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This paper introduces harmonic control Lyapunov barrier functions (harmonicCLBF) that aid in constrained control problems such as reach-avoid problems.Harmonic CLBFs exploit the maximum principle that harmonic functions satisfy toencode the properties of control Lyapunov barrier functions (CLBFs). As aresult, they can be initiated at the start of an experiment rather than trainedbased on sample trajectories. The control inputs are selected to maximize theinner product of the system dynamics with the steepest descent direction of theharmonic CLBF. Numerical results are presented with four different systemsunder different reach-avoid environments. Harmonic CLBFs show a significantlylow risk of entering unsafe regions and a high probability of entering the goalregion.

Harmonic Control Lyapunov Barrier Functions for Constrained Optimal Control with Reach-Avoid Specifications