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The value function of an asymptotic exit-time optimal control problem

  • M. Motta
  • C. SartoriEmail author
Article

Abstract

We consider a class of exit-time control problems for nonlinear systems with a nonnegative vanishing Lagrangian. In general, the associated PDE may have multiple solutions, and known regularity and stability properties do not hold. In this paper we obtain such properties and a uniqueness result under some explicit sufficient conditions. We briefly investigate also the infinite horizon problem.

Mathematics Subject Classification (2000)

49J15 93C10 49L20 49L25 93D20 

Keywords

Optimal control Exit-time problems Viscosity solutions Asymptotic controllability 

References

  1. 1.
    Aubin, J.P., Frankowska, H.: Set Valued Analysis. Birkhäuser, Boston (1992)Google Scholar
  2. 2.
    Bacciotti, A., Rosier, L.: Liapunov Functions and Stability in Control Theory. Communications and Control Engineering Series, 2nd edn. Springer, Berlin (2005)Google Scholar
  3. 3.
    Bardi, M., Capuzzo Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. Birkhäuser, Boston (1997)Google Scholar
  4. 4.
    Cannarsa P., Da Prato G.: Nonlinear optimal control with infinite horizon for distributed parameter systems and stationary Hamilton–Jacobi equations. SIAM J. Control Optim. 27, 861–875 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Camilli F., Siconolfi A.: Maximal subsolution for a class of degenerate Hamilton–Jacobi problems. Indiana Univ. Math. J. 48, 1111–1131 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Cannarsa P., Sinestrari C.: Convexity properties of the minimum time function. J. Calc. Var. Partial Differ. Equ. 3, 273–298 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Da Lio F.: On the Bellman equation for infinite horizon problems with unbounded cost functional. Appl. Math. Optim. 41(2), 171–197 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Goebel R.: Convex optimal control problems with smooth Hamiltonians. SIAM J. Control Optim. 43(5), 1787–1811 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Guerra M., Sarychev A.: Measuring singularity of generalized minimizers for control-affine problems. J. Dyn. Control Syst. 15(2), 177–221 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Ishii H., Ramaswamy M.: Uniqueness results for a class of Hamilton–Jacobi equations with singular coefficients. Commun. Partial Diff. Equ. 20, 2187–2213 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Malisoff M.: Further results on the Bellman equation for optimal control problems with exit times and nonnegative Lagrangians. Syst. Control Lett. 50(1), 65–79 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Malisoff M.: Bounded-from-below solutions of the Hamilton–Jacobi equation for optimal control problems with exit times: vanishing Lagrangians, eikonal equations, and shape-from-shading. NoDEA Nonlinear Differ. Equ. Appl. 11(1), 95–122 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Motta M.: Viscosity solutions of HJB equations with unbounded data and characteristic points. Appl. Math. Optim. 49(1), 1–26 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Motta M., Rampazzo F.: Asymptotic controllability and optimal control. J. Differ. Equ. 254(7), 274–2763 (2013)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Motta, M., Sartori, C.: On asymptotic exit-time control problems lacking coercivity. ESAIM (2014). doi: 10.1051/cocv/2014003
  16. 16.
    Soravia P.: Pursuit-evasion problems and viscosity solutions of Isaacs equations. SIAM J. Control Optim. 1(3), 604–623 (1993)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Soravia, P.: Optimality principles and representation formulas for viscosity solutions of Hamilton–Jacobi equations I: Equations of unbounded and degenerate control problems without uniqueness. Differ. Integral Equ. 12(2), 275–293 (1999)Google Scholar
  18. 18.
    Soravia P.: Boundary value problems for Hamilton–Jacobi equations with a discontinuous Lagrangian. Indiana Univ. Math. J. 51(2), 451–477 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Trélat, E., Zuazua, E.: The turnpike property in finite-dimensional nonlinear optimal control. arXiv:1402.3263 (2014)
  20. 20.
    Zaslavski, A.J.: Turnpike Properties in the Calculus of Variations and Optimal Control. Nonconvex Optimization and Its Applications, vol. 80. Springer, New York (2006)Google Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Dipartimento di MatematicaPaduaItaly

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