Application of a family of Lyapunov functions. (Q2761355)

The author considers the perturbed motion equations in the form NEWLINE\[NEWLINE \begin{aligned} &A_1(x_1,\dot x_1,x_2,t)\ddot x_1 = B_1(x_1,\dot x_1,x_2,t);\\ &N_1(x_1,\dot x_1,x_2,t)\ddot x_2 = K_1(x_1,\dot x_1,x_2,t), \end{aligned}\tag{1} NEWLINE\]NEWLINE where \(A_1\) and \(N_1\) are \((n\times n)\)- and \((m\times m)\)-matrices, \(x_1\) and \(B_1\) are \(n\)-dimensional vectors and \(x_2\) and \(K_1\) are \(m\)-dimensional vectors. It is assumed that the elements of the matrices \(A_1\) and \(N_1\) and those of the vectors \(B_1\) and \(K_1\) are bounded, continuous and continuously differentiable in the bounded domain \(G\) containing points \(\,x_1=0\), \(\,\dot x_1=0\,\) and \(\,x_2=0\,\) for \(\,t\geq t_0\). Under some assumptions the author proposes a procedure constructing the family of Lyapunov functions which allow the investigation of stability problem for systems of system (1) type with respect to the first approximation. Examples of application of the proposed algorithm in the investigation of some mechanical problems are presented.