Lyapunov Function Construction using Constrained Least Square Optimization

In this study, the Lyapunov function is constructed for polynomial dynamical systems using Constrained Least Square Optimization. In literature, the least square method is used to determine the coefficients of the Lyapunov function to make its derivative negative semi-definite. Then the coefficients are inserted in the Lyapunov function to check its positive definiteness. This way requires many iterations for finding the Lyapunov function. However, in the proposed method, a single optimization program is used to search the optimal coefficients to make the Lyapunov function positive definite and its derivative negative semi-definite. In the proposed method the polynomial Lyapunov function of a particular degree is selected with unknown coefficients. The least-square problem is solved to ensure that the coefficients of monomials that have odd power are zero and the coefficients of monomials that have even power are positive. Moreover, the least-square problem is solved under the constraint to make the derivative of the Lyapunov function along the system trajectories negative semi-definite. The proposed method is illustrated with few examples, and the results show that it successfully identifies the Lyapunov function for globally asymptotically stable systems.