Abstract
The use of the second Lyapunov method in many problems in the theory of stability of motion leads to the problem of sign definiteness of a quadratic form whose variables are defined in a convex polyhedral cone C ⊂R n. A method of obtaining the necessary and sufficient conditions is given for this problem. The conditions imposed on the elements of the third- and fourth-order matrices are given. The problem of asymptotic stability of a system with resonance /1/ is solved as an example. A number of problems of the theory of the stability of motion require that the sign definiteness of the quadratic form be established, with conditions written in the form of linear inequalities. Usually, the conditions are those of non-negativity /1–3/, and the more general conditions can be reduced to them. The problem of sign definiteness of a quadratic form under the conditions of non-negativeness was considered for an arbitrary number of variables in /4/. However, the results obtained there can be reduced to the problem of the compatibility of systems of inequalities and pose well-known difficulties when used to solve specific problems. The problem of the sign definiteness of a quadratic form in a convex cone (in general, infinitely-sided) belonging to a Hubert space is considered in /5/, and the necessary and sufficient conditions are obtained, but in the finite-dimensional case discussed below the above result is the same as that obtained in /4/.
Published Version
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