This proposal introduces a novel framework for the analysis and control of nonlinear systems exhibiting complex dynamics. The underlying idea of the framework is to study the evolution of sets in the phase space as opposed to the conventional point-wise approach. The advantage of the approach is that the evolution of sets is linear and is described by a linear transfer Perron-Frobenius operator. The linear nature of the framework allows us to carry our intuition from linear systems, a mature area of research, to nonlinear systems. A perfect example is the introduction of Lyapunov measure as a new tool to verify a set-theoretic notion of ?almost everywhere? stability in nonlinear systems. Lyapunov measure is shown to be dual to the Lyapunov function and a linear program using the finite dimensional approximation of the Perron-Frobenius operator is proposed for its computation. The proposed research exploits the linearity of the framework for the model order reduction and parameter identification in systems exhibiting complex dynamics with applications in jet engines. The framework also provides a systematic procedure based on the linear programming for the control of complex dynamics. The proposed research uses this procedure for the problem of control of mixing in fluid flows with potential application in analyzing the transport and mixing properties of large-scale oceanographic and atmospheric flows.
The success of this research will result in a new set of tools and methods that are general enough to be applied in diverse fields, such as fluid dynamics and mechanical systems. The reduced order model in jet engines will be used in better control design, which ultimately results in less fuel consumption and increased life of engines. The improved control algorithm for mixing will benefit the pharmaceutical, petrochemical, transportation, and power generation industries among others. The PI?s collaboration with United Technologies Research Center will help in transferring the know-how from industry to academia, and vice versa. As the part of the education plan, the PI will introduce interdisciplinary courses on nonlinear dynamics at the undergraduate and graduate levels. By serving as a faculty mentor for the special interest group in robotics, the PI will help increase the involvement of undergraduate students in research.
