Construction of Control Lyapunov-Barrier Functions from CLF-CBF Pairs

This paper studies the construction of control Lyapunov-barrier functions (CLBFs) from a given control Lyapunov function (CLF) $V$ and control barrier function (CBF) $h$. We consider functions of the form $W=F(V,h)$ that increase with the CLF value and do not increase with the barrier value, and show that the CLBF decrease condition is completely characterized by a nonnegative scalar weight that governs the relative contributions of the CLF and CBF. Because this weight depends only on $(V,h)$, the same value must satisfy the decrease condition at all states sharing the same CLF and CBF values, which leads to a joint-level-set admissibility condition. We then construct CLBFs from admissible weights through a first-order partial differential equation (PDE) and obtain explicit multiplicative and power-type families as special cases. We further identify an integrability obstruction showing that admissibility alone does not guarantee properness. Two nonlinear examples illustrate the constructions and demonstrate safe stabilization in cases where feedback based on the original CLF violates the safety constraint.