Weighted Phase Volume Method in Stability Analysis: Integral Criteria and Ellipsoidal Reachable Sets
A method for analysing the stability of dynamical systems is proposed, based on the introduction of a weighted phase volume and time rescaling by a positive function. The advantage of the method is the ability to set the contraction properties of the phase volume by choosing the weighting function and the scaling factor, while preserving the topology of the phase portrait. Integral dissipativity conditions are derived, leading to new definitions of integral stability, asymptotic stability, and exponential stability. For quadratic weighting functions, covering and inner ellipsoids are constructed, providing geometric estimates of reachable sets. The connection between the proposed approach and classical Lyapunov stability is established. The efficiency of the method is demonstrated through numerical examples.