This letter presents an approach to develop a variable impedance controller with considerations of the optimality and stability. Firstly, an original optimal variable law is designed via demonstration learning, through which the Gaussian mixture model/Gaussian mixture regression (GMM/GMR) algorithm is employed to transfer the human impedance functions to the robot. By using the formulation of the GMR result, the regression task can be completed without using the ground-truth information of the human impedance parameters. To ensure the stability, a minimal complementary input is designed for the learned second-order impedance system. We transform the design problem to a constrained convex optimization problem, of which the constraints are related to a Lyapunov function. A criterion for choosing the Lyapunov functions is presented to ensure the feasibility of the problem, and an analytical solution is computed. The proposed approach is verified by the robotic-assisted rehabilitation and trajectory reproduction experiments conducted on a 7-DOF Franka Panda robot.