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This study mainly focuses on the stability of a class of linear neutral systems in a critical case, that is, the spectral radius of the principal neutral term (matrix H) is equal to 1. It is difficult to determine the stability of such systems by using existing methods. In this study, a sufficient stability criterion for the critical case is given in terms of the existence of solutions to a linear matrix inequality (LMI). Moreover, it is also shown that the proposed stability criterion conforms with a fact that the considered linear neutral systems are unstable when H has a Jordan block corresponding to the eigenvalue of modulus 1. An illustrative example is presented to determine the stability of a linear neutral system whose principal neutral term H has multiple eigenvalues of modulus 1 without Jordan chains. This is difficult in existing studies.

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