One major thrust of current research concerns implicit mathematical models. Depending on the area of application, such systems are called differential algebraic equations (DAEs), singular systems, descriptor systems, semi-state equations, constrained systems, or differential equations on a manifold. Usually the term DAE refers to systems of ordinary differential equations F(x', x, t)=0 with the Jacobian of F with respect to x' being singular. Many physical systems are most naturally and easily first modeled as an implicit system. This is often the case with computer generated models. A goal of our research is to be able to carry out analysis and simulation from the original implicit description thus reducing the time between formulation and numerical results. An implicit formulation also allows one to consider more complex phenomena and to explore different designs or parameter values from one model. We are interested in mathematical theory, numerical analysis and design of algorithms, and application.

Control applications are of special interest. This research currently includes both numerical approaches to optimal control problems and also problems in failure detection and failure identification

Among the topics being investigated are

- The mathematical analysis of nonlinear DAEs.

- All aspects of the development and analysis of numerical
algorithms, especially for higher index, fully implicit
nonlinear DAEs. Of particular interest is the development of
algorithms which can be included in design and simulation
packages both for simulation and for model diagnostics. These
algorithms are both to increase the robustness of the
software and to assist users.

- Control theory, especially for implicitly modeled systems.

- The incorporation of DAE integrators into nonlinear control
algorithms. One example is the design of observers for complex
nonlinear systems. Another is using the DAE integrator to
numerically compute a stabilizing nonlinear feedback.

- Simulation of physical systems including how to exploit
problem structure. In the past, particular attention has been
given to constrained mechanical systems and nonlinear
circuits. The results are also of interest in areas such as
chemical process control but to date that classes of
applications has not been carefully considered by our group.

- Numerical algorithms for various questions in nonlinear
control. Of particular interest has been prescribed path
control and optimal control.

- Failure detection and failure identification. In particular, we have been investigating the use of auxiliary signals to be used either when controller action masks the onset of system deterioration or when there has been a failure of the monitoring system itself but it is still necessary to continue operations.

There are always collaborations in progress with groups at other institutions. All of these collaborations have several different aspects not all of which are given here. Current collaborations are:

- Groups of Peter Kunkel at University Leipzig and Volker Mehrmann at Technische Universitaet Berlin. This research focuses on higher index DAEs, their numerical solutions, and applications to control.

- Applied Mathematical Analysis. This project includes the optimization and simulation of large complex DAE modeled systems with applications to manufacturing. In particular, we are investigating direct transcription methods and their use on problems with state and control delays. This work is part of their SOS (Sparse Optimization Suite).