Simulation, Optimization, and Analysis of High Dimensional Higher Index DAEs

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Optimization is used in everything from the control of spacecraft to optimizing chemical processes. Optimization usually takes place in the presence of equality and inequality constraints. Differential algebraic equations (DAEs) are implicit systems of differential equations. Usually it is also assumed that the Jacobian with respect to the highest derivative is singular. An integer quantity called the index is one measure of how different a DAE is from being an explicit ordinary differential equation. Many problems are most naturally initially modeled as a DAE particularly those that are analyzed and simulated using computer generated mathematical models. DAEs occur in optimization because the original problem is a DAE, because of the activation of constraints, and when partial differential equations are numerically solved using the method of lines. Many problems also involve delays either because delays are inherently present in the process or because of delays in control or feedback loops. Direct transcription methods are a popular approach for the solution of optimal control problems. Recently it has been shown that the usual theory of DAE integrators needs to be substantially modified when DAEs occur during the numerical solution by direct transcription methods of inequality constrained optimal control problems. The primary focus of this proposal is the numerical solution of inequality constrained optimal control problems including those with delays using direct transcription methods. Theory will be developed, the theory will be applied to numerical algorithms and their implementation in production codes, and the algorithms will be applied to a number of applications. The interaction between optimization, delays, direct transcription, numerical discretizations, and state inequality and equality constraints is far from fully understood. This proposal will carefully examine this interaction. In addition, more general DAE simulation algorithms will be developed. Designing controls and process behavior which are optimal in some sense is fundamental to designing efficient and effective processes. The optimization can include minimizing energy consumption, producing a part in minimum time, maximizing part quality or maximizing product production. In general, the availability of robust DAE simulation, analysis, and optimization software will result in more accurate models and predictions of performance, which in turn translates to more efficient processes and their more efficient operation resulting in better performance. The particular theoretical and computational questions to be examined are motivated by, and will be applied to, applications arising at the Boeing Corporation. The specific applications each of which may have delays, are real time control, chemical processes, and aerospace vehicles. However, the results will be publicly distributed and will find use in a variety of software packages and in a number of similar applications that affect a wide range of industries. High performance and efficient use of complex processes is a basic need in modern engineering and science. The training of young scientists in the areas of this project and the establishment of both industrial and international aspects to their research will be a significant contribution of this research project for years to come.