On the optimal parameter of the composite Laplacian quadratics function

Recently, the composite quadratic Lyapunov function has been extended to study of multi-agent systems, leading to the so-called composite Laplacian quadratics (CLQs) function. Compared with quadratic Lyapunov functions, the CLQs function can yield a larger convergence region and is particularly useful in stabilization of multi-agent systems with complex dynamics, such as differential inclusions. In the definition of the CLQs function, an optimal vector parameter plays a critical role in determining the value of the CLQs function and in constructing stabilization laws derived from the CLQs function. This paper focuses on the properties of the optimal parameter of the CLQs function. The uniqueness of the optimal parameter is established. A distributed computation approach is further proposed, which is useful in computing the optimal parameter. The robustness issue of the optimal parameter is also investigated for a multi-agent system described by linear differential inclusions. Finally, a numerical example is provided to validate the proposed theoretical results.