optimal controls have been found. We find an excellent agreement between the maximum transfer efficiency produced by our computational method and the analytical expressions. Moreover, our method permits us to extend the analysis and address practical concerns, including smoothing discontinuous controls as well as deriving minimum energy controls. The method is not restricted to the systems studied in this article but is universal to every open quantum system whose performance is limited by dissipation.

physics.chem-ph↗
arXiv2008

Ensemble Control of Finite Dimensional Time-Varying Linear Systems

In this article, we investigate the problem of simultaneously steering an uncountable family of finite dimensional time-varying linear systems. We call this class of control problems Ensemble Control, a notion coming from the study of spin dynamics in Nuclear Magnetic Resonance (NMR) spectroscopy and imaging (MRI). This subject involves controlling a continuum of parameterized dynamical systems with the same open-loop control input. From a viewpoint of mathematical control theory, this class of problems is challenging because it requires steering a continuum of dynamical systems between points of interest in an infinite dimensional state space by use of the same control function. The existence of such a control raises fundamental questions of ensemble controllability. We derive the necessary and sufficient controllability conditions and an accompanying analytical optimal control law for ensemble control of time-varying linear systems. We show that ensemble controllability is in connection with singular values of the operator characterizing the system dynamics. In addition, we study the problem of optimal ensemble control of harmonic oscillators to demonstrate our main results. We show that the optimal solutions are pertinent to the study of time-frequency limited signals and prolate spheroidal wave functions. A systematic study of ensemble control systems has immediate applications to systems with parameter uncertainties as well as to broad areas of quantum control systems as arising in coherent spectroscopy and quantum information processing. The new mathematical structures appearing in such problems are an excellent motivation for new developments in control theory.

math.DS↗
arXiv2005

Noncommuting vector fields, polynomial approximations and control of inhomogeneous quantum ensembles

Finding control fields (pulse sequences) that can compensate for the dispersion in the parameters governing the evolution of a quantum system is an important problem in coherent spectroscopy and quantum information processing. The use of composite pulses for compensating dispersion in system dynamics is widely known and applied. In this paper, we make explicit the key aspects of the dynamics that makes such a compensation possible. We highlight the role of Lie algebras and non-commutativity in the design of a compensating pulse sequence. Finally we investigate three common dispersions in NMR spectroscopy, the Larmor dispersion, rf-inhomogeneity and strength of couplings between the spins.

quant-ph↗