Hard monotone stabilization of nonlinear systems by direct Lyapunov method

Stabilization problem is formulated as a problem of synthesis of asymptotically stable systems. It is indicated that the second Lyapunov method is highly efficient for the solution of stabilization problems. The matrix algorithm of synthesis of nonlinear regulator is proposed. This algorithm is based on the solution of Lyapunov's equation relative to unknown control matrix. Asymptotic stability of nonlinear system is ensured with the help of some positively defined quadratic form and its total derivative which is negatively defined for any trajectory of this system. Proposed algorithm enables to obtain the control law in explicit form and to construct quadratic form and its derivative simultaneously. It allows to apply the method for stabilization of unstable nonlinear systems. The example of hard monotone stabilization of nonlinear oscillator is considered.