The viability kernel of dynamical systems with mixed constraints: A level-set approach

Abstract This paper deals with control systems with mixed (state-control) constraints and dynamics in discrete-time. We aim at providing a practical method for computing the viability kernel for such systems. We take a standard approach via optimal control theory, that is, we identify the viability kernel as the effective domain of the value function of a suitable optimal control problem. A dynamic programming principle can be established in order to compute that value function. However, from a practical point of view this may turn out in a slow scheme for such computation, because at each step the mixed constraints need to be verified. In this work we take a different path which has been inspired by the level-set approach studied for optimal control problems with (pure) state constraints. An algorithm is proposed for computing the viability kernel and some numerical examples are shown to demonstrate the proposed method.

[1]  Alexandre M. Bayen,et al.  A time-dependent Hamilton-Jacobi formulation of reachable sets for continuous dynamic games , 2005, IEEE Transactions on Automatic Control.

[2]  Jacek B. Krawczyk,et al.  Computation of viability kernels: a case study of by-catch fisheries , 2013, Comput. Manag. Sci..

[3]  Christian Mullon,et al.  VIABILITY MODEL OF TROPHIC INTERACTIONS IN MARINE ECOSYSTEMS , 2004 .

[4]  Computing the viability kernel in large state dimension , 2006 .

[5]  Luc Doyen,et al.  Monotonicity properties for the viable control of discrete-time systems , 2007, Syst. Control. Lett..

[6]  Christophe Béné,et al.  A viability analysis for a bio-economic model , 2001 .

[7]  Alain Rapaport,et al.  Viability analysis for the sustainable management of renewable resources , 2006, Math. Comput. Model..

[8]  Luc Doyen,et al.  Sustainable Management of Natural Resources: Mathematical Models and Methods , 2008 .

[9]  Georges Zaccour,et al.  A survey of applications of viability theory to the sustainable exploitation of renewable resources , 2017 .

[10]  S. Gourguet,et al.  A bio-economic model for the ecosystem-based management of the coastal fishery in French Guiana , 2013, Environment and Development Economics.

[11]  J. Scheffran,et al.  Viability analysis of management frameworks for fisheries , 2005, math/0510115.

[12]  José A. De Doná,et al.  On Barriers in State and Input Constrained Nonlinear Systems , 2013, SIAM J. Control. Optim..

[13]  L Doyen,et al.  Sustainability of fisheries through marine reserves: a robust modeling analysis. , 2003, Journal of environmental management.

[14]  U. Rashid Sumaila,et al.  Economic viability and small-scale fisheries — A review , 2016 .

[15]  Yu. S. Ledyaev,et al.  Qualitative properties of trajectories of control systems: A survey , 1995 .

[16]  P. Saint-Pierre.  Approximation of the viability kernel , 1994 .

[17]  Jaroslav Dolezal.  Existence of optimal solutions in general discrete systems , 1975, Kybernetika.

[18]  Jean Lévine,et al.  From Pure State and Input Constraints to Mixed Constraints in Nonlinear Systems , 2017 .

[19]  Pierre-Yves Hardy,et al.  Ecological-economic modelling for the sustainable management of biodiversity , 2013, Computational Management Science.

[20]  Héctor Ramírez Cabrera,et al.  Viable states for monotone harvest models , 2011, Syst. Control. Lett..

[21]  Luc Doyen,et al.  The triple bottom line: Meeting ecological, economic and social goals with individual transferable quotas , 2012 .

[22]  Ian M. Mitchell,et al.  Lagrangian methods for approximating the viability kernel in high-dimensional systems , 2013, Autom..

[23]  Sean Pascoe,et al.  Ecoviability for ecosystem-based fisheries management , 2017 .

[24]  Hasnaa Zidani,et al.  A general Hamilton-Jacobi framework for non-linear state-constrained control problems , 2013 .

[25]  Jean-Pierre Aubin,et al.  Viability theory , 1991 .

[26]  M. Lara,et al.  Satisficing Versus Optimality: Criteria for Sustainability , 2015, Bulletin of mathematical biology.

[27]  Sophie Martin,et al.  Approximating Viability Kernels With Support Vector Machines , 2007, IEEE Transactions on Automatic Control.

[28]  Eladio Ocaña Anaya,et al.  Ecosystem Viable Yields , 2011, Environmental Modeling & Assessment.

[29]  Marc Quincampoix,et al.  An Algorithm for Viability Kernels in H?lderian Case: Approximation by Discrete Dynamical Systems , 1993 .

[30]  Christophe Béné,et al.  Storage and Viability of a Fishery with Resource and Market Dephased Seasonalities , 2000 .