Optimal Control of State Constrained Systems via Measure Relaxations and Polynomial Optimization
Résumé
We address the optimal control problem for a class of dynamical systems with constrained state trajectories. These systems are modeled by a differential inclusion with a drift term and a normal cone mapping associated with the constraint set. The optimal control problem is considered in continuous-time and discrete-time, where the latter provides a computational advantage over the former. In both cases, the nonlinear problem is reformulated as an infinite-dimensional linear program over occupation measures. We show that this does not introduce any relaxation gap, that is, the optimal value remains the same for the reformulated linear program. Using appropriate tools from functional analysis and optimal transport, we also show the convergence of the optimal value of the discrete problem to the optimal value of the continuous problem. We propose finite-dimensional convex optimization algorithms based on the moment-sum-of-squares hierarchy to provide numerical approximations of the proposed infinite-dimensional linear programs.
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