In a wide variety of important areas of modern technology from the control of aerospace systems to guiding vehicles to determining drug therapies to producing industrial products it is desired to find the optimal or best way to perform different tasks. The goal is improved efficiency and performance and lower cost. The physical system can be described mathematically. This project will develop numerical algorithms, computer software, and supporting mathematics that will enable scientists and engineers to take these mathematical descriptions and solve a number of optimal control problems that they currently have great difficulty in solving. This will contribute to a number of areas of national need including more efficient industrial processes. The initial areas of application of this research will be chemical processes, aerospace applications, and biomedical problems since those collaborations are already under way. The long term impact is even wider than these areas since the need for increased efficiency and better performance is a driving need throughout science and technology.
Mathematical descriptions often involve differential equations. Frequently there are physical or operational constraints on either the physical process or the means being used to control this process. One of the more successful numerical approaches that is often used for complex optimal control problems is direct transcription where the problem is converted to a discrete problem by various programing techniques, and then iteratively solved on specially chosen refined grids until a high quality solution is found. Direct transcription is a popular technique in industry. Many challenging problems also have delays in either the state dynamics or the controls or both. Delays can occur for several reasons including transmission delays, physical processes, actuator delays, or human operator induced delays. State dependent delays are common in the chemical industry, for example. These delays, to date, have posed challenges that have hindered the development of industrial grade direct transcription algorithms for the computation of the optimal solution of state and control constrained nonlinear systems with delays. This project will address this difficulty. It is proposed to develop algorithms, efficient implementations, and supporting mathematical analysis for the direct transcription solution of state and control constrained optimal control problems with delays. The proposed research will result in new algorithms and new theoretical understanding of numerical methods. These types of algorithms tend to be very sophisticated. Theory backed guidelines will be developed for both users and developers of these types of algorithms. The proposed project will result in substantial advancement on a number of important problems in optimal control, the training of the next generation of researchers to work on these problems, and contributions to several areas of national need. Among the broad impacts are the capability to more widely use constrained mathematical descriptions with delays which will permit easier simulation and optimization of a wide variety of physical systems resulting in reduced time needed for design and improved performance. The implementation of the projects results into an industrial grade code will greatly hasten and broaden their impact across a number of industries.
