Contraction analysis of switched Filippov systems via regularization

We study incremental stability and convergence of switched (bimodal) Filippov systems via contraction analysis. In particular, by using results on regularization of switched dynamical systems, we derive sufficient conditions for convergence of any two trajectories of the Filippov system between each other within some region of interest. We then apply these conditions to the study of different classes of Filippov systems including piecewise smooth (PWS) systems, piecewise affine (PWA) systems and relay feedback systems. We show that contrary to previous approaches, our conditions allow the system to be studied in metrics other than the Euclidean norm. The theoretical results are illustrated by numerical simulations on a set of representative examples that confirm their effectiveness and ease of application.Comment: Preprint submitted to Automatic

On a tropicalization of planar polynomial ODEs with finitely many structurally stable phase portraits

Recently, concepts from the emerging field of tropical geometry have been used to identify different scaling regimes in chemical reaction networks where dimension reduction may take place. In this paper, we try to formalize these ideas further in the context of planar polynomial ODEs. In particular, we develop a theory of a tropical dynamical system, based upon a differential inclusion, that has a set of discontinuities on a subset of the associated tropical curve. The development is inspired by an approach of Peter Szmolyan that uses the connection of tropical geometry with logarithmic paper. In this paper, we define a phaseportrait, a notion of equivalence and characterize structural stability. Furthermore, we demonstrate the results on several examples, including a(n) (generalized) autocatalator model. Our main result is that there are finitely many equivalence classes of structurally stable phase portraits and we enumerate these ($15$ in total) in the context of the generalized autocatalator model. We believe that the property of finitely many structurally stable phase portraits underlines the potential of the tropical approach, also in higher dimension, as a method to obtain and identify skeleton models in chemical reaction networks in extreme parameter regimes

Breaking Dense Structures: Proving Stability of Densely Structured Hybrid Systems

Abstraction and refinement is widely used in software development. Such techniques are valuable since they allow to handle even more complex systems. One key point is the ability to decompose a large system into subsystems, analyze those subsystems and deduce properties of the larger system. As cyber-physical systems tend to become more and more complex, such techniques become more appealing. In 2009, Oehlerking and Theel presented a (de-)composition technique for hybrid systems. This technique is graph-based and constructs a Lyapunov function for hybrid systems having a complex discrete state space. The technique consists of (1) decomposing the underlying graph of the hybrid system into subgraphs, (2) computing multiple local Lyapunov functions for the subgraphs, and finally (3) composing the local Lyapunov functions into a piecewise Lyapunov function. A Lyapunov function can serve multiple purposes, e.g., it certifies stability or termination of a system or allows to construct invariant sets, which in turn may be used to certify safety and security. In this paper, we propose an improvement to the decomposing technique, which relaxes the graph structure before applying the decomposition technique. Our relaxation significantly reduces the connectivity of the graph by exploiting super-dense switching. The relaxation makes the decomposition technique more efficient on one hand and on the other allows to decompose a wider range of graph structures.Comment: In Proceedings ESSS 2015, arXiv:1506.0325

Synthesis of Switching Protocols from Temporal Logic Specifications

We propose formal means for synthesizing switching protocols that determine the sequence in which the modes of a switched system are activated to satisfy certain high-level specifications in linear temporal logic. The synthesized protocols are robust against exogenous disturbances on the continuous dynamics. Two types of finite transition systems, namely under- and over-approximations, that abstract the behavior of the underlying continuous dynamics are defined. In particular, we show that the discrete synthesis problem for an under-approximation can be formulated as a model checking problem, whereas that for an over-approximation can be transformed into a two-player game. Both of these formulations are amenable to efficient, off-the-shelf software tools. By construction, existence of a discrete switching strategy for the discrete synthesis problem guarantees the existence of a continuous switching protocol for the continuous synthesis problem, which can be implemented at the continuous level to ensure the correctness of the nonlinear switched system. Moreover, the proposed framework can be straightforwardly extended to accommodate specifications that require reacting to possibly adversarial external events. Finally, these results are illustrated using three examples from different application domains

Wall-Crossing in Coupled 2d-4d Systems

We introduce a new wall-crossing formula which combines and generalizes the Cecotti-Vafa and Kontsevich-Soibelman formulas for supersymmetric 2d and 4d systems respectively. This 2d-4d wall-crossing formula governs the wall-crossing of BPS states in an N=2 supersymmetric 4d gauge theory coupled to a supersymmetric surface defect. When the theory and defect are compactified on a circle, we get a 3d theory with a supersymmetric line operator, corresponding to a hyperholomorphic connection on a vector bundle over a hyperkahler space. The 2d-4d wall-crossing formula can be interpreted as a smoothness condition for this hyperholomorphic connection. We explain how the 2d-4d BPS spectrum can be determined for 4d theories of class S, that is, for those theories obtained by compactifying the six-dimensional (0,2) theory with a partial topological twist on a punctured Riemann surface C. For such theories there are canonical surface defects. We illustrate with several examples in the case of A_1 theories of class S. Finally, we indicate how our results can be used to produce solutions to the A_1 Hitchin equations on the Riemann surface C.Comment: 170 pages, 45 figure