Optimal control with budget constraints and resets.
Alexander Vladimirsky, Cornell
(Joint work with R. Takei, W. Chen, Z. Clawson, and S. Kirov)

Consider a model problem: given a room with multiple obstacles and a stationary enemy observer,  find the fastest path to the target for a robot, with the constraint that the observer should not be able to see that robot for more than five seconds in a row.

Many realistic control problems involve multiple criteria for optimality and/or integral constraints on allowable controls. This can be conveniently modeled by introducing a budget for each secondary criterion/constraint. An augmented Hamilton-Jacobi-Bellman equation is then solved on an expanded state space, and its discontinuous viscosity solution yields the value function for the primary criterion/cost. This formulation was previously used by Kumar & Vladimirsky to build a fast (non-iterative) method for problems in which the resources/budgets are monotone decreasing. We currently address a more challenging case, where the resources can be instantaneously renewed (& budgets can be "reset") upon entering a pre-specified subset of the state space. This leads to a hybrid control problem with more subtle causal properties of the value function & additional challenges in constructing efficient numerical methods.