常系数线性微分方程组的ляпунов函数的公式

<正> §1.引言 我们考虑实常系数线性微分方程组(?)Ляпунов早已证明:如果(1)的特征方程(?)所有的根皆具负实部,那末对于任意给定的负定(正定)m 次齐次多项式 U(x_1,…,x_n),恒存在唯一正定(负定)m 次齐次多项式 V(x_1,…,x_n)满足方程

THE FORMULA OF LIAPOUNOFF FUNCTION OF SYSTEM OF LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS

We consider system of linear differential equations with constant coefficients(?)Liapounoff proved that:if real parts of all roots of the characteristic equation(?)arare negative,then,for any given negative definite m-th homogeneous polynomialu(x_1…,x_n),there exists an unique positive definite m-th homogemeous polynomialV(x_1,…,x_n),which satisfies the equation(?)In this paper,we shall give the explict form of Liapounoff function V.V can bewritten in the form of a sum of square terms and its coefficients are functions of thefollowing Routh-Hurwitz determinants:(?)We introduce the following symbols:1°Take the n—th order determinant(?),replaceits j-th column by(?)and denote it by M~(i)(x_1,…,x_n),then take a minor ofM~(i)(x_1…,x_n),the elements of this minor are situated on v_1,…-th columns andV1,…,Vk-th rows(v1<…