An Efficient Approach for Estimating Domain of Attraction of Complex Network

This article investigates the estimate (i.e., the invariant subset) of domain of attraction (DOA) for complex network. Starting with the quadratic Lyapunov function of isolated node, we construct a quadratic Lyapunov function of complex network for estimating the DOA of network. In this way, if the largest spherical estimate of the DOA for isolated node can be obtained, we can directly obtain the largest spherical estimate of the DOA for network. Then, for improving the existing estimate, we directly utilize the Laplacian matrix to ingeniously construct a new Lyapunov-like function, relaxing the constraint that the derivative of the Lyapunov function is negative definite in a neighborhood of the origin. Moreover, we iteratively compute Lyapunov-like functions to maximize the obtained estimate as far as possible. Afterward, for polynomial networks, the estimate problem of the DOA is transformed into a classical sum of squares (SOS) programming problem. Particularly, we use the properties of isolated node and network topology to significantly reduce the computational complexity of the above SOS programming problem, such that our estimate can be effectively obtained even for large-scale networks. Finally, four examples are given to illustrate the validity of our theoretical results and the efficiency of our computable approach.