Numerical synthesis of pontryagin optimal control minimizers using sampling-based methods

We present a theoretical formulation, and a corresponding numerical algorithm, that can find Pontryagin-optimal inputs for general dynamical systems by using a direct method. Optimal control remains as a versatile and relevant framework in systems theory applications, many decades after being formally defined. Pontryagin-optimal inputs can be found for some classes of problems using indirect methods, but these are often slow or lack robustness. On the other hand, convergent direct optimal control methods are fast, but their solutions usually converge to first-order optimality conditions, which are weaker. Our result, founded on the theory of relaxed inputs as defined by J. Warga, establishes an equivalence between Pontryagin-optimal inputs and optimal relaxed inputs. We also formulate a sampling-based numerical method to approximate the Pontryagin-optimal relaxed inputs using an iterative direct method. Finally, using a provably-convergent numerical method, we synthesize approximations of the Pontryagin-optimal inputs from the sampled relaxed inputs.

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