David L. Russell

Exact controllability and stabilizability of the Korteweg-de Vries equation

In this paper, we consider distributed control of the system described by the Korteweg-de Vries equation (i) atu + uaxu + e3 u = f on the interval 0 0, with periodic boundary conditions (ii) &au(27r, t) = &ku(O, t), k = 0, 1, 2, where the distributed control f _ f (x, t) is restricted so that the volume f u(x, t)dx of the solution is conserved. Both exact controllability and stabilizibility questions are studied for the system. In the case of open loop control, if the control f is allowed to act on the whole spatial domain (0, 27r), it is shown that the system is globally exactly controllable, i.e., for given T > 0 and functions q(x), +(x) with the same volume, one can alway find a control f so that the system (i)-(ii) has a solution u(x, t) satisfying u(x, 0) 0(x), u(x, T) (x). If the control f is allowed to act on only a small subset of the domain (0, 27r), then the same result still holds if the initial and terminal states, b and 0, have small amplitude in a certain sense. In the case of closed loop control, the distributed control f is assumed to be generated by a linear feedback law conserving the volume while monotonically reducing f u(x, t)2dx. The solutions of the resulting closed loop system are shown to have uniform exponential decay to a constant state. As in the open loop control case, a small amplitude assumption is needed if the control is allowed to act on only a small subdomain. The smoothing property of the periodic (linear) KdV equation discovered recently by Bourgain has played an important role in establishing the exact controllability and stabilizability results presented in this paper.

Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain

In this paper, solutions of the third-order linear dispersion equations [frac{{partial w}}{{partial t}} + frac{{partial ^3 w}}{{partial x^3 }} = f(x,t)qquad {text{and}}qquad frac{{partial w}}{{partial t}} + frac{{partial ^3 w}}{{partial x^3 }} = 0] are studied for

Control and optimal design of distributed parameter systems

t geqq 0

Exponential bases in Sobolev spaces in control and observation problems for the wave equation

,

Stabilization of the Korteweg-de Vries Equation on a Periodic Domain

0 leqq x leqq 2pi

AN ELEMENTARY NONLINEAR BEAM THEORY WITH FINITE BUCKLING DEFORMATION PROPERTIES

. In the first case, periodic boundary conditions are imposed at 0 and

Formulation and validation of dynamical models for narrow plate motion

2pi

An Extended Beam Theory for Smart Materials Applications Part I: Extended Beam Models, Duality Theory, and Finite Element Simulations ⁄

and the distributed control f, which may, however, have support smaller than

The Betti reciprocity principle and the normal boundary component control problem for linear elastic systems

[0,2pi ]

Exponential Bases in Sobolev Spaces in Control and Observation Problems

, is assumed to be generated by a linear feedback law conserving the “volume”