Minimal time function and viscosity solutions

  • V. Staicu
Contributed Papers

Abstract

Two theorems in Ref. 1 are generalized. It is proved that, ifV(A,Γ) is the set of points that can be steered to the origin along a solution of the control systemx′=Ax−c, ifc(t)∈Γ, Γ is a compact subset ofR n , 0∈ intrelco Γ, and if a rank condition holds, then the minimal time functionT(·) is a viscosity solution of the Bellman equation
$$\max \{ \left\langle {DT(x),\gamma - Ax} \right\rangle :\gamma \varepsilon co\Gamma \} - 1 = 0,x\varepsilon V(A,\Gamma )\backslash \{ 0\} ,$$
and of the Hàjek equation
$$1 - \max \{ \left\langle {DT(x),\exp [ - AT(x)]} \right\rangle :\gamma \varepsilon co\Gamma \} = 0,x\varepsilon V(A,\Gamma ).$$

Key Words

Minimal time function Bellman equation Hàjek equation viscosity solutions linear control system 

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References

  1. 1.
    Hàjek, O.,On Differentiability of Minimal Time Function, Funkcialaj Ekvacioj, Vol. 20, pp. 97–114, 1976.Google Scholar
  2. 2.
    Crandall, M. G., andLions, P. L.,Viscosity Solutions of Hamilton-Jacobi Equations, Transactions of the American Mathematical Society, Vol. 277, pp. 1–42, 1983.Google Scholar
  3. 3.
    Crandall, M. G., Evans, L. C., andLions, P. L.,Some Properties of Viscosity Solutions of Hamilton-Jacobi Equations, Transactions of the American Mathematical Society, Vol. 282, pp. 1–42, 1984.Google Scholar
  4. 4.
    Lions, P. L.,Equations de Hamilton-Jacobi et Solutions de Viscosité, Ennio De Giorgi Colloquium, Edited by P. Krée, Pitman, Boston, Massachusetts, 1985.Google Scholar
  5. 5.
    Lions, P. L.,Generalized Solutions of Hamilton-Jacobi Equations, Pitman, Boston, Massachusetts, 1982.Google Scholar
  6. 6.
    Crandall, M. G., andLions, P. L.,On Existence and Uniqueness of Solutions of Hamilton-Jacobi Equations, Nonlinear Analysis, Theory, Methods and Applications, Vol. 10, pp. 353–370, 1984.Google Scholar
  7. 7.
    Ishii, H.,Existence and Uniqueness of Solutions of Hamilton-Jacobi Equations, Funkcialaj Ekvacioj, Vol. 29, pp. 267–388, 1986.Google Scholar
  8. 8.
    Mignanego, F., andPieri, G.,On a Generalized Bellman Equation for the Optimal Time Problem, System and Control Letters, Vol. 3, pp. 235–241, 1983.Google Scholar
  9. 9.
    Mignanego, F., andPieri, G.,On the Sufficiency of the Hamilton-Jacobi-Bellman for the Optimality in a Linear Optimal Time Problem. System and Control Letters, Vol. 6, pp. 357–363, 1986.Google Scholar
  10. 10.
    Mirica, S., Staicu, V., andAngelescu, N.,Equivalent Definitions and Basic Properties of Fréchet Semidifferentials, SIAM Journal on Control and Optimization (to appear).Google Scholar
  11. 11.
    Athans, M., andFalb, P. L.,Optimal Control, McGraw-Hill, New York, New York, 1966.Google Scholar
  12. 12.
    Conti, R.,Processi di Controllo Lineari in R n, Pitagora Editrice, Bologna, Italy, 1985.Google Scholar
  13. 13.
    Staicu, V.,Uniqueness of Minimal Time Function as Viscosity Solution of Bellman Equation (to appear).Google Scholar

Copyright information

© Plenum Publishing Corporation 1989

Authors and Affiliations

  • V. Staicu
    • 1
  1. 1.International School for Advanced Studies (SISSA)TriesteItaly

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