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

Optimal Control Problem Optimal Synthesis Time Optimal Control Switching Surface Universal Structure
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.

References

1. [1]
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
2. [2]
Isaacs, R., Differential games, Wiley, New-York, 1965, Russian transl., Mir, Moscow, 1967.
3. [3]
Thorn, R., Local topological properties of differentiate mappings, Differential Analysis (Papers presented at the Bombay Colloquium, 1964).Google Scholar
4. [4]
Картан Э. Внешние дифференциальные системы и иҳ геометрические приложения. Издательство Московского Университета, 1962.Google Scholar