The Asymptotic Stability of a Class of Second Order Differential Equations

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Abstract
In Section 1 of this note a necessary and sufficient condition for the asymptotic stability of a class of ordinary differential equations is developed. These equations appear frequently in problems of elasticity (see, e.g. Genin & Maybee, 1966, 1967). They are second order vector differential equations of the form x+Gx+Sx = 0, where G and S are n×n matrices and x is an n-vector. Here G is called the damping matrix and S the stiffness matrix. As Genin & Maybee (1966) have pointed out, the Routh-Hurwitz criterion (see, e.g., Gantmacher, 1959), although giving necessary and sufficient conditions for asymptotic stability, is not able to directly relate the stability of the system to the matrices G and S. The criterion, based on a theorem of A. M. Lyapunov, which is developed in this note does this and has the further advantage of being completely feasible for high speed computers. Furthermore, for many important cases, asymptotic stability may be determined by inspection. In Section 2 we generalize the sufficiency conditions of Section 1 to non-linear equations of the form x+f(x, x)x+g(x) = 0. Here f(x, x) is an n×n matrix whose Hessian is positive definite and g(x) is an n-vector which has a positive definite Jacobian. Before proceeding, some notation will be developed. (a) x and y will denote n-vectors. (b) Capital letters A, B,… etc. will always denote constant n×n real matrices and I will always stand for the n×n unit matrix. (c) xT, yT and AT will denote respectively the transposes of the vectors x, y and the matrix A. (d) x will represent differentiation of a time dependent vector x(t). (e) B > 0, C > 0,… etc. will denote real positive definite symmetric matrices. (f) A real matrix A will be called Hurwitzian or stable if it has all its characteristic roots with real parts strictly negative.

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