Weijiu Liu

BOUNDARY FEEDBACK STABILIZATION OF AN UNSTABLE HEAT EQUATION

In this paper we study the problem of boundary feedback stabilization for the unstable heat equation ut(x,t) = uxx(x,t)+a(x) u(x,t). This equation can be viewed as a model of a heat conducting rod in which not only is the heat being diffused (mathematically due to the diffusive term uxx) but also the destabilizing heat is generating (mathematically due to the term a u with a >0). We show that for any given continuously differentiable function a and any given positive constant

Boundary control of an unstable heat equation via measurement of domain-averaged temperature

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Backstepping boundary control of Burgers’ equation with actuator dynamics ☆

we can explicitly construct a boundary feedback control law such that the solution of the equation with the control law converges to zero exponentially at the rate of

Modeling a simplified regulatory system of blood glucose at molecular levels

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The exponential stability of the problem of transmission of the wave equation

. This is a continuation of the recent work of Boskovic, Krstic, and Liu [IEEE Trans. Automat. Control, 46 (2001), pp. 2022--2028] and Balogh and Krstic [European J. Control, 8 (2002), pp. 165--176].

A molecular mathematical model of glucose mobilization and uptake.

In this note, a feedback boundary controller for an unstable heat equation is designed. The equation can be viewed as a model of a thin rod with not only the heat loss to a surrounding medium (stabilizing) but also the heat generation inside the rod (destabilizing). The heat generation adds a destabilizing linear term on the right-hand side of the equation. The boundary control law designed is in the form of an integral operator with a known, continuous kernel function but can be interpreted as a backstepping control law. This interpretation provides a Lyapunov function for proving stability of the system. The control is applied by insulating one end of the rod and applying either Dirichlet or Neumann boundary actuation on the other.

Closed-loop separation control: An analytic approach

In this paper, we propose a backstepping boundary control law for Burgers’ equation with actuator dynamics. While the control law without actuator dynamics depends only on the signals u(0;t) and u(1;t), the backstepping control also depends on ux(0;t), ux(1;t), uxx(0;t) and uxx(1;t), making the regularity of the control inputs the key technical issue of the paper. With elaborate Lyapunov analysis, we prove that all these signals are suciently regular and the closed-loop system, including the boundary dynamics, is globally H 3 stable and well posed. c 2000 Elsevier Science B.V. All rights reserved.

Designing dynamical output feedback controllers for store-operated Ca2+ entry

In this paper, we propose a new mathematical control system for a simplified regulatory system of blood glucose by taking into account the dynamics of glucose and glycogen in liver and the dynamics of insulin and glucagon receptors at the molecular level. Numerical simulations show that the proposed feedback control system agrees approximately with published experimental data. Sensitivity analysis predicts that feedback control gains of insulin receptors and glucagon receptors are robust. Using the model, we develop a new formula to compute the insulin sensitivity. The formula shows that the insulin sensitivity depends on various parameters that determine the insulin influence on insulin-dependent glucose utilization and reflect the efficiency of binding of insulin to its receptors. Using Lyapunov indirect method, we prove that the new control system is input-output stable. The stability result provides theoretical evidence for the phenomenon that the blood glucose fluctuates within a narrow range in response to the exogenous glucose input from food. We also show that the regulatory system is controllable and observable. These structural system properties could explain why the glucose level can be regulated.

Stabilization and controllability for the transmission wave equation

The problem of exponential stability of the problem of transmission of the wave equation with lower-order terms is considered. Making use of the classical energy method and multiplier technique, we prove that this problem of transmission is exponentially stable.

Mixing Enhancement by Optimal Flow Advection

A new molecular mathematical model is developed by considering the kinetics of GLUT2, GLUT3, and GLUT4, the process of glucose mobilization by glycogen phosphorylase and glycogen synthase in liver, and the dynamics of the insulin signaling pathway. The new model can qualitatively reproduce the experimental glucose and insulin data. It also enables us to use the Bendixson criterion about the existence of periodic orbits of a two-dimensional dynamical system to mathematically predict that the oscillations of glucose and insulin are not caused by liver, instead they would be caused by the mechanism of insulin secretion from pancreatic beta cells. Furthermore it enables us to conduct a parametric sensitivity analysis. The analysis shows that both glucose and insulin are most sensitive to the rate constant for conversion of PI(3,4,5)P(3) to PI(4,5)P(2), the multiplicative factor modulating the rate constant for conversion of PI(3,4,5)P(3) to PI(4,5)P(2), the multiplicative factor that modulates insulin receptor dephosphorylation rate, and the maximum velocity of GLUT4. Moreover, the sensitivity analysis predicts that an increase of the apparent velocity of GLUT4, a combination of elevated mobilization rate of GLUT4 to the plasma membrane and an extended duration of GLUT4 on the plasma membrane, will result in a decrease in the needs of plasma insulin. On the other hand, an increase of the GLUT4 internalization rate results in an elevated demand of insulin to stimulate the mobilization of GLUT4 from the intracellular store to the plasma membrane.