Construction of a family of Lyapunov functions for nonmechanical systems (Q2719391)

The author considers systems of equations which are reduced to the form NEWLINE\[NEWLINEA(x_1,\dot x_1,x_2,t)\ddot x_1 = B(x_1,\dot x_1,x_2,t), \quad N(x_1,\dot x_1,x_2,t)\dot x_2 = K(x_1,\dot x_1,x_2,t),\tag{1} NEWLINE\]NEWLINE where \(A\) and \(N\) are \(n\times n\)- and \(m\times m\)-nondegenerated matrices. It is shown that the function NEWLINE\[NEWLINE V = \tfrac 12(y^TAy + x_2^TNx_2 + x_1^TCx_1) NEWLINE\]NEWLINE can be taken as the desired family of Lyapunov functions, where \( y = \dot x_1 - f(x_1,t) \) and \( f(x_ 1,t)\) is an arbitrary \(n\)-dimensional vector function with bounded and differential components admitting infinitely small upper limit. Some applications are shown for the above mentioned Lyapunov functions for controlled systems.