Ensemble Control

State-of-the-art quantum technology can trap and experiment with individual atoms, image brains as well as generate the structural and dynamical information of biological macromolecules. Numerous applications arising from such emerging techniques involve controlling a large quantum ensemble, e.g., on the order of Avogadro number, by use of the same control field. In many cases, the elements of the ensemble show variations in the values of the parameters characterizing the system dynamics. For example, in magnetic resonance experiments, nuclear spins of an ensemble may have dispersion in their natural frequencies, so that the spins with different frequencies have distinct dynamics. A canonical problem among these applications is to develop excitations (control signals) that can steer such an ensemble of systems with different dynamics from an initial state to a desired final state. From the perspective of mathematical control theory, the challenge is to simultaneously steer a continuum of systems between points of interest with the same control signal. We define such a class of problems Ensemble Control. This new subject raises some unexplored and challenging questions including controllability and optimal control of such systems. We are pursuing a fundamental investigation for general ensemble control systems.

Optimal Pulse Sequence for MRI

Molecular imaging with paramagnetic chemical exchange saturation transfer (PARACEST) nanoparticles has emerging clinical applications to early detection and noninvasive monitoring of cancer and cardiovascular disease. PARACEST relies on the chemical exchange of nuclei between two or more different environments or pools, e.g., between bulk water and bound water. The application of presaturation pulses can saturate the exchangeable protons, and the image contrast can be turned on and off at will by gating the pulse sequence parameters. Optimal presaturation pulse design can be formulated as a problem of optimal control of dissipative bilinear system. We employ pseudospectral methods to solve this problem, by which the continuous time optimal control problem is converted to a finite dimensional nonlinear programming using Legendre polynomial approximations. We will then study the convergence of this method. This is an ongoing project with the Center for Translational Research in Advanced Imaging and Nanomedicine (C-TRAIN) at Washington University.

Optimization and Adaptive Design of Intensity Modulated Radiation Therapy

Intensity Modulated Radiation Therapy (IMRT) is a technique used to deliver precise radiation doses to a targeted tumor while avoiding other structures. Mathematically, the IMRT forms a large scale optimization problem, which involves a large number of variables, attempting to maximize the radiation dose given to a tumor while minimizing radiation to other healthy and important tissue. Consequently, it requires significant computational power. Even more challenging, tumors can change in size, shape, and position throughout the treatment process. To effectively reduce the damage to healthy tissue and to achieve the desired dose for the tumor, the optimization must be refined before each session. This can be done by consistently repeating the entire optimization procedure based on the new geometry and motion of the tumors. However, this method is computationally expensive and does not adapt previous optimized information. We view the IMRT planning as an adaptive control design and solve it using the dynamical system approach. This is an ongoing research project with the Division of Bioinformatics and Outcomes Research at Washington University.

DNA Computation

The Central Dogma of molecular biology is a framework for understanding the transfer of sequence information between sequential information-carrying biopolymers in living organisms. To date, the description of this process is based on conventional molecular biology through a character-based model. Several mathematical scenarios have been proposed for biological computation which convert character-based models into numerical computation problems. However, there so far exist no unified mathematical models to describe the entire process of the central dogma.

We develop a mathematical representation of the Central Dogma of molecular biology by which genetic information flows from DNA to RNA and proteins are mathematically expressed. We employ matrix theory for the representation of key concepts related to DNA hybridization, RNA self-hybridization, and polarities of the amino acids. Group theory is used to represent concepts and properties of gene mutation, protein syntheses, and protein secondary structures. We are constructing a mathematical foundation for biology and DNA computation by introducing systems and control theory concepts.

Dynamic Pricing

We study dynamic pricing and inventory control of substitute products for a retailer who faces a long supply lead time and a short selling season. Within a multinomial logit model of consumer choice over substitutes, we can develop a stochastic intensity control formulation and derive the optimal dynamic pricing policy. Extensive numerical study of the effects of time and inventory depletion on the optimal pricing policy reveals two fundamental underlying driving forces of the complex price behavior: the level of inventory scarcity and the quality difference among products. Works have been done for a homogeneous customer arrival rate. Now, we want to apply this model to an environment where the customer arrival rate is non-homogenous in time.

Nonlinear Dynamics and Feedback

Nonlinear phenomenon is ubiquitous in both physical and engineered systems such as structures and climate dynamics. We are currently collaborating with structural engineers on nonlinear control of earthquake damping systems. Magnetorheological (MR) dampers are semiactive control devices used to dampen and mitigate the vibrations in buildings during earthquakes. Models for both the building and damper dynamics are cascaded to develop a control system, where the voltage input to the MR damper is the control that is designed to stabilize the combined system. Systems with linear building and linear MR damper models have been studied in the past. To advance this research, we propose to work on the system with a more accurate, nonlinear model for the MR damper dynamics. The ideas of feedback linearization are implemented to remove the nonlinearity with a compensating control. We will then study the stability and design a robust control for the linearized system to reduce the disturbance caused by an earthquake.

We are also interested in global climate change research. Decades of research into the phenomenon of global climate change have brought us step by step closer to piecing together a fully integrated Earth model. Numerous research groups have spent years refining models that characterize the components of the climate system, from carbon cycling in the ocean to atmospheric dynamics. We propose to investigate the carbon/climate system which inherits a "feedback loop" that the carbon in the atmosphere leads to a warming trend and then increased temperatures result in less absorption by carbon reservoirs further increasing the amount of carbon in the atmosphere. In collaboration with the Sustainable Development Laboratory at National Taiwan University, we look to analyze this environmental system as driven by an economic input. Policy changes and requirements will act as shaping control variables. We will also characterize the fundamental behavior, e.g. stability, of the system using a simplified carbon-climate model, which will motivate the use of systems science for the study of climate change. This is an ongoing project supported by the McDonnell Academy.