  This paper proposes new quadratic constraints (QCs) to bound a quadratic
polynomial. Such QCs can be used in dissipation ineqaulities to analyze the
stability and performance of nonlinear systems with quadratic vector fields.
The proposed QCs utilize the sign-indefiniteness of certain classes of
quadratic polynomials. These new QCs provide a tight bound on the quadratic
terms along specific directions. This reduces the conservatism of the QC bounds
as compared to the QCs in previous work. Two numerical examples of local
stability analysis are provided to demonstrate the effectiveness of the
proposed QCs.
