Sufficient conditions for bang-bang control in Hilbert space

Sufficient conditions for bang-bang and singular optimal control are established in the case of linear operator equations with cost functionals which are the sum of linear and quadratic terms, that is, Ax = u , J ( u )=( r,x )+β( x,x ), β>0. For example, if A is a bounded operator with a bounded inverse from a Hilbert space H into itself and the control set U is the unit ball in H , then an optimal control is bang-bang (has norm l) if 0⩽β1/2∥ A −1 * r ∥·∥ A ∥ 2 .Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45225/1/10957_2004_Article_BF00928120.pd

On the validity of the geometrical theory of diffraction by convex cylinders

In this paper we consider the scattering of a wave from an infinite line source by an infinitely long cylinder C. The line source is parallel to the axis of C , and the cross section C of this cylinder is smooth, closed and convex. C is formed by joining a pair of smooth convex arcs to a circle C 0 , one on the illuminated side, and one on the dark side, so that C is circular near the points of diffraction. By a rigorous argument we establish the asymptotic behavior of the field at high frequencies, in a certain portion of the shadow S that is determined by the geometry of C in S. The leading term of our asymptotic expansion is the field predicted by the geometrical theory of diffraction.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46181/1/205_2004_Article_BF00248157.pd