Algorithms and software for the hamilton-jacobi equation and applications in control, robotics and verification

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I propose development of a Hamilton-Jacobi Workshop (HJW), which is a collection of algorithms and implementations for approximating the solution of both the time dependent and stationary versions of the Hamilton-Jacobi (HJ) partial differential equation (PDE). This equation shows up in such diverse fields as fluid and combustion simulation, graphics and animation, image processing and computer vision, financial mathematics, robotics and optimal control, and verification; for this reason it has received considerable research attention. My research agenda is distinct from and complementary to the study of this equation undertaken by numerical analysts. While that community has developed powerful, efficient and accurate numerical methods, researchers and practitioners from other fields find it difficult to choose among and implement the various schemes, and students new to HJ PDEs must replicate an enormous software infrastructure to demonstrate their own contributions. Therefore, my research will focus on developing easy to use and extensible implementations at the cutting edge of accuracy and power, yet factorized in such a manner as to provide graceful flexibility for trying various alternative schemes. Where necessary the code may not achieve optimal execution efficiency, but with access to all of the source code users can tune or reimplement once they have determined the best schemes for their particular application. Such a collection of techniques cannot be developed in isolation, and this research will be informed by feedback from HJW users and by personal study in application fields, such as provably safe control synthesis and filtering for vehicles, path planning under dynamic motion constraints and multiple objectives, and automated tools for reachability in continuous and hybrid systems.