# Discontinuous dynamical systems: theory, numerics and applications

PRIN: PROGETTI DI RICERCA DI RILEVANTE INTERESSE NAZIONALE – Bando 2017.

Prot. 2017E844SL

Principal Investigator: Nicola Guglielmi del Gran Sasso Science Institute

Associated Investigator: Luciano Lopez

ABSTRACT DEL PROGETTO.

The numerical treatment of differential equations usually relies on smoothness assumptions which play a fundamental role in the development of suitable schemes and in their convergence analysis. However, an emergent field of interest, especially in some recent engineering applications in control systems, is that of systems of ODEs with discontinuous right hand side. In most mathematical models described by discontinuous ODEs the vectorfield is piecewise smooth in regions which are separated by manifolds across which take place the discontinuities. The existence and uniqueness of the solution are not anymore guaranteed at such interfaces and the classical Filippov and Utkin theories become ambiguous when the interfaces have a codimension higher than 1. Moreover any efficient numerical integrator has to detect and accurately compute the time instants when the solution reaches a discontinuity interface and possibly switch to a different formulation (describing the hidden dynamics).

Our research project focuses on getting new theoretical insight into numerical approaches, and on exploiting this insight for the design and implementation of efficient algorithms. Moreover in those cases where the theory remains ambiguous and the solution is not uniquely defined, one of the main aim of the project is twofold: i) investigate physical regularizations, which transform the discontinuous ODE into a continuous singularly perturbed problem, in order to select a solution with a most physical meaning; ii) study stable trajectories of the original discontinuous system in order to recognize solutions that persist under perturbations that do not belong to the interface. A special class of discontinuous ODEs, widely investigated in the control theory community, is that of switched systems, where the switching mechanism is usually managed by a controller or an external uncontrollable event generator. For linear switched systems (LSS), the use of the theory concerned with the joint spectral radius of a set of matrices, and that concerned with common Lyapunov exponents, provide powerful tools for the stability and stabilizability analysis. However there remain many important open issues and in particular there do not exist general purpose algorithms able to detect the stabilizability of a LSS. For nonlinear SSs stability can be studied in terms of switching Lyapunov functions. A main aim of this project is to advance in these directions and develope new algorithms and related software for the analysis of switched systems, exploiting the HPC resources at the University of L'Aquila (http://caliban.dm.univaq.it/wordpress/)