Optimization and Simulation of Constrained Dynamical Systems

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Stephen L. Campbell will perform research on optimization, numerical methods for constrained differential equations, and constrained optimal control. This research will result in improved algorithms and new theoretical understanding of numerical methods for differential algebraic equations (DAEs) and the solution of inequality constrained optimal control problems by direct transcription approaches. More specifically, a DAE is an implicit differential equation. 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 when 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. Because of the importance of DAEs in applications, a variety of numerical techniques have been developed in recent years to simulate and analyze DAEs although most of these methods only work on specific classes of problems with special structure and low index. The development of more general DAE integrators is one focus of this proposal. 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. A second primary focus of this proposal is the numerical solution of inequality constrained optimal control problems using direct transcription methods. Throughout industry there is a need for the development of more efficient processes and systems. This is essential for improved performance, conserving of resources, and industrial and military competitiveness. This increased performance requires working with complex mathematical models of the systems in question. In many areas it is becoming necessary to utilize larger, more complex, and more complete models of the physical process or system. Two important parts of systems design are simulation and optimization. In applications there are many constraints. These constraints range from mission constraints (such as flying within certain flight corridors or intercepting specific targets) to physical constraints (such as contact between a robotic arm and a tool) to financial or cost constraints. These constraints pose both theoretical and computational challenges which increase with the model's complexity. This project is to develop both mathematical theory and numerical algorithms for the numerical simulation and optimal control of complex systems with constraints. The proposer has ongoing collaborations with developers of software used in a number of industries including the aerospace industry. The results of this research will both be incorporated into production computer codes as part of collaborations of the Principal Investigator and also shared with others developing numerical software. These algorithms and software will have very wide applicability. The current collaborations involve the aerospace industry including space and aircraft mission planning, robotic manufacturing, and optimization of chemical processes. In addition to the significant research to be accomplished, graduate students will play a fundamental role in this research. The training of the next generation of researchers in this area is another important contribution of this project. As with prior students of the Principle Investigator, the students trained on this project will move on to work in Industry, National Laboratories and Research Centers, the Armed Forces, and Academia.