Gravity and the Quantum

The goal of this article is to present a broad perspective on quantum gravity for \emph{non-experts}. After a historical introduction, key physical problems of quantum gravity are illustrated. While there are a number of interesting and insightful approaches to address these issues, over the past two decades sustained progress has primarily occurred in two programs: string theory and loop quantum gravity. The first program is described in Horowitz's contribution while my article will focus on the second. The emphasis is on underlying ideas, conceptual issues and overall status of the program rather than mathematical details and associated technical subtleties.Comment: A general review of quantum gravity addresed non-experts. To appear in the special issue `Space-time Hundred Years Later' of NJP; J.Pullin and R. Price (editors). Typos and an attribution corrected; a clarification added in section 2.

A note on systems with ordinary and impulsive controls

We investigate an everywhere defined notion of solution for control systems whose dynamics depend nonlinearly on the control $u$ and state $x,$ and are affine in the time derivative $\dot u.$ For this reason, the input $u,$ which is allowed to be Lebesgue integrable, is called impulsive, while a second, bounded measurable control $v$ is denominated ordinary. The proposed notion of solution is derived from a topological (non-metric) characterization of a former concept of solution which was given in the case when the drift is $v$-independent. Existence, uniqueness and representation of the solution are studied, and a close analysis of effects of (possibly infinitely many) discontinuities on a null set is performed as well.Comment: Article published in IMA J. Math. Control Infor