We investigate the properties of systems on lattices with spatially distributed sensors and actuators. These systems arise in a variety of applications such as the control of vehicular platoons, formation of unmanned aerial vehicles, arrays of microcantilevers, and satellites in synchronous orbit. We use a Lyapunov-based framework as a tool for stabilization/regulation/asymptotic tracking. We first present results for nominal design and then describe the design of adaptive controllers in the presence of parametric uncertainties. These uncertainties are assumed to be temporally constant, but are allowed to be spatially varying. We show that the design yields decentralized distributed controllers with the passage of information determined by the interactions between different plant units. We also provide several examples and validate derived results using computer simulations of systems containing a large number of units.