A Generalized Hamilton—Jacobi—Bellman Equation for Deterministic Optimal Control Problems

  • Leonard D. Berkovitz
Chapter
Part of the Systems & Control: Foundations & Applications book series (SCFA)

Abstract

We shall be concerned with the following problem. Minimize g(t 1, x 1) subject to
$$\frac{{dx\left( t \right)}}{{dt}} = f\left( {t,x\left( t \right),u\left( t \right)} \right) x\left( \tau \right) = \xi $$
(1)
$$u\left( t \right) \in Z {\text{a}}{\text{.e}}{\text{.}}$$
(2)
$$\left( {{t_1}{x_1}} \right) \in \mathcal{T}{\text{,}}$$
(3)
where t 1 is the first time that the solution x(·) of (1) hits a preassigned set T and x 1 = x(t 1).

Keywords

Hull Controle 

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References

  1. Aubin, J.P. and Cellina, A. (1984). Differential Inclusions, Springer-Verlag, Berlin.MATHCrossRefGoogle Scholar
  2. Berkovitz, L.D. (1970). A Hamilion-Jacobi theory for a class of control problems, Colloque sur la Théorie Mathématique du Contrôle Optimal, CBRM, Vander, Louvain, 1-23.Google Scholar
  3. Berkovitz, L.D. (1974). Optimal Control Theory, Springer-Verlag, New York.MATHCrossRefGoogle Scholar
  4. Berkovitz, L.D. (1989). Optimal feedback controls, SIAM Journal on Control and Optimization, 27, 991–1006.MathSciNetMATHCrossRefGoogle Scholar
  5. Bruckner, A. (1994). Differentiation of Real Functions, CRM Monograph Series, Vol. 5, American Mathematical Society, Providence, RI.Google Scholar
  6. Cesari, L. (1983). Optimization-Theory and Applications. Problems with Ordinary Differential Equations, Springer-Verlag, New York.MATHGoogle Scholar
  7. Clarke, F.H. (1983). Optimization and Nonsmooth Analysis. John Wiley, New York.MATHGoogle Scholar
  8. Crandall, M.G., Ishii, H., and Lions, P-L. (1992). User’s guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society, 27, 1–67.MathSciNetMATHCrossRefGoogle Scholar
  9. Fleming, W.H. and Rishel, R.W. (1975). Deterministic and Stochastic Optimal Control, Springer-Verlag, New York.MATHCrossRefGoogle Scholar
  10. Fleming, W.H. and Soner, H.M. (1993). Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, New York.MATHGoogle Scholar
  11. Frankowska, H. (1989). Optimal trajectories associated with a solution of the contingent Hamilton-Jacobi equation, Applied Mathematics and Optimization, 19, 291–311.MathSciNetMATHCrossRefGoogle Scholar
  12. Frankowska, H. (1993). Lower semi continuous solutions of Hamilton-Jacobi-Bellman equation, SIAM Journal on Control and Optimization, 31, 257–272.MathSciNetMATHCrossRefGoogle Scholar
  13. Mirica, S. (1990). Bellman-Isaacs equation in deterministic optimal control, Studii Si Cercetari Mathematice, 42, 437–447.MathSciNetMATHGoogle Scholar
  14. Rowland, J.D.L. and Vinter, R.B. (1991). Construction of optimal feedback controls, Systems and Control Letters, 16, 357–367.MathSciNetMATHCrossRefGoogle Scholar
  15. Vinter, R. (1994). Uniqueness of solutions to the Hamilton-Jacobi equation: a system theoretic proof, Systems and Control Letters, 32, 267–275.MathSciNetCrossRefGoogle Scholar

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© Springer Science+Business Media New York 1999

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  • Leonard D. Berkovitz

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