The goal here is to give a simple approach to a quadrature formula based on the divided diffierences of the integrand at the zeros of the nth Chebyshev polynomial of the first kind,and those of the(n-1)st Chebyshev po...The goal here is to give a simple approach to a quadrature formula based on the divided diffierences of the integrand at the zeros of the nth Chebyshev polynomial of the first kind,and those of the(n-1)st Chebyshev polynomial of the second kind.Explicit expressions for the corresponding coefficients of the quadrature rule are also found after expansions of the divided diffierences,which was proposed in[14].展开更多
Let X<sub>n</sub>={x<sub>nk</sub>}<sub>k</sub><sup>n</sup>=1 be a set of real numbers satisfying -1【x<sub>nn</sub>【…【x<sub>n1</sub>【1, q’, ...Let X<sub>n</sub>={x<sub>nk</sub>}<sub>k</sub><sup>n</sup>=1 be a set of real numbers satisfying -1【x<sub>nn</sub>【…【x<sub>n1</sub>【1, q’, q: 0≤q’≤q be two integers. For a non-negative integer r, we denote by C<sub>[</sub>-1,1]<sup>r</sup> the set of all the rth continuously differentiable real-valued functions on [-1, 1], and by ∏<sub>r</sub> the set of the polynomials of degree at most r. For a given f ∈C<sub>[</sub>-1,1]<sup>q’</sup>, we call the only S<sub>N</sub>（f, x）∈∏<sub>N</sub> （N = （q+1） n-1）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
基金Supported by the National Natural Science Foundation of China（10571121） Supported by the Natural Science Foundation of Guangdong Province（5010509）
文摘The goal here is to give a simple approach to a quadrature formula based on the divided diffierences of the integrand at the zeros of the nth Chebyshev polynomial of the first kind,and those of the(n-1)st Chebyshev polynomial of the second kind.Explicit expressions for the corresponding coefficients of the quadrature rule are also found after expansions of the divided diffierences,which was proposed in[14].
文摘Let X<sub>n</sub>={x<sub>nk</sub>}<sub>k</sub><sup>n</sup>=1 be a set of real numbers satisfying -1【x<sub>nn</sub>【…【x<sub>n1</sub>【1, q’, q: 0≤q’≤q be two integers. For a non-negative integer r, we denote by C<sub>[</sub>-1,1]<sup>r</sup> the set of all the rth continuously differentiable real-valued functions on [-1, 1], and by ∏<sub>r</sub> the set of the polynomials of degree at most r. For a given f ∈C<sub>[</sub>-1,1]<sup>q’</sup>, we call the only S<sub>N</sub>（f, x）∈∏<sub>N</sub> （N = （q+1） n-1）
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.
