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Extra resources for Lectures on n-Dimensional Quasiconformal Mappings
Example text
72) The index a runs over the N initial and final, respectively, boundary conditions and α runs over the N components of the state vector. 71) can be explicitly written as P ∗ (0) = b (X ∗ (0))Λ and P ∗ (T ) = −b (X ∗ (T ))Λ . 73) These relations are usually called transversality conditions. 53) describing the evolution of the state variables. 63) ∂L ∂ ∂ = 0 or (F (X ∗ , u∗ , t) | P ∗ ) − φ(t, X ∗ , u∗ ) = 0 . 66). 55) for the state vector X. The complete set of equations defining the extremals of the general control problem consists of N first-order differential equations for the N components of the state vector and N first-order differential equations for the N components of the adjoint state.
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49) 0 ¨ The Euler–Lagrange equation now reads X(t) + ω 2 X(t) = 0. The extremal ∗ ∗ solution is X (t) = 0 for ωT < π and X (t) = X0 sin(ωt) for ωT = π. Since the Lagrangian is of the standard form L = T − U , the Weierstrass criterion suggests a strong minimum for these extremal solutions. We obtain for both types of extremals S[X ∗ , T ] = 0. 50) 0 ∗ show that S[X , T ] = 0 is in fact the lower limit of the action. 50) holds only for ωT ≤ π, because the expression X(t) tan−1 ωt has no relevant singularities as long as 0 ≤ ωT ≤ π.