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12), when one end of the curve is fixed, say x(ti ) = xi , and the other end is free. Then it can be shown that the curve x satisfies the Euler-Lagrange equation, the transversality condition ∂F (x∗ (t), x∗ (t), t) ∂x h(ti ) = 0 t=ti at the free end point, and x(ti ) = xi serves as the other boundary condition. 14) and moreover there holds ∂F (x∗ (t), x∗ (t), t) = 0 at the free end point. ∂x Exercises. 1. Find the curve y = y(x) which has minimum length between (0, 0) and the line x = 1. 2. Find critical curves for the following functionals: (a) I(x) = (b) I(x) = (c) I(x) = π 2 0 π 2 0 1 0 (x(t))2 − (x (t))2 dt, x(0) = 0 and x (x(t))2 − (x (t))2 dt, x(0) = 1 and x cos(x (t))dt, x(0) = 0 and x(1) is free.

Let F (x, u, t) be a continuously differentiable function of each of their arguments. If u∗ ∈ (C[ti , tf ])m is an optimal control for the functional tf Ixi (u) = F (x(t), u(t), t)dt, ti subject to the differential equation x(t) ˙ = Ax(t) + Bu(t), t ∈ [ti , tf ], x(ti ) = xi , x(tf )k = xf,k , k ∈ {1, . . , r}, and if x∗ denotes the corresponding state, then there exists a p∗ ∈ (C 1 [ti , tf ])n such that ∂H (p∗ (t), x∗ (t), u∗ (t), t) = ∂x ∂H (p∗ (t), x∗ (t), u∗ (t), t) = ∂u −p˙ ∗ (t), t ∈ [ti , tf ], p∗ (tf )k = 0, k ∈ {r + 1, .

What if x(0) = 0 and x(1) = 2? 1 0 1 0 (t))2 dt where x(1) = 5 and x(2) = 2. where x(1) = 1 and x(2) = 7. 2tx(t) − (x (t))2 + 3x (t)(x(t))2 dt where 2(x(t))3 + 3t2 x (t) dt where x(0) = 0 and 7. A strip-mining company intends to remove all of the iron ore from a region that contains an estimated Q tons over a fixed time interval [0, T ]. As it is extracted, they will sell it for processing at a net price per ton of p(x(t), x (t)) = P − αx(t) − βx (t) for positive constants P , α, and β, where x(t) denotes the total tonnage sold by time t.

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