Robustness of Operational Matrices of Differentiation for Solving State-Space Analysis and Optimal Control Problems
نویسندگان
چکیده
and Applied Analysis 3 subject to the nonlinear system (8), for Q ∈ R, R ∈ Rm×m positive semidefinite and positive definite matrices, respectively. Since the performance index (9) is convex, the following extreme necessary conditions are also sufficient for optimality [28]: ?̇? = f (t, x) + g (t, x) u ∗ , ̇ λ = −H x (x, u ∗ , λ) , u ∗ = arg min u H(x, u, λ) , x (t 0 ) = x 0 , λ (t f ) = 0, (10) where H(x, u, λ) = (1/2)[xTQx + uTRu] + λT[f(t, x) + g(t, x)u] is referred to theHamiltonian. Equivalently, (10) can be written in the form of ?̇? = f (t, x) + g (t, x) [−R −1 g T (t, x) λ] ̇ λ = − (Qx + ( ∂f (t, x)
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