Near-optimal search in continuous domains

We investigate search problems in continuous state and action spaces with no uncertainty. Actions have costs and can only be taken at discrete time steps (unlike the case with continuous control). Given an admissible heuristic function and a starting state, the objective is to find a minimum-cost plan that reaches a goal state. As the continuous domain does not allow the tight optimality results that are possible in the discrete case (for example by A*), we instead propose and analyze an approximate forward-search algorithm that has the following provable properties. Given a desired accuracy ε, and a bound d on the length of the plan, the algorithm computes a lower bound L on the cost of any plan. It either (a) returns a plan of cost L that is at most ε more than the optimal plan, or (b) if, according to the heuristic estimate, there may exist a plan of cost L of length > d, returns a partial plan that traces the first d steps of such plan. To our knowledge, this is the first algorithm that provides optimality guarantees in continuous domains with discrete control and without uncertainty.