Walter Littman

Fourier transforms of surface-carried measures and differentiability of surface averages

This is related to the behavior at oo of the Fourier transform of the measure p. Our main result in that direction is the following: ESTIMATE OF FOURIER TRANSFORMS. Let 5 be a sufficiently smooth compact w-surface (possibly with boundary) embedded in R, JJL a sufficiently smooth mass distribution on S vanishing near the boundary of 5. Suppose that at each point of 5, k of the n principal curvatures are different from zero. Then

Exact boundary controllability of a hybrid system of elasticity

A hybrid control system is presented: an elastic beam, governed by a partial differential equation, linked to a rigid body which is governed by an ordinary differential equation and to which control forces and torques are applied. The entire system, elastic beam plus rigid body, is proved to be exactly controllable by smooth open-loop controllers applied to the rigid body only, and in arbitrarily short durations. This system is modeled as a two-dimensional space-structure.

Blow-up Surfaces for Nonlinear Wave Equations, I

We introduce a systematic procedure for reducing nonlinear wave equations to characteristic problems of Fuchsian type. This reduction is combined with an existence theorem to produce solutions blowing up on a prescribed hypersurface. This first part develops the procedure on the example □u = exp(u); we find necessary and sufficient conditions for the existence of a solution of the form ln(2/⊘2) + v, where {⊘ = 0} is the blow-up surface, and v is analytic. This gives a natural way of continuing solutions after blow-up.

Smoothing evolution equations and boundary control theory

We establish a relationship between the local smoothing properties of evolution equations and boundary control theory. This relationship extends to hyperbolic equations, as well as equations of the Schrödinger type.

Boundary control theory for beams and plates

We present a general method for exact boundary controllability for a class of partial differential (evolution) equations. Particular examples such as the Euler-Bernoulli vibrating beam equation and the vibrating plate equation are mentioned.

Null boundary controllability for semilinear heat equations

We consider null boundary controllability for one-dimensional semilinear heat equations. We obtain null boundary controllability results for semilinear equations when the initial data is bounded continuous and sufficiently small. In this work we also prove a version of the nonlinear Cauchy-Kowalevski theorem.

Decay at infinity of solutions to higher order partial differential equations: Removal of the curvature assumption

In an earlier paper a generalization of Rellich’s theorem on the Helmholz equation was obtained for a large class of higher order equationsP(1/i∂/∂x)u=f, subject to the condition that the Gaussian curvature ofP(ξ)=0 never vanish. This restriction is removed in this note.

Boundary Feedback Stabilization of a Vibrating String with an Interior Point Mass

We study the boundary stabilization of a vibrating string with an interior point mass, zero Dirichlet condition at the left end and velocity feedback at the right end. Assuming finite energy initially, we show that the energy to the right of the point mass decays like C/t while that of the point mass decays like C/√t. The energy to the left of the point mass approaches zero but at no specific rate.

Spectral properties of operators arising in acoustic wave propagation in an ocean of variable depth

AbstractWe prove ann-dimensional version of the following theorem: Letu(x, y) be a solution to