Given a polynomial ordinary differential equation (ODE), we devise generic polynomial reduction algorithms to automatically investigate the intertwined relationship between the total degree of (nontrivial) Darboux polynomials and the polynomials defining the ODE. By generic we mean that both the coefficients and the multidegree of the involved polynomials are symbolic. We use Newton polytopes as a light-weight abstraction to select optimal weight monomial orders improving the efficiency of the involved computations. The method works by inferring necessary conditions on both the coefficients and the multidegree for the polynomial to be Darboux. These conditions are then used, via constants’ propagation, to restrict the shape of the generic candidate, pinpointing which monomials ought to be preserved by removing the superfluous ones. In some relevant cases, we are able to automatically prove the nonexistence of (nontrivial) Darboux polynomials providing a new toolbox to prove and formally certify that some limit cycles are not algebraic.

Differential Algebraic Equation (DAE) systems constitute the mathematical model supporting physical modeling languages such as Modelica, VHDL-AMS, or Simscape. Unlike ODEs, they exhibit subtle issues because of their implicit latent equations and related differentiation index. Multi-mode DAE (mDAE) systems are much harder to deal with, not only because of their mode-dependent dynamics, but essentially because of the events and resets occurring at mode transitions. Unfortunately, the large literature devoted to the numerical analysis of DAEs does not cover the multi-mode case. It typically says nothing about mode changes. This lack of foundations cause numerous difficulties to the existing modeling tools. Some models are well handled, others are not, with no clear boundary between the two classes. In this paper we develop a comprehensive mathematical approach to the structural analysis of mDAE systems which properly extends the usual analysis of DAE systems. We define a constructive semantics based on nonstandard analysis and show how to produce execution schemes in a systematic way.

In this article we study ways of constructing meaningful operational models of piecewise-smooth systems (PWS). The systems we consider are described by polynomial vector fields defined on non-overlapping semi-algebraic sets, which form a partition of the state space. Our approach is to give meaning to motion in systems of this type by automatically synthesizing operational models in the form of hybrid automata (HA). Despite appearances, it is in practice often difficult to arrive at satisfactory HA models of PWS. The different ways of building operational models that we explore in our approach can be thought of as defining different semantics for the underlying PWS. These differences have a number of interesting nuances related to phenomena such as chattering, non-determinism, so-called mythical modes and sliding behaviour.

Formal verification of industrial systems is very challenging, due to reasons ranging from scalability issues to communication difficulties with engineering-focused teams. More importantly, industrial systems are rarely designed for verification, but rather for operational needs. In this paper we present an overview of our experience using hybrid systems theorem proving to formally verify ACAS X, an airborne collision avoidance system for airliners scheduled to be operational around 2020. The methods and proof techniques presented here are an overview of the work already presented in [8], while the evaluation of ACAS X has been significantly expanded and updated to the most recent version of the system, run 13. The effort presented in this paper is an integral part of the ACAS X development and was performed in tight collaboration with the ACAS X development team.