This paper introduces harmonic control Lyapunov barrier functions (harmonic CLBF) that aid in constrained control problems such as reach-avoid problems. Harmonic CLBFs exploit the maximum principle that harmonic functions satisfy to encode the properties of control Lyapunov barrier functions (CLBFs). As a result, they can be initiated at the start of an experiment rather than trained based on sample trajectories. The control inputs are selected to maximize the inner product of the system dynamics with the steepest descent direction of the harmonic CLBF. Numerical results are presented with four different systems under different reach-avoid environments. Harmonic CLBFs show a significantly low risk of entering unsafe regions and a high probability of entering the goal region.

翻译：本文提出谐波控制李雅普诺夫障碍函数（Harmonic Control Lyapunov Barrier Functions, Harmonic CLBF），用于解决如rech-avoid问题等约束控制问题。谐波CLBF利用谐波函数满足的极大值原理，编码控制李雅普诺夫障碍函数（CLBF）的特性。因此，它们可在实验开始时直接初始化，而无需基于样本轨迹进行训练。控制输入通过最大化系统动力学与谐波CLBF最速下降方向的内积进行选取。针对四种不同系统在多种reach-avoid环境下的数值实验结果表明，谐波CLBF进入不安全区域的风险显著降低，且进入目标区域的概率显著提高。