Application of the second Lyapunov method for getting the conditions of stability in systems with quadratic right-hand side (Q6539984)

The authors consider an autonomous system with quadratic nonlinearity \(\dot{x}=Ax+S_2(x)\), where \(x\in\mathbb{R}^n\), \(A\) is an asymptotically stable matrix (i.e., all eigenvalues \(\lambda_i(A)\) have a negative real part), \(S_2(x)=X^TBx\) is a quadratic vector-polynome written in the universal vector-matrix form: \(X^T=(X_1^T\cdots X_n^T)\), the matrix \(X_i^T\) has the vector \((x_1,\dots,x_n)\) in \(i\)-th row and all other elements are equal to zero, \(B=\begin{bmatrix} B_1 \\\NB_2\\\N\vdots \\\NB_n \end{bmatrix}\), \(B_i\) is a symmetric matrix, \(i=1,\dots,n\).\N\NAlgebraic sufficient conditions for the global asymptotic stability are presented for the systems of the second and third orders.