Arbitrarily precise abstractions for optimal controller synthesis

We study a class of leavable, undiscounted, minimax optimal control problems for perturbed, continuous-valued, nonlinear control systems. Leaving or “stopping” is mandatory and the costs are assumed to be non-negative, extended real-valued functions. In a previous contribution, we have shown that this class of optimal control problems is amenable to the solution based on symbolic models of the plant in the sense that an arbitrarily precise upper bound on the value function (measured in terms of its hypograph) can be computed from a given abstraction with prescribed precision on every compact subset of state space. In this work, we propose an algorithm to compute arbitrarily precise abstractions of discrete-time plants that represent the sampled behavior of continuous-time, perturbed, nonlinear control systems and establish the convergence rate of the precision in dependence of the discretization parameters of the algorithm. We illustrate the algorithm by approximately solving an optimal control problem involving a two dimensional version of the cart-pole swing-up problem.