Analysis and control of a scalar conservation law modeling a highly re-entrant manufacturing system

In this paper, we study a scalar conservation law that models a highly re-entrant manufacturing system as encountered in semi-conductor production. As a generalization of \cite{CKWang}, the velocity function possesses both the local and nonlocal character. We prove the existence and uniqueness of the weak solution to the Cauchy problem with initial and boundary data in $L^{\infty}$. We also obtain the stability (continuous dependence) of both the solution and the out-flux with respect to the initial and boundary data. Finally, we prove the existence of an optimal control that minimizes, in the $L^p$-sense with $p\in [1,\infty)$, the difference between the actual out-flux and a forecast demand over a fixed time period.Comment: 32 pages, 12 figure

Exact boundary observability for nonautonomous quasilinear wave equations

By means of a direct and constructive method based on the theory of semiglobal $C^2$ solution, the local exact boundary observability is shown for nonautonomous 1-D quasilinear wave equations. The essential difference between nonautonomous wave equations and autonomous ones is also revealed.Comment: 18 pages, 5 figure

Phase Retrieval for Sparse Signals

The aim of this paper is to build up the theoretical framework for the recovery of sparse signals from the magnitude of the measurement. We first investigate the minimal number of measurements for the success of the recovery of sparse signals without the phase information. We completely settle the minimality question for the real case and give a lower bound for the complex case. We then study the recovery performance of the $\ell_1$ minimization. In particular, we present the null space property which, to our knowledge, is the first sufficient and necessary condition for the success of $\ell_1$ minimization for $k$-sparse phase retrievable.Comment: 14 page