Optimization and Flow-Invariance via High Order Tangent Cones

The goals of this dissertation are: 1) to present some results on the flow-invariance of a closed set S of a Banach space with respect to a differential equation, and to discuss optimization problems on S, as well; 2) to point out their unifying effect in the theory of differential equations and optimization.

For the following optimization problem, one establishes necessary conditions of extremum in terms of the high order tangential directions to the constraint set at the extremum point:

F(x₀)=Local Minimum F(x), subject to x ∈ S,

where X is a normed space, F: X→ ℝ is a function of class C^p in a neighborhood of x₀ ∈  S ⊆ X, S≠∅, p≥ 1.

It is analyzed in detail the case when S is the kernel D_G of a function G : X→ℝ^m, m≥ 1. To this aim, one describes the high order tangent cones to the set D_Gat x ∈ D_G, and then derives some sufficient conditions for the optimality of F on D_G.

The characterizations of the high order tangent cones are also used to obtain some necessary and sufficient conditions for the flow-invariance of a subset D_G= {x ∈ X; G (x)= 0}, of a Banach space X with respect to the differential equation u⁽ⁿ⁾(t) = F (u(t)),t ≥ 0, where G : U→ ℝ^m, m≥ 1, is a n-times Fréchet differentiable mapping on an open subset U of X, n ≥ 3, and F : U → X is locally Lipschitz.

The examples discussed illustrate some applications of the results presented.