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Resilience
Published in Andrew Cook, Damián Rivas, Complexity Science in Air Traffic Management, 2016
Henk A.P. Blom, Soufiane Bouarfa
Viability theory (Aubin, 1991) was originally developed to study dynamical systems which collapse or badly deteriorate if they leave a given subset of the state space. Therefore the objective is to keep the system in the part of the state space where it can survive, that is, where it is viable. In follow-up research by Aubin et al. (2002), viability theory has been extended to hybrid dynamical systems. Recently, Martin et al. (2011) have explained that viability theory provides a natural mathematical framework for the modelling and analysis of resilience in complex systems. In general, viability theory can be applied in a wide range of domains, including cognitive sciences, finance, economics and the sociological sciences. An example application in air transport is obstacle avoidance, which also appears in numerous application fields. Other examples include using viability algorithms to compute wind optimal routes to reach an airport in minimal time, and computing safety envelopes of an aircraft in different phases of flight (Aubin et al., 2011).
Risk-averse decision-making to maintain supply chain viability under propagated disruptions
Published in International Journal of Production Research, 2023
While risky situations are typically modelled as outcomes with known probabilities, viability theory aims at offering resilience to 'unknown unknowns'. A typical problem of viability theory is looking for a feasible portfolio of available controls that maintain evolutions of a dynamic system in a closed set of constraints. Such a portfolio of controls, which can be called a viable portfolio, is obtained as a solution to a viability problem. In this paper, the viable portfolio of controls has been obtained by solving a stochastic quadratic optimisation problem aiming at keeping the production trajectory in a viability space between the two risk-averse boundaries. The risk-averse boundaries are associated with the two conflicting performance measures of a supply chain under worst-case disruption scenarios. Given a viable portfolio of controls, greater losses may be caused only by a more severe disruption capable of pushing production trajectory outside the viability space.
Partial asymptotic null-controllability with mixed state-input constraints
Published in Journal of Control and Decision, 2022
Viability theory offers a solid framework of dynamical systems obeys to constraints in a set-valued form (Aubin, 2009). It has been used in many applications such as air traffic management (Margellos & Lygeros, 2009) and aircraft collision avoidance (Bayen et al., 2007). The main concern of viability theory is the computation of the set of initial states (known as a viability kernel) from which the evolutions of a given system remains within a viability constraint set. In this paper, the characterisation of this set is given in term of contingent cone (see Definition A.2). Motivated by the difficulty on kernel computation, some works proposes algorithms to compute numerically this set. See for instance (Bonneuil, 2006).
Fault detection and isolation using viability theory and interval observers
Published in International Journal of Systems Science, 2018
Majid Ghaniee Zarch, Vicenç Puig, Javad Poshtan, Mahdi Aliyari Shoorehdeli
Viability theory goal is to prove if the dynamical system evolution (1) can be maintained inside a viability constraint set . Any trajectory of system (1) that leaves the set K at some point in time is considered to be no longer viable.