Lyapunov and exponential dichotomy spectral theory is extended from ordinary differential equations (ODEs) to nonautonomous differential-algebraic equations (DAEs). By using orthogonal changes of variables, the original DAE system is transformed into appropriate condensed forms, for which concepts such as Lyapunov exponents, Bohl exponents, exponential dichotomy and spectral intervals of variou...

Mathematical control theory provides the theoretical foundations that undergird many modern technologies, including aeronautics, biotechnology, communications networks, manufacturing, and models of climate change. During the past fifteen years, there have been numerous exciting developments at the interface of control engineering and mathematical control theory. Many of these advances were base...

We present a novel method to compute Lyapunov functions for continuous-time systems with multiple local attractors. In the proposed method one first computes an outer approximation of the local attractors using a graphtheoretic approach. Then a candidate Lyapunov function is computed using a Massera-like construction adapted to multiple local attractors. In the final step this candidate Lyapuno...

It has been veri"ed that a controllable series capacitor with a suitable control scheme can improve transient stability and help to damp electromechanical oscillations. A question of great importance is the selection of the input signals and a control strategy for this device in order to damp power oscillations in an e!ective and robust manner. Based on Lyapunov theory a control strategy for da...

The ergodic properties of SDEs, and various time discretizations for SDEs, are studied. The ergodicity of SDEs is established by using techniques from the theory of Markov chains on general state spaces. Application of these Markov chain results leads to straightforward proofs of ergodicity for a variety of SDEs, in particular for problems with degenerate noise and for problems with locally Lip...

In this paper we introduce a new modeling paradigm for shortest path games representation with Petri nets. Whereas previous works have restricted attention to tracking the net using Bellman’s equation as a utility function, this work uses a Lyapunov-like function. In this sense, we change the traditional cost function by a trajectory-tracking function which is also an optimal cost-to-target fun...

Introduction 1 1. Lyapunov exponents of dynamical systems 3 2. Examples of systems with nonzero exponents 6 3. Lyapunov exponents associated with sequences of matrices 18 4. Cocycles and Lyapunov exponents 24 5. Regularity and Multiplicative Ergodic Theorem 31 6. Cocycles over smooth dynamical systems 46 7. Methods for estimating exponents 54 8. Local manifold theory 62 9. Global manifold theor...

The theory of Lyapunov exponents and methods from ergodic theory have been employed by several authors in order to study persistence properties of dynamical systems generated by ODEs or by maps. Here we derive sufficient conditions for uniform persistence, formulated in the language of Lyapunov exponents, for a large class of dissipative discrete-time dynamical systems on the positive orthant o...

We consider the relationship between stationary distributions for stochastic models of reaction systems and Lyapunov functions for their deterministic counterparts. Specifically, we derive the well-known Lyapunov function of reaction network theory as a scaling limit of the non-equilibrium potential of the stationary distribution of stochastically modeled complex balanced systems. We extend thi...

In this paper, the problem of robust filter design for networked systems with time-varying sampling rate is investigated. The design conditions are obtained by using the Lyapunov theory and the Finsler’s Lemma. A robust filter, that minimizes an upper bound to the H∞ performance of the estimation error, is obtained as the solution of an optimization problem. A path-dependent Lyapunov function i...