Role of the Generalized Lipschitz Condition in Finite-Time Stability and in the Derivation of the Maximum Principle

摘要

The main purpose of the report was to show how important the generalized Lipschitz condition is in proving certain properties of varied solutions of differential equations. These are particularly useful in consideration of finite-time stability and in deriving the Pontryagin Maximum Principle. It was shown that if a system satisfies a generalized Lipschitz condition in the state variables, it is finite-time stable with respect to the initial state. If it satisifies a generalized Lipschitz condition in the control, it is finite-time stable with respect to the control. In deriving the Maximum Principle using the Calculus of Variations approach, an implicit assumption was made that for sufficiently small variations of the optimum control, the terminal conditions of the problem can still be met. This assumption was shown to be valid if a generalized Lipschitz condition is satisfied. (Author)