Optimal control with budget constraints and resets.
A. Vladimirsky (Math, Cornell)

The model problem: given a robot in a room with multiple obstacles, what is that robot's fastest path to the target, with the constraint that a stationary enemy observer should not be able to see it for more than 5 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 HJB 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.

(Joint work with R. Takei, W. Chen, Z. Clawson, and S. Kirov)