Stratified Universal Manifolds and Turnpike Theorems for a Class of Optimal Control Problems

  • L. F. Zelikina
Part of the Lecture Notes in Computer Science book series (LNCIS)

Abstract

Consider the following optimal control problem
$$\overset{\centerdot }{\mathop{x}}\,=uQ(x)$$
(1)
where x ∈ R + n , i.e. x = (x 1 , ..., x n ), x i < 0; u- is n-dimensional control vector belonging to the simplex: u i ≥ 0, \( \sum\limits_{i = 1}^n {{u_i} = 1} \) for any 1 ≤ i ≤ n. Q is a scalar function: Q(x) ≠ 0 in R + n , Q ∈ C 2 (R + n ). Our goal is to minimize the functional
$$ R(x(T),T) $$
(2)
where T is the smallest value of t such that (x(t),t) ∈ M, M being a given set: M (x(t)), t) = 0 (called the terminal set). Here R and M are scalar functions: R, M ∈ C1(R + n + 1 ). For example, in the time optimal problem of transferring the point from the state (x 0, t 0) to the terminal set M(x) = 0 we have to minimize R(x(T), T) = T - t 0

Keywords

Manifold Univer 

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References

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    L. S. Pontrjagin, V. G. Boltjanskiī, R. V. Gamkrelidze and E. F. Misčenko, The mathematical theory of optimal processes, Fizmatgiz, Moscow, 1961; English transl., Wiley, New York, 1962, and Macmillan, New York, 1964.Google Scholar
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Copyright information

© Springer-Verlag Berlin Heidelberg 1975

Authors and Affiliations

  • L. F. Zelikina
    • 1
  1. 1.Central Economical Mathematical InstituteUSSR Academy of SciencesMoscowUSSR

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