Pareto Fronts of the Set of Sustainable Thresholds for Constrained Control Systems


This paper is concerned with a dual object of the so-called viability kernel of a control system with mixed (state-control) constraints. This object, called the set of sustainable thresholds, describes the possible thresholds for which a given initial condition is viable. In this work, we are concerned with characterizing the weak and strong Pareto fronts of the set of sustainable thresholds and providing a practical method for computing such objects based on optimal control theory and a level-set approach. A numerical example, relying on renewable resource management, is shown to demonstrate the proposed method.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4


  1. 1.

    Following Theorem 2, for vectors \(\bar{c} \notin {\mathbb {S}}(\xi )\), one has \(\omega _{\xi }\left( {\bar{c}}\right) > 0\). In this example, we are considering constraints in the opposite sense (see Remark 7), so we obtain the same result with the opposite sign for the function \(\omega _{\xi }\left( {\cdot }\right) \).


  1. 1.

    Altarovici, A., Bokanowski, O., Zidani, H.: A general hamilton-jacobi framework for nonlinear state-constrained control problems. ESAIM: COCV 19(2), 337–357 (2013)

    MATH  Google Scholar 

  2. 2.

    Aubin, J.P.: Viability Theory. Systems & Control: Foundations & Applications. Birkhäuser Boston Inc., Boston, MA (1991)

    Google Scholar 

  3. 3.

    Barrios, E., Gajardo, P., Vasilieva, O.: Sustainable thresholds for cooperative epidemiological models. Math. Biosci. 302, 9–18 (2018)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Béné, C., Doyen, L.: Storage and viability of a fishery with resource and market dephased seasonalities. Environ. Resour. Econ. 15(1), 1–26 (2000)

    Article  Google Scholar 

  5. 5.

    Béné, C., Doyen, L., Gabay, D.: A viability analysis for a bio-economic model. Ecol. Econ. 36(3), 385–396 (2001)

    Article  Google Scholar 

  6. 6.

    Cissé, A., Gourguet, S., Doyen, L., Blanchard, F., Péreau, J.C.: A bio-economic model for the ecosystem-based management of the coastal fishery in french guiana. Environ. Dev. Econ. 18(3), 245–269 (2013)

    Article  Google Scholar 

  7. 7.

    Clark, C.W.: Mathematical Bioeconomics, second edn. Pure and Applied Mathematics (New York). Wiley, New York (1990). The optimal management of renewable resources, With a contribution by Gordon Munro, A Wiley-Interscience Publication

  8. 8.

    De Doná, J.A., Lévine, J.: On barriers in state and input constrained nonlinear systems. SIAM J. Control Optim. 51(4), 3208–3234 (2013)

    MathSciNet  Article  Google Scholar 

  9. 9.

    De Lara, M., Doyen, L.: Sustainable Management of Natural Resource: Mathematical Models and Methods. Springer, New York (2008)

    Book  Google Scholar 

  10. 10.

    De Lara, M., Doyen, L., Guilbaud, T., Rochet, M.J.: Monotonicity properties for the viable control of discrete-time systems. Syst. Control Lett. 56(4), 296–302 (2007)

    MathSciNet  Article  Google Scholar 

  11. 11.

    De Lara, M., Gajardo, P., Ramírez, C.H.: Viable states for monotone harvest models. Syst. Control Lett. 60(3), 192–197 (2011)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Doyen, L., Béné, C.: Sustainability of fisheries through marine reserves: a robust modeling analysis. J. Environ. Manag. 69(1), 1–13 (2003)

    Article  Google Scholar 

  13. 13.

    Doyen, L., Gajardo, P.: Viability standards and multi-criteria maximin. Cahiers du GREThA, Groupe de Recherche en Economie Théorique et Appliquée (2018).

  14. 14.

    Doyen, L., Cissé, A., Gourguet, S., Mouysset, L., Hardy, P.Y., Béné, C., Blanchard, F., Jiguet, F., Pereau, J.C., Thébaud, O.: Ecological-economic modelling for the sustainable management of biodiversity. Comput. Manag. Sci. 10(4), 353–364 (2013)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Doyen, L., Béné, C., Bertignac, M., Blanchard, F., Cissé, A.A., Dichmont, C., Gourguet, S., Guyader, O., Hardy, P.Y., Jennings, S., Little, L.R., Macher, C., Mills, D.J., Noussair, A., Pascoe, S., Pereau, J.C., Sanz, N., Schwarz, A.M., Smith, T., Thébaud, O.: Ecoviability for ecosystem-based fisheries management. Fish Fish. 18(6), 1056–1072 (2017)

    Article  Google Scholar 

  16. 16.

    Eisenack, K., Scheffran, J., Kropp, J.P.: Viability analysis of management frameworks for fisheries. Environ. Model. Assess. 11(1), 69–79 (2006)

    Article  Google Scholar 

  17. 17.

    Esterhuizen, W., Lévine, J.: From pure state and input constraints to mixed constraints in nonlinear systems. In: Feedback Stabilization of Controlled Dynamical Systems, Lect. Notes Control Inf. Sci., vol. 473, pp. 125–141. Springer, Cham (2017)

  18. 18.

    Gajardo, P., Olivares, M., Ramírez, C.H.: Methods for the sustainable rebuilding of overexploited natural resources. Environ. Model. Assess. 23(6), 713–727 (2018)

    Article  Google Scholar 

  19. 19.

    Krawczyk, J.B., Pharo, A., Serea, O.S., Sinclair, S.: Computation of viability kernels: a case study of by-catch fisheries. Comput. Manag. Sci. 10(4), 365–396 (2013)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Martinet, V.: A characterization of sustainability with indicators. J. Environ. Econ. Manag. 61, 183–197 (2011)

    Article  Google Scholar 

  21. 21.

    Martinet, V., Gajardo, P., De Lara, M., Ramırez C.H.: Bargaining with intertemporal maximin payoffs. EconomiX Working Papers 2011-7 (2011)

  22. 22.

    Mitchell, I.M., Bayen, A.M., Tomlin, C.J.: A time-dependent Hamilton-Jacobi formulation of reachable sets for continuous dynamic games. IEEE Trans. Autom. Control 50(7), 947–957 (2005)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Mullon, C., Cury, P., Shannon, L.: Viability model of trophic interactions in marine ecosystems. Nat. Resour. Model. 17(1), 71–102 (2004)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Péreau, J.C., Doyen, L., Little, L., Thébaud, O.: The triple bottom line: meeting ecological, economic and social goals with individual transferable quotas. J. Environ. Econ. Manag. 63(3), 419–434 (2012)

    Article  Google Scholar 

  25. 25.

    Rapaport, A., Terreaux, J.P., Doyen, L.: Viability analysis for the sustainable management of renewable resources. Math. Comput. Model. 43(5–6), 466–484 (2006)

    MathSciNet  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Pedro Gajardo.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported by FONDECYT grants N 1160567 (P. Gajardo) and N 3170485 (C. Hermosilla), both CONICYT-Chile programs, and also by project MathAmsud 18-MATH-05.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Gajardo, P., Hermosilla, C. Pareto Fronts of the Set of Sustainable Thresholds for Constrained Control Systems. Appl Math Optim 83, 1103–1121 (2021).

Download citation


  • Set of sustainable thresholds
  • Discrete-time systems
  • Mixed constraints
  • Level-set approach
  • Dynamic programming

Mathematics Subject Classification

  • 49L20
  • 93C55
  • 93C10