Two sample applications of a classical isoperimetric problem of the calculus of variations to fluid mechanical problems in fluidization and spouting

This paper is devoted to outlining precisely the basic mathematics of a classical isoperimetric problem of the calculus of variations and showing how significant fluid mechanical problems in fluidization and spouting can be addressed using this approach.     

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.     

Stark mixing spectroscopy in cesium

We present a new technique for selectively populating excited states which are inaccessible by dipole excitation from the ground state. The method uses a static electric field to introduce a component of a dipole-allowed state into the state of interest. We have applied the method to cesium to measure lifetimes and a Stark mixing coefficient. The results are $\text{τ(6}{}^2\text{D}{}_{\mathrm{<mtext>5</mtext><mtext>2</mtext>}}\text{)=64(2) ns}$, $\text{τ(7}{}^2\text{D}{}_{\mathrm{<mtext>5</mtext><mtext>2</mtext>}}\text{)=92.5(15) ns}$, and <6²D_{$\text{5}\text{2}$}|;ez |7²P_{$\text{3}\text{2}$}>/(E_7P?E_6D)=0.7(3)×10^?3 where is in kV/cm. 141     

Boundary Control of PDEs via Curvature Flows: the View from the Boundary, II

We describe some results on the exact boundary controllability of the wave equation on an orientable two-dimensional Riemannian manifold with nonempty boundary. If the boundary has positive geodesic curvature, we show that the problem is controllable in finite time if (and only if) there are no closed geodesics in the interior of the manifold. This is done by solving a parabolic problem to construct a convex function. We exhibit an example for which control from a subset of the boundary is possible, but cannot be proved by means of convex functions. We also describe a numerical implementation of this method.     

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.This research was partially supported by NSF grant DMS 84-13129, and Markus also received support from SERC.     

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.     

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.W. Littman was partially supported by NSF Grant DMS 90-02919. The results of this paper were presented by Yung-Jen Lin Guo at the P.D.E. seminar at the University of Minnesota on January 27, 1993 and by W. Littman at the First International Conference on Dynamics Systems and Applications held in Atlanta in May 1993.     

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

We prove ann-dimensional version of the following theorem: Letu(x, y) be a solution to $$c^2 (y)\rho (y)\left( {\frac{1}{{\rho (y)}}u_y } \right)_y + c^2 (y)u_{xx} + k^2 u = 0 (k > 0)$$ in Ω≡{y>0}?B, continuous in \(\bar \Omega \) ,B being a disc centered at the origin, andρ(y) andc(y) being strictly positive functions constant outside of a bounded set,C ⁽²⁾ except for a finite number of jumps. Then ifu(x,0)→0 exponentially as |x|→∞ andu∈L ²(Ω),u≡0 in Ω.