Let P_n(z) be the Legendre polynomial satisfing P_n(1)=1. For a function f(z) we denote by L_n(f;z) the Lagrange interpolation polynomial based on the zeros of the P_n(z), we denote by G_d the elliptic region with foc...Let P_n(z) be the Legendre polynomial satisfing P_n(1)=1. For a function f(z) we denote by L_n(f;z) the Lagrange interpolation polynomial based on the zeros of the P_n(z), we denote by G_d the elliptic region with foci at—1, 1, where d is sum of its semi-axes. Theorem 1. If f(z) is analytic in G_d and continuous on G_d then where a, b are semimajor and semiminor axis of the ellipse G_d respectively. By P_n(z) denote the Legendre polynomial whose highest coefficient is 1. Let A be any fixed posititve number, By L_n^((A))(f;z) and L_n^((A))(f;z) denote interpolation polynomial based on the zeros of the P_n(z)-A and the P_n(z)-A respectively. Theorem2. If f(z) is analytic in G_d, then when d>2, where d^2/2 cannot be improved. If f(z) is analytic in G_d and continuous on G_d(d>2), then If f(z) is analytic in G_d and continuous on G_d, with modulus of continuity ω(f;δ)=o(δ^(1/2), then展开更多
In this paper, a numerical method for solving the optimal control （OC） problems is presented. The method is enlightened by the Chebyshev-Legendre （CL） method for solving the partial differential equations （PDEs）...In this paper, a numerical method for solving the optimal control （OC） problems is presented. The method is enlightened by the Chebyshev-Legendre （CL） method for solving the partial differential equations （PDEs）. The Legendre expansions are used to approximate both the control and the state functions. The constraints are discretized over the Chebyshev-Gauss-Lobatto （CGL） collocation points. A Legendre technique is used to approximate the integral involved in the performance index. The OC problem is changed into an equivalent nonlinear programming problem which is directly solved. The fast Legendre transform is employed to reduce the computation time. Several further illustrative examples demonstrate the efficiency of the proposed method.展开更多
文摘Let P_n(z) be the Legendre polynomial satisfing P_n(1)=1. For a function f(z) we denote by L_n(f;z) the Lagrange interpolation polynomial based on the zeros of the P_n(z), we denote by G_d the elliptic region with foci at—1, 1, where d is sum of its semi-axes. Theorem 1. If f(z) is analytic in G_d and continuous on G_d then where a, b are semimajor and semiminor axis of the ellipse G_d respectively. By P_n(z) denote the Legendre polynomial whose highest coefficient is 1. Let A be any fixed posititve number, By L_n^((A))(f;z) and L_n^((A))(f;z) denote interpolation polynomial based on the zeros of the P_n(z)-A and the P_n(z)-A respectively. Theorem2. If f(z) is analytic in G_d, then when d>2, where d^2/2 cannot be improved. If f(z) is analytic in G_d and continuous on G_d(d>2), then If f(z) is analytic in G_d and continuous on G_d, with modulus of continuity ω(f;δ)=o(δ^(1/2), then
基金supported by the National Natural Science Foundation of China (Grant Nos.10471089,60874039)the Shanghai Leading Academic Discipline Project (Grant No.J50101)
文摘In this paper, a numerical method for solving the optimal control （OC） problems is presented. The method is enlightened by the Chebyshev-Legendre （CL） method for solving the partial differential equations （PDEs）. The Legendre expansions are used to approximate both the control and the state functions. The constraints are discretized over the Chebyshev-Gauss-Lobatto （CGL） collocation points. A Legendre technique is used to approximate the integral involved in the performance index. The OC problem is changed into an equivalent nonlinear programming problem which is directly solved. The fast Legendre transform is employed to reduce the computation time. Several further illustrative examples demonstrate the efficiency of the proposed method.
