Proximal Analysis and the Minimal Time Function of a Class of Semilinear Control Systems

  • Yi Jiang
  • Yiran He
  • Jie Sun


The minimal time function of a class of semilinear control systems is considered in Banach spaces, with the target set being a closed ball. It is shown that the minimal time functions of the Yosida approximation equations converge to the minimal time function of the semilinear control system. Complete characterization is established for the subdifferential of the minimal time function satisfying the Hamilton–Jacobi–Bellman equation. These results extend the theory of finite dimensional linear control systems to infinite dimensional semilinear control systems.


Hamilton–Jacobi–Bellman equation Minimal time function Subdifferential Time optimal control 

Mathematics Subject Classification

93C23 90C48 49J52 



This work was partially supported by National Natural Science Foundation of China (11201324), Fok Ying Tuny Education Foundation (141114), Outstanding Youth Academic Technology Leader Training Plan of Sichuan province (2013JQ0027), the Provost’s Chair Fund of National University of Singapore, and a research grant from Faculty of Science and Engineering, Curtin University.


  1. 1.
    Bardi, M., Cannatsa-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. Birkh\(\ddot{a}\)user Boston, Cambridge (1997)Google Scholar
  2. 2.
    Fattorini, H.O.: Infinite Dimensional Optimization and Control Theory. Cambridge University Press, New York (1996)MATHGoogle Scholar
  3. 3.
    Li, X.J., Yong, J.M.: Optimal Control Theory for Infinite Dimensional Systems. Birkh\(\ddot{a}\)user Boston, Cambridge (1995)Google Scholar
  4. 4.
    Bardi, M.: A boundary value problem for the minimal time problem. SIAM J. Control Optim. 27, 776–785 (1989)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Soravia, P.: Discontinuous viscosity solutions to Dirichlet problems for Hamilton–Jacobi equations with convex Hamiltonians. Commun. Partial Differ. Equ. 18, 1493–1514 (1993)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cannarsa, P., Cârjǎ, O.: On the Bellman equation for the minimum time problem in infinite dimensions. SIAM J. Control Optim. 43, 532–548 (2004)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Wolenski, P.R., Yu, Z.: Proximal analysis and the minimal time function. SIAM J. Control Optim. 36, 1048–1072 (1998)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Colombo, G., Wolenski, P.R.: The subgradient formula for the minimal time function in the case of constant dynamics in Hilbert space. J. Glob. Optim. 28, 269–282 (2004)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Colombo, G., Wolenski, P.R.: Variational analysis for a class of minimal time functions in Hilbert spaces. J. Convex Anal. 11, 335–361 (2004)MathSciNetMATHGoogle Scholar
  10. 10.
    He, Y.R., Ng, K.F.: Subdifferentials of a minimum time function in Banach spaces. J. Math. Anal. Appl. 321, 896–910 (2006)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Jiang, Y., He, Y.R.: Subdifferentials of a minimum time function in normed spaces. J. Math. Anal. Appl. 358, 410–418 (2009)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Jiang, Y., He, Y.R.: Subdifferential properties for a class of minimal time functions with moving target sets in normed spaces. Appl. Anal. 91, 491–502 (2012)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Mordukhovich, B.S., Nguyen, M.N.: Limiting subgradients of minimal time functions in Banach spaces. J. Glob. Optim. 46, 615–633 (2009)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Mordukhovich, B.S., Nguyen, M.N.: Subgradients of minimal time functions under minimal requirements. J. Convex Anal. 18, 915–947 (2011)MathSciNetMATHGoogle Scholar
  15. 15.
    Jiang, Y., He, Y.R., Sun, J.: Subdifferential properties of the minimal time function of linear control systems. J. Glob. Optim. 51, 395–412 (2011)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Albano, P., Cannarsa, P., Sinestrari, C.: Regularity results for the minimum time function of a class of semilinear evolution equations of parabolic type. SIAM J. Control Optim. 38, 916–946 (2000)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Rudin, W.: Functional Analysis. McGraw-Hill Book Inc, New York (1991)MATHGoogle Scholar
  18. 18.
    Cârjǎ, O.: On the minimum time function and the minimum energy problem a nonlinear case. Syst. Control Lett. 55, 543–548 (2006)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)MATHGoogle Scholar
  20. 20.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I. Springer, Berlin (2006)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.VC/VR Lab and Department of MathematicsSichuan Normal UniversityChengduChina
  2. 2.Department of Decision Sciences, National University of Singapore and Department of Mathematics and StatisticsCurtin UniversityBentleyAustralia

Personalised recommendations