Consider again the dynamics

In this section we discuss another variant problem, one for which the initial position is constrained to lie in a given set X₀ ⊂ Rⁿ and the final position is also constrained to lie within a given set X₁ ⊂ Rⁿ.

So in this model we get to choose the starting point x⁰ ∈ X⁰ in order tomaximize

where τ = τ [α(.)] is the first time we hit X₁.

NOTATION. We will assume that X0,X1 are in fact smooth surfaces in Rⁿ.

We let T₀ denote the tangent plane to X₀ at x⁰, and T₁ the tangent plane to X₁ at x¹.

THEOREM 1.1 (MORE TRANSVERSALITY CONDITIONS). Let α^∗(.) and x^∗(.) solve the problem above, with

                        x⁰= x^∗ (0), x¹= x^∗ (τ ^∗).

Then there exists a function p^∗(.) : [0, τ ^∗] → Rⁿ, such that (ODE), (ADJ) and  (M) hold for 0 ≤ t ≤ τ ^∗. In addition,

We call (T) the transversality conditions.

REMARKS AND INTERPRETATIONS. (i) If we have T > 0 fixed and

in agreement with our earlier form of the terminal/transversality condition.

(ii) Suppose that the surface X₁ is the graph X₁ = {x | g_k(x) = 0, k = 1, . . . , l}.

Then (T) says that p^∗(τ ^∗) belongs to the “orthogonal complement” of the subspace T₁. But orthogonal complement of T₁ is the span of ∇_gk(x1) (k = 1, . . . , l). Thus

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