Application of Higher Order Derivatives of Lyapunov Functions in Stability Analysis of Nonlinear Homogeneous Systems

The Lyapunov stability analysis method for nonlinear dynamic systems needs a positive definite function whose time derivative is at least negative semi-definite in the direction of the system's solutions. However merging the both properties in a single function is a challenging task. In this paper some linear combination of higher order derivatives of the Lyapunov function with non-negative coefficients is resulted. If the resultant summation is negative definite and all the derivatives are decrescent then the zero equilibrium state of the nonlinear system is asymptotically stable. If the higher order time derivatives of the Lyapunov function are not well-defined, then some well-defined smooth functions may be used instead. In this case a linear combination of time derivatives of all functions, with non-negative coefficients, must be negative definite. The new conditions are then reformed to be applied for stability analysis of nonlinear homogeneous systems. Some examples are presented to describe the approach.