The overarching aim of my research is to analyse nonlinear dynamical system and their properties through a combination of rigorous mathematical analysis and numerical optimisation. I am interested both in the theoretical formulation methods that allow characterizing various properties of nonlinear system using convex optimization, and in their practical applications. Some of my work also aims at the development of structure- and sparsity-exploiting algorithms for the solution of large-scale optimisation problems, particularly semidefinite and sum-of-squares programs.
For more details, please expand the tabs below or see the list of my publications.
Nonlinear systems analysis via auxiliary functions
Nonlinear effects play a major role in determining the behaviour of physical processes and engineering systems across a wide range of fields, ranging from aviation and climate modelling to smart energy management and bioengineering. Understanding, predicting and ultimately controlling the properties of such system is essential to design efficient and effective technological solutions, but remains difficult because, often, nonlinear effects give rise to chaotic or otherwise complex dynamics.
Fundamental yet challenging questions about the dynamical behaviour of nonlinear systems include:
- Given an equilibrium state, is it stable with respect to perturbations, either arbitrary (global stability) or within a certain class (local stability)?
- If the system is unsteady, what are the average or extreme values of various quantities of interest, such as energy or power consumption?
Direct numerical simulations of the equations of motion that describe the system at hand are inherently unsuitable to answer such questions: one must characterize whole classes of solutions, rather than particular ones.
My research aims to develop indirect methods to study nonlinear systems, which do not rely on numerical simulations of the dynamics. Key to my approach is the recent realization that many properties of nonlinear systems can be analysed using so-called auxiliary functions – functions of the system's state that satisfy suitable equality and inequality constraints. For instance, the global stability of an equilibrium state can be established by constructing a Lyapunov function, which achieves a global minimum at the equilibrium and decays monotonically when the system evolves away form equilibrium. Auxiliary functions can be used to design optimal controls, and place rigorous bounds on the average or extreme values of particular quantities of interests (e.g., power consumption).
The overarching aims of my work are to further develop these approaches, and to explore other ways in which auxiliary functions can help study nonlinear dynamics.
Scalable algorithms for semidefinite and sum-of-squares programming
Coming soon – in the meantime, please visit the GitHub page related to this work.