Small Grant for Exploratory Research

Harvard University, Cambridge MA

Investigators

Abstract

This proposal describes a new more comprehensive approach to the design of feedback controls for a wide variety of systems. The basic idea is to replace the usual differential equation description of the system being controlled by a corresponding Louiville equation and to formulate an optimization problem that takes into account a range of initial conditions and the way in which the control depends on time and state. The mathematical problems that arise do not seem to have been explored up until now and so an important intellectual challenge to be overcome in this research will be to develop methods for solving the type of variational problems involving partial differential equations that arise in these problems. The potential for this work to benefit society rests on two main lines of thought. First of all, control systems are now found in almost all durable goods, including dishwashers, heaters, toasters, automobiles, etc. and are becoming common in building management involving lighting, heating and security. The use of conventional optimal control often defines a control law that is too expensive to implement and so that ad hoc schemes are used that yield a performance level that is not all that it could be. The ideas discussed here are directed toward developing a methodology for finding less expensive, higher performance solutions. Secondly, although biological systems are replete with feedback control mechanisms, conventional control theory has had only limited success in predicting and explaining the structures identified so far. Arguing that living systems, through evolution, are attempting to optimize a trade-off between the performance of the system and its complexity, we make the case that better success will be had in explaining biological phenomena using theories of the type proposed here. The proposer has a long history of educating students at both the graduate and undergraduate level. For example, this spring one of his undergraduate students received one of only three ``Fay" prizes given to graduating seniors for research work. The student designed and fabricated a VLSI chip implementing a very sophisticated motor controller. This is the second undergraduate student of his in the last two years that has received University wide recognition for undergraduate research. The funds requested are absolutely essential if he is to make these kinds of opportunities available to tomorrow's students, both at the graduate and undergraduate level.