Viscosity solutions to evolution problems of star-shaped reachable sets



The article deals with Lipschitz continuous differential inclusions that yield star-shaped reachable sets. The purpose of the paper is to show that the radial function of such reachable sets is a viscosity solution to a certain partial differential equation. As a result, the existing theory of viscosity solutions to first-order partial differential equations was applied to resolve the existence, uniqueness, and some calculation aspects. Several relaxations concerning the forms of the inclusion and the initial set were also considered.


Differential inclusion Generalized solutions Radial function Minkowski function Gauge function 

Mathematics Subject Classification

35D40 93B03 34A60 


  1. 1.
    Kurzhanski, A.B., Varaiya, P.: Dynamics and Control of Trajectory Tubes. Theory and Computation. Birkhauser, Basel (2014)CrossRefMATHGoogle Scholar
  2. 2.
    Mazurenko, S.S.: Partial differential equation for evolution of star-shaped reachability domains of differential inclusions. Set Valued Var. Anal. 24(2), 333–354 (2016)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Springer, Berlin (1988)CrossRefGoogle Scholar
  4. 4.
    Panasyuk, A.I., Panasyuk, V.I.: An equation generated by a differential inclusion. Math. Notes Acad. Sci. USSR 27(3), 213–218 (1980). (original Russian text published in Matematicheskiye Zametki, 27:3, 429–437, 1980)MathSciNetMATHGoogle Scholar
  5. 5.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefMATHGoogle Scholar
  6. 6.
    Lions, P.L.: Generalized Solutions of Hamilton–Jacobi Equations, vol. 69. Pitman Publishing, London (1982)MATHGoogle Scholar
  7. 7.
    Crandall, M.G., Ishii, H., Lions, P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27(1), 1–67 (1992)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Souganidis, P.E.: Existence of viscosity solutions of Hamilton–Jacobi equations. J. Differ. Equ. 56(3), 345–390 (1985)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Clarke, F.H.: Optimization and Nonsmooth Analysis, vol. 5. SIAM, Philadelphia (1990)CrossRefMATHGoogle Scholar
  10. 10.
    Azagra, D., Ferrera, J., LÓpez-Mesas, F.: Nonsmooth analysis and Hamilton–Jacobi equations on Riemannian manifolds. J. Funct. Anal. 220(2), 304–361 (2005)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Ledyaev, Y., Zhu, Q.: Nonsmooth analysis on smooth manifolds. Trans. Am. Math. Soc. 359(8), 3687–732 (2007)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Azagra, D., Ferrera, J., Sanz, B.: Viscosity solutions to second order partial differential equations on Riemannian manifolds. J. Differ. Equ. 245(2), 307–36 (2008)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Crandall, M.G., Lions, P.L.: Viscosity solutions of Hamilton–Jacobi equations. Trans. Am. Math. Soc. 77(1), 1–42 (1983)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kurzhanski, A.B., Filippova, T.F.: On the Theory of Trajectory Tubes–A Mathematical Formalism for Uncertain Dynamics, Viability and Control. Advances in Nonlinear Dynamics and Control, pp. 122–188. Birkhauser, Boston (1993)MATHGoogle Scholar
  15. 15.
    Kurzhanski, A.B., Varaiya, P.: Dynamic optimization for reachability problems. J. Optim. Theory Appl. 108(2), 227–251 (2001)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Crandall, M.G., Lions, P.L.: Two approximations of solutions of Hamilton–Jacobi equations. Math. Comput. 43(167), 1–19 (1984)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Souganidis, P.E.: Approximation schemes for viscosity solutions of Hamilton–Jacobi equations. J. Differ. Equ. 59(1), 1–43 (1985)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Wang, S., Gao, F., Teo, K.L.: An upwind finite-difference method for the approximation of viscosity solutions to Hamilton–Jacobi–Bellman equations. IMA J. Math. Control Inf. 17(2), 167–178 (2000)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Sinyakov, V.V.: Method for computing exterior and interior approximations to the reachability sets of bilinear differential systems. Differ. Equ. 51(8), 1097–1111 (2015)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Filippova, T.F., Matviychuk, O.G.: Estimates of reachable sets of control systems with bilinear–quadratic nonlinearities. Ural Math. J. 1(1), 45–54 (2015)CrossRefGoogle Scholar
  21. 21.
    Sethian, J.A.: Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge University Press, Cambridge (1999)MATHGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Science, Research Centre for Toxic Compounds in the Environment RECETOX, Loschmidt LaboratoriesMasaryk UniversityBrnoCzech Republic

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