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

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