On Two-Point Boundary Conditions in Optimal Control Problems

Abstract

Let x=g(t,x(t),u(t)) be the governing equation of an optimal control problem with two-point boundary conditions h ₀(x(a))+h ₁(x(b)) = 0, where x: [a,b] → ℝⁿ is continuous, u: [a,b] → ℝ^k-n is piecewise continuous and left continuous, h₀,h₁: ℝⁿ → ℝ^q are continuously differentiable, and g:[a,b]× ℝ^k → ℝⁿ is continuous. The paper finds functions ξ _i ∈ C¹([a,b]× ℝⁿ) such that (x(t),u(t)) is a solution of the governing equation if and only if

$$\int {_a^b [(\partial \xi _i /\partial x)g + \partial \xi _i /\partial t]} dt = 0, i = 1,2,3,....$$