My work is in the area of numerical solution of optimal control problems for general, low-dimensional nonlinear systems, with particular application to guidance in coastal ocean flows.
The energy efficiency of buoyancy-driven ocean gliders makes them very well-suited for gathering oceanographic data on missions lasting months at a time and traversing hundreds of kilometers. But in the presence of ocean currents inherent limitations on maximum speed make even simple tasks like station-keeping difficult. Our goal is to develop a framework for high-level planning of both long missions and simple behaviors like station-keeping under constraints on maximum speed, energy, time, and communication, which utilizes the sophisticated ocean current data that is available.
Our main approach has been an optimal control approach. Two opposing optimal control techniques have been applied to obtain numerical solutions: iterative 2PBVP methods, and backward-integration, or extremal field, methods. The former involves solving a 2PBVP by somehow varying a trajectory until it is optimal - yielding an extremal trajectory which satisfies a necessary condition for optimality. The latter assumes a final state or a terminal manifold and involves adaptively filling the state space of initial conditions with a "field of extremals" - yielding the global solution input and corresponding HJB value function, i.e. optimal cost.
In addition to the application of these optimal feedback controllers, the resulting value functions for various sets of cost objectives and constraints serve as a useful tool for analyzing transport in complicated ocean flows. In addition, they offer a straightforward way of determining how the controllability of a vehicle to or around a target position varies in space and time. We also study the relationship between these control-theoretic tools and simulation-based tools like Finite Time Lyapunov Exponents or the "exit time function" defined inside a target set.