Mayer cost functional - the AOE home page

AOE 5244 - E.M. Cliff 4✬➤ Let˜x =⎛⎜⎝⎞x ⎟⎠ and ũ =˜x n+1⎛⎜⎝⎞u ⎟⎠ũ m+1✩➤ The new state satisfies thedifferential equation˙˜x n+1 =ũ m+1✫and the initial condition✪

AOE 5244 - E.M. Cliff 5✬✩˜x n+1 (0) = 0. Note that˜x n+1 (t 1 )=∫ t10ũ m+1 (t) dt➤ The new control is unrestrictedso we have˜Ω =Ω× [−∞, ∞]➤ We also impose the additional✫✪

AOE 5244 - E.M. Cliff 8✬✩Analyze the Transformed Problem➤ The transformed problem hasn + 1 state variables and hencen + 1 adjoint variables. Thevariational Hamiltonian can be✫✪

AOE 5244 - E.M. Cliff 9✬written as˜H(˜λ, ˜x, ũ) =H(λ, x, u) +(˜λ n+1 + λ 0 )ũ m+1➤ It’s clear that the adjointdifferential equations for thefirst n components of ˜λ areunaltered. The final adjoint✩✫✪

AOE 5244 - E.M. Cliff 10✬equation is✩˙˜λ n+1 (t) =0while the optimality equation is∂ ˜H=0=(˜λ n+1 + λ 0 )∂ũ m+1Hence, the new adjoint variable˜λ n+1 (t) is constant and equal to✫−λ 0 .✪

AOE 5244 - E.M. Cliff 11✬✩➤ The final transversalitycondition includes an additionaltermµ q+1⎛⎜⎝ −∇g1⎞⎟⎠Since this is the onlyend-condition that depends on✫✪

AOE 5244 - E.M. Cliff 13✬✩adds the termλ 0 (∇g) .➤ In summary, the terminaltransversality condition we hadearlier is replaced withCondition (c)’:There exists scalars✫✪