Chapter 1 Stabilization and Control over Gaussian Networks Ali A. Zaidi, Tobias J. Oechtering, Serdar Y¨ uksel and Mikael Skoglund Abstract We provide an overview and some recent results on real-time commu- nication and control over Gaussian channels. In particular, the problem of remote stabilization of linear systems driven by Gaussian noise over Gaussian relay chan- nels is considered. Necessary and sufficient conditions for mean-square stabiliza- tion are presented, which reveal signal-to-noise ratio requirements for stabilization which are tight in certain class of settings. Optimal linear policies are constructed, global optimality and sub-optimality of such policies are investigated in a variety of settings. We also consider the design of low-delay sensing and transmit schemes for real-time communication. 1.1 Introduction In this chapter we consider a setup where a linear time invariant system (plant) with a random initial state and driven by Gaussian noise has to be remotely stabilized. A group of sensor nodes monitor the plant and communicate their observations (mea- surements) to a remotely situated control unit over wireless links, that are modeled as additive white Gaussian channels. The common goal of the sensors and the con- troller is to stabilize the plant in closed-loop. Usually in remote control applica- tions, sensing and transmission under strict delay and power constraints is required. Ali A. Zaidi KTH Royal Institute of Technology, Stockholm, Sweden, e-mail: zaidi@kth.se Tobias J. Oechtering KTH Royal Institute of Technology, Stockholm, Sweden, e-mail: oech@kth.se Serdar Y¨ uksel Queen’s University, Kingston, Canada, e-mail: yuksel@mast.queensu.ca Mikael Skoglund KTH Royal Institute of Technology, Stockholm, Sweden, e-mail: skoglund@kth.se 1