COVID-19 Advisory: In abidance with Mayor Garcetti’s “Safer at Home” emergency order, IPAM will hold all workshops that are part of our current program (High Dimensional Hamilton-Jacobi PDEs), including High Dimensional Hamilton-Jacobi Methods in Control and Differential Games, via zoom. Workshop registrants will receive the Zoom link a few days prior to the workshop, along with instructions on how to participate. The video of the recorded sessions will be made available on IPAM website.
Workshop Overview: Solving Hamilton-Jacobi-Bellman equation is central to problems in optimal control, differential games, path planning and formal verification of reachability sets. In practical problems with nonlinear dynamics, subject to unpredictable disturbances with high-dimensional state spaces, solution of the Hamilton-Jacobi-Bellman equation becomes computationally intractable as the state space dimension increases.
As theoretical developments have advanced, computational challenges remain. For example, Hamilton-Jacobi reachability analysis is a verification method used to guarantee performance and safety properties of systems. Traditional reachable set computations involve solving an Hamilton-Jacobi partial differential equation on a discretized state space grid, which results in an exponential scaling of computational complexity with respect to system dimensionality.
Recently proposed techniques include decomposing the computation of the reachable set into several small dimensional computations; using convex optimization applied to the Hopf-Lax formula in conditions of state and time independence, generalizations of Hopf-Lax to more complicated Hamiltonians, and applying newly developed numerical methods to convert continuous optimal control problems into nonlinear programming problems, which can then be solved using advanced nonlinear programming solvers. The choice of problem structure, conversion technique, and underlying solver makes all the difference in the quality and efficiency of the method. In particular, pseudospectral methods have proven to be extremely powerful.
Ideas to be explored in the workshop include different computational methodologies for efficient real-time solution of nonlinear HJ equations in reachability, optimal control, and differential games. Ideas such as efficient reduced-complexity computation of optimal control solutions, exploiting structure to decompose the solution space to scalable computations and reduced-complexity feedback structures for efficient implementation of optimal controllers based on available data will be considered.
This workshop will focus on how such new methods may be used to broaden the classes of control and differential game problems that may be treated.
Fariba Fahroo (Air Force Office of Scientific Research (AFOSR), Computational Math)
Stanley Osher (University of California, Los Angeles (UCLA), Institute for Pure and Applied Mathematics (IPAM))
Claire Tomlin (University of California, Berkeley (UC Berkeley), Computer Science Engineering)