Scott W. Hansen

Exponential energy decay in a linear thermoelastic rod

Abstract We examine the stability of solutions to a pair of coupled linear partial differential equations which describe the temperature distribution and displacement within a one-dimensional thermoelastic rod. For particular sets of natural boundary conditions, the eigenfunctions are shown to form a Riesz basis on the Hilbert space of finite energy states. For these cases, asymptotic eigenvalue estimates show that the spectrum is to the left of some vertical line Re λ = −β contained in the open left half of the complex plane. An energy decay estimate of the form E(t) ⩽ Me−βt E(0) is then obtained.

Boundary Control of a One-Dimensional Linear Thermoelastic Rod

Boundary control of a linear partial differential equation that describes the temperature distribution and displacement within a one-dimensional thermoelastic rod is examined. In particular, it is shown that temperature or heat flux control at an endpoint is sufficient to obtain exact null-controllability. This improves earlier results for similar systems in which only partial null-controllability is obtained. Sharp regularity results for the controlled system are also obtained.

SEVERAL RELATED MODELS FOR MULTILAYER SANDWICH PLATES

Mathematical models for multilayer sandwich plates consisting of alternating stiff and compliant layers are derived. Two main types of models are described. First an initial model (analogous to the three-layer Rao–Nakra model) is derived under Kirchhoff plate assumptions for the stiff layers and Mindlin shear-deformable displacement assumptions for the compliant layers. The second type of model can be obtained from the original model by dropping the in-plane and rotational inertia. The resulting model is a generalization of the well-known model of Mead and Markus. Well-posedness and continuous parameter dependence results are described. Some variations of the initial model corresponding to thin compliant layers are described and shown to be regular perturbations of the initial model.

ANALYTICITY, HYPERBOLICITY AND UNIFORM STABILITY OF SEMIGROUPS ARISING IN MODELS OF COMPOSITE BEAMS

We examine the stability properties of a sandwich beam consisting of two outer layers and a thin core. The outer layers are modeled as Euler Bernoulli beams and the inner core provides both elastic and viscous resistance to shearing. We show for both clamped and hinged boundary conditions that (i) if rotational inertia terms are neglected, the model is described by an analytic semigroup, and (ii) if rotational inertia is retained in the outer layers, the model is uniformly exponentially stable.

Exact Boundary Controllability Results for a Multilayer Rao--Nakra Sandwich Beam

We study the boundary controllability problem for a multilayer Rao--Nakra sandwich beam. This beam model consists of a Rayleigh beam coupled with a number of wave equations. We consider all combinations of clamped and hinged boundary conditions with the control applied to either the moment or the rotation angle at an end of the beam. We prove that exact controllability holds provided the damping parameter is sufficiently small. In the undamped case, exact controllability holds without any restriction on the parameters in the system. In each case, optimal control time is obtained in the space of optimal regularity for

Exact controllability of a Rayleigh beam with a single boundary control

L^2(0,T)

Analysis of a plate with a localized piezoelectric patch

controls. A key step in the proof of our main result is the proof of uniqueness of the zero solution of the eigensystem with the homogeneous boundary conditions together with zero boundary observation.

Simultaneous boundary control of a Rao-Nakra sandwich beam

We prove exact boundary controllability for the Rayleigh beam equation

Null boundary controllability of a 1-dimensional heat equation with an internal point mass

{\varphi_{tt} -\alpha\varphi_{ttxx} + A\varphi_{xxxx} = 0, 0 < x < l, t > 0}