In this paper we consider the problem of the sign-definiteness of a quadratic form (QF) in the domain defined by quadratic constraints under quadratic constraints. Each constraint is determined by an inequality on a QF. Well known and widely applicable in the control theory approach consists of using so called S–procedure. The semidefinite relaxation approach investigated in this paper allows us to derive an S–procedure from duality conditions. However, the S–procedure, which gives necessary and sufficient conditions for sign-definiteness for the relaxed problem, gives only sufficient conditions for sign-definiteness for the initial problem if the number of quadratic constraints is two or more. In this paper the new approach is proposed, allowing establishment of conditional sign definiteness in some cases, when the S–procedure doesn’t give an answer. The results are illustrated by an example.
