Two types of control problems are considered where the information available for controller design is limited. In one type of problem, output data is available only at discrete times, so sampled-data control is used instead of continuous-time control. Projects include how to design sampled-data controllers for partial differential equations; an analysis of which systems can be stabilized by a simple sampled-data controller; how to modify continuous-time feedback controllers so that they are effective sampled-data controllers; an analysis of how the performance of a sampled-data controller compares to that of related continuous time controllers; and the design of sampled-data controllers which track an external reference signals, even when very little information about the plant is available. In these projects, the interest lies not only in what can be done with sampled-data control design, but also its limitations. Since a discrete time controller can operate with only limited frequency response, and infinite-dimensional systems often have high frequency effects which cannot be ignored, sampled-data and discrete-time control of infinite-dimensional systems is very delicate. In addition, population dynamics problems modeled by infinite-dimensional discrete-time systems, such as integro-difference equations, are considered. Most such problems in population ecology have highly uncertain data, and we will analyze how growth or decay is affected by the data uncertainties.
The development of sampled-data control is motivated in part by advances in digital electronics, which has led increasingly to sampled-data design and implementation of control algorithms. Furthermore, in many practical applications output data from a system is often only available at discrete times rather than in continuous time, so it is of practical value to know how to deal with such data. The proposed research will address basic questions about the capabilities of sampled-data control design for infinite-dimensional systems. Conservation problems in population ecology typically have very uncertain data, and the growth or decay of an endangered or invasive species can be dramatically affected by this uncertainty. Infinite-dimensional models are used to model spatial spread of a population or in cases where the growth stages are continuously distributed. This research will develop new techniques, and use techniques from robust control theory, to analyze the effect of uncertain data on long-term population growth. These techniques will be available and accessible for use on similar population problems. Undergraduate and graduate researchers will be incorporated into this research program.
