Differential equations model many important processes in biology, physics and other sciences. The theory of Dynamical Systems studies the qualitative properties and long-time behaviour of solutions. Starting with special solutions such as equilibria (constant solutions) or periodic solutions, one investigates whether the solution is stable and attractive, i.e. nearby solutions tend to the special one, and tries to determine its basin of attraction, which consists of all solutions which tend to the special one as time goes to infinity. I am particularly interested in numerical methods for the construction of Lyapunov functions and contraction metrics. I have published a Review on Computational Methods for Lyapunov Functions in 2015 and a Review on contraction analysis and computation of contraction metrics in 2023.
Construction of (complete) Lyapunov Functions using meshless collocation
In order to determine the basin of attraction of an equilibrium for a concrete equation, the method of a Lyapunov function can be used. A Lyapunov function is characterised by its negative orbital derivative, i.e. the derivative along solutions of the differential equation. In general, the explicit construction of a Lyapunov function is very difficult. I have developed a general method to construct a Lyapunov function. This method uses meshless collocation, in particular Radial Basis Functions, to approximate Lyapunov functions with a certain orbital derivative. In particular, one approximately solves a linear partial differential equation. The research includes the description and analysis of the method as well as error bounds. The method has been developed for equilibria of autonomous and time-periodic ordinary differential equations, as well as for discrete dynamical systems, non-smooth systems and asymptotically autonomous systems. I have also studied grid refinement and the verification of the constructed Lyapunov function. These methods have been incorporated into a computer tool and have also been extended to compute complete Lyapunov functions for ODEs and discrete-time systems.Construction of Lyapunov Functions using Linear Programming (Optimisation)
Another method for the construction of Lyapunov functions uses linear programming. The Lyapunov function in this case is a piecewise linear function and the optimisation conditions guarantee that its orbital derivative is negative. In a series of paper, we have revised the CPA method to construct Lyapunov functions with negative orbital derivative also near the equilibrium. We have combined the advantages of the construction methods using meshless collocation and linear programming, using the CPA interpolation as a verification tool. This can also be applied to Lyapunov functions derived by other methods such as numerical integration.Construction of (complete) Lyapunov Functions by solving Differential Inequalities with Quadratic Programming
It is often more appropriate to find Lyapunov functions by solving differential inequalities rather than differential equations. We have extended the framework of meshfree collocation to solve differential inequalities with application to the computation of Lyapunov functions. After discretisation, the computational problem becomes a quadratic programming problem. In this context, I have also worked on the existence theory of complete Lyapunov functions with prescribed orbital derivative. Currently, we are applying this method to switched systems.Contraction metrics
I further developed a contraction criterion for autonomous differential equations which measures whether adjacent solutions will approach each other with respect to a given Riemannian metric. With the criterion, one can prove existence, uniqueness and stability of a equilibrium or periodic orbit, and determine subsets of its basin of attraction. I have generalized this criterion for unbounded sets of the phase space and proved its necessity. Moreover, I have applied it to time-periodic, almost periodic, as well as to nonsmooth differential equations. I have established converse theorems, proving the existence of contraction metrics with various properties for an equilibrium as well as for a periodic orbit. This enabled me to develop constructive methods to find a contraction metric using semi-definite optimisation on the one hand, and meshless collocation on the other hand. In more detail, I have generalized meshless collocation to solve a matrix-valued PDE and thus to construct a contraction metric for an equilibrium and a periodic orbit, respectively.
Page last updated 11/12/2024