Abstract
This letter provides an approximate online adaptive solution to the infinite-horizon optimal control problem for control-affine continuous-time nonlinear systems while formalizing system safety using barrier certificates. The use of a barrier function transform provides safety certificates to formalize system behavior. Specifically, using a barrier function, the system is transformed to aid in developing a controller which maintains the system in a pre-defined constrained region. To aid in online learning of the value function, the state-space is segmented into a number of user-defined segments. Off-policy trajectories are selected in each segment, and sparse Bellman error extrapolation is performed within each respective segment to generate an optimal policy within each segment. A Lyapunov-like stability analysis is included which proves uniformly ultimately bounded regulation in the presence of the barrier function transform and discontinuities. Simulation results are provided for a two-state dynamical system to compare the performance of the developed method to existing methods.
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