Algrím til reikninga á Lyapunov föllum - verkefni lokið
Fréttatilkynning verkefnisstjóra
The results of the project were presented at international conferences and published in two peer reviewed ISI journals articles and six peer reviewd conference proceedings contributions.
A dynamical system is a concept in mathematics that describes how quantities may change in time. The mathematical abstraction is to define the system as a fixed rule that describes the time dependence of a point in geometrical space. The applications and usefulness of the theory of dynamical systems is virtually limitless, as it is used to model systems in physics, biology, chemistry, economics, all branches of engineering, etc. A central subject in the study of dynamical systems are invariant sets and attractors, for they determine the asymptotic behavior and robustness of systems. These concepts are closely related to the stability theory of dynamical systems. Linear systems are quite well understood and therefore very often real-world nonlinear systems are approximated by such systems in an attempt to derive information on the systems behavior. However, this approach is often very limiting because the dynamics are only linear in the first approximation and all nonlinear effects are ignored. To reach new levels in engineering and truly understand the physical world a better understanding of nonlinear systems is therefore indispensable.
The Lyapunov theory is generally accepted to be the most useful theory for studying stability of attractors in dynamical systems. Its centerpiece, the Lyapunov function, is an energy-like function from the state-space to the real number that is decreasing along all solution trajectories. If a Lyapunov function for a dynamical system is known most qualitative properties of the system can be derived from it. Unfortunately, the computation of a Lyapunov function for a nonlinear system is a very difficult task.
Within the project novel methods to compute Lyapunov functions were developed, combining classical converse theorems with a previous method to compute Lyapunov functions using linear programming. The novel method is not only orders of magnitudes faster than previous methods, but can also be used for systems with multiple stable equilibira and/or multiple attractors. Software to compute Lyapunov functions using severeal diffrent methods was developed, coded and made available for download free of charge. Different computational methods were used to construct Lyapunov functions for a series of interesting examples using the software. The results of the project were presented at international conferences and published in two peer reviewed ISI journals articles and six peer reviewd conference proceedings contributions.
Heiti verkefnis: Algrím til reikninga á Lyapunov föllum / Algorithms to compute Lyapunov functions
Verkefnisstjóri: Sigurður Freyr Hafstein, Háskólanum í Reykjavík ehf.
Tegund styrks: Verkefnisstyrkur
Styrkár: 2013-2015
Fjárhæð styrks: 10,44 millj. kr. alls
Tilvísunarnúmer Rannís: 13067705
