THE PONTRYAGIN MAXIMUM PRINCIPLE-MAXIMUMPRINCIPLEWITH TRANSVERSALITY CONDITIONS |
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date: 9-10-2016
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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 | gk(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
for some unknown constants λ1, . . . , λl.
References
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