A Compact Uncertainty Set

  • Vladimir G. Boltyanski
  • Alexander S. Poznyak
Part of the Systems & Control: Foundations & Applications book series (SCFA)


This chapter extends the possibilities of the MP approach for a class of Min-Max control problems for uncertain models given by a system of stochastic differential equations with a controlled diffusion term and unknown parameters within a given measurable compact set. For simplicity, we consider the Min-Max problem belonging to the class of optimization problems with a fixed finite horizon where the cost function contains only a terminal term (without an integral part). The proof is based on the Tent Method in a Banach space, discussed in detail in Part II; it permits us to formulate the necessary conditions of optimality in the Hamiltonian form.


Admissible Control Polar Cone Complementary Slackness Nontriviality Condition Stochastic Maximum Principle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Fleming, W.H., & Rishel, R.W. (1975), Optimal Deterministic and Stochastic Control, Applications of Mathematics, Springer, Berlin. CrossRefGoogle Scholar
  2. Kushner, H. (1972), ‘Necessary conditions for continuous parameter stochastic optimization problems’, SIAM J. Control Optim. 10, 550–565. MathSciNetCrossRefGoogle Scholar
  3. Neustadt, L. (1969), ‘A general theory of extremals’, J. Comput. Syst. Sci. 3(1), 57–92. MathSciNetCrossRefGoogle Scholar
  4. Pontryagin, L.S., Boltyansky, V.G., Gamkrelidze, R.V., & Mishenko, E.F. (1969), Mathematical Theory of Optimal Processes, Nauka, Moscow (in Russian). Google Scholar
  5. Poznyak, A.S. (2009), Advanced Mathematical Tools for Automatic Control Engineers, Vol. 2: Stochastic Technique, Elsevier, Amsterdam. Google Scholar
  6. Yong, J., & Zhou, X.Y. (1999), Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer, Berlin. CrossRefGoogle Scholar
  7. Yoshida, K. (1979), Functional Analysis, Narosa Publishing House, New Delhi. Google Scholar
  8. Zhou, X.Y. (1991), ‘A unified treatment of maximum principle and dynamic programming in stochastic controls’, Stoch. Stoch. Rep. 36, 137–161. MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Vladimir G. Boltyanski
    • 1
  • Alexander S. Poznyak
    • 2
  1. 1.CIMATGuanajuatoMexico
  2. 2.Automatic Control DepartmentCINVESTAV-IPNMéxicoMexico

Personalised recommendations