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Finite-Horizon Control of Nonlinear Discrete-Time Systems with Terminal Cost of Wasserstein Distance

This study explores a finite-horizon optimal control problem of nonlinear discrete-time systems for steering a probability distribution of initial states as close as possible to a desired probability distribution of terminal states. The problem is formulated as an optimal control problem of the Mayer form, with the terminal cost given by the Wasserstein distance, which provides a metric on probability distributions. For this optimal control problem, this paper provides a necessary condition of the optimality as a variation of the minimum principle of standard optimal control problems. The motivation for exploring this optimal control problem was to provide a control-theoretic viewpoint of a machine-learning algorithm called "the normalizing flow". The obtained necessary condition is employed for developing a simple variation of the normalizing flow approach, and a gradient descent-type numerical algorithm is also provided.