Fast-acting smart inverters that utilize preset operating conditions to determine real and reactive power injection/consumption can create voltage instabilities (over-voltage, voltage oscillations and more) in an electrical distribution network if set-points are not properly configured. In this work, linear distribution power flow equations and droop-based Volt-Var and Volt-Watt control curves are used to analytically derive a stability criterion using Lyapunov analysis that includes the network operating condition. The methodology is generally applicable for control curves that can be represented as Lipschitz functions. The derived Lipschitz constants account for smart inverter hardware limitations for reactive power generation. A local policy is derived from the stability criterion that allows inverters to adapt their control curves by monitoring only local voltage, thus avoiding centralized control or information sharing with other inverters. The criterion is independent of the internal time-delays of smart inverters. Simulation results for inverters with and without the proposed stabilization technique demonstrate how smart inverters can mitigate voltage oscillations locally and mitigate real and reactive power flow disturbances at the substation under multiple scenarios. The study concludes with illustrations of how the control policy can dampen oscillations caused by solar intermittency and cyberattacks. NOMENCLATURE: , Set of all lines; , Set of all nodes; p c i , p g i , Real power demand, Real power generation at node i; q c i , q g i , Reactive power demand, Real power generation at node i; r i j , x i j , Line resistance, Line reactance of line between nodes i and j ; P i j , Q i j , Real power flow, Reactive power flow of line between nodes i and j ; v i , Voltage of node i; c i j , Current of line between nodes i and j ; s i , Rated apparent power of inverter at node i; p i , Maximum real power output of inverter at node i; q lim i , Hardware limit of reactive power generation of inverter at node i; f p,i , Volt-Watt control function of inverter at node i; f q,i , Volt-Var control function of inverter at node i; C p,i , Lipschitz constant for Volt-Watt control function of inverter at node i; C q,i , Lipschitz constant for Volt-Var control function of inverter at node i This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.