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## Section: New Results

### Hybrid Systems Modeling

Participants : Ayman Aljarbouh, Albert Benveniste, Benoît Caillaud, Khalil Ghorbal.

#### Robust Simulation for Hybrid Systems: Chattering Path Avoidance

The sliding mode approach is recognized as an efficient tool for treating the chattering behavior in hybrid systems. However, the amplitude of chattering, by its nature, is proportional to magnitude of discontinuous control. A possible scenario is that the solution trajectories may successively enter and exit as well as slide on switching mani-folds of different dimensions. Naturally, this arises in dynamical systems and control applications whenever there are multiple discontinuous control variables. The main contribution of [9] is to provide a robust computational framework for the most general way to extend a flow map on the intersection of $p$ intersected $\left(n-1\right)$-dimensional switching manifolds in at least $p$ dimensions. We explored a new formulation to which we can define unique solutions for such particular behavior in hybrid systems and investigate its efficient computation/simulation. An extended version of this work has been presented at the Baltic Young Scientists Conference [8] .

#### A Hierarchy of Proof Rules for Checking Positive Invariance of Algebraic and Semi-Algebraic Sets

In [6] , we studied sound proof rules for checking positive invariance of algebraic and semi-algebraic sets, that is, sets satisfying polynomial equalities and those satisfying finite boolean combinations of polynomial equalities and inequalities, under the flow of polynomial ordinary differential equations. Problems of this nature arise in formal verification of continuous and hybrid dynamical systems, where there is an increasing need for methods to expedite formal proofs. We study the trade-off between proof rule generality and practical performance and evaluate our theoretical observations on a set of benchmarks. The relationship between increased deductive power and running time performance of the proof rules is far from obvious; we discuss and illustrate certain classes of problems where this relationship is interesting.

#### A Formally Verified Hybrid System for Safe Advisories in the Next-Generation Airborne Collision Avoidance System

The Next-Generation Airborne Collision Avoidance System (ACAS X) is intended to be installed on all large aircraft to give advice to pilots and prevent mid-air collisions with other aircraft. It is currently being developed by the Federal Aviation Administration (FAA). In [16] we determined the geometric configurations under which the advice given by ACAS X is safe under a precise set of assumptions and formally verify these configurations using hybrid systems theorem proving techniques. We considered subsequent advisories and showed how to adapt our formal verification to take them into account. We examined the current version of the real ACAS X system and discussed some cases where our safety theorem conflicts with the actual advisory given by that version, demonstrating how formal, hybrid systems proving approaches are helping to ensure the safety of ACAS X. Our approach is general and could also be used to identify unsafe advice issued by other collision avoidance systems or confirm their safety.

#### Domain Globalization: Using Languages to Support Technical and Social Coordination

When a project is realized in a globalized environment, multiple stakeholders from different organizations work on the same system. Depending on the stakeholders and their organizations, various (possibly overlapping) concerns are raised in the development of the system. In this context a Domain Specific Language (DSL) supports the work of a group of stakeholders who are responsible for addressing a specific set of concerns. We contributed to a book chapter [11] , identifying the open challenges arising from the coordination of globalized domain-specific languages. We identified two types of coordination: technical coordination and social coordination. After presenting an overview of the current state of the art, we discussed first the open challenges arising from the composition of multiple DSLs, and then the open challenges associated to the collaboration in a globalized environment.