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The Bellman pseudospectral method is a pseudospectral method for optimal control based on Bellman's principle of optimality. It is part of the larger theory of pseudospectral optimal control, a term coined by Ross. The method is named after Richard E. Bellman. It was introduced by Ross et al.〔I. M. Ross, Q. Gong and P. Sekhavat, The Bellman pseudospectral method, AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Honolulu, Hawaii, AIAA-2008-6448, August 18–21, 2008.〕 first as a means to solve multiscale optimal control problems, and later expanded to obtain suboptimal solutions for general optimal control problems. ==Theoretical foundations== The multiscale version of the Bellman pseudospectral mehod is based on the spectral convergence property of the Ross–Fahroo pseudospectral methods. That is, because the Ross–Fahroo pseudospectral method converges at an exponentially fast rate, pointwise convergence to a solution is obtained at very low number of nodes even when the solution has high-frequency components. This aliasing phenomenon in optimal control was first discovered by Ross et al.〔 Rather than use signal processing techniques to anti-alias the solution, Ross et al. proposed that Bellman's principle of optimality can be applied to the converged solution to extract information between the nodes. Because the Gauss–Lobatto nodes cluser at the boundary points, Ross et al. suggested that if the node density around the initial conditions satisfy the Nyquist–Shannon sampling theorem, then the complete solution can be recovered by solving the optimal control problem in a recursive fashion over piecewise segments known as Bellman segments.〔 In an expanded version of the method, Ross et al.,〔 proposed that method could also be used to generate feasible solutions that were not necessarily optimal. In this version, one can apply the Bellman pseudospectral method at even lower number of nodes even under the knowledge that the solution may not have converged to the optimal one. In this situation, one obtains a feasible solution. A remarkable feature of the Bellman pseudospectral method is that it automatically determines several measures of suboptimality based on the original pseudospectral cost and the cost generated by the sum of the Bellman segments.〔〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Bellman pseudospectral method」の詳細全文を読む スポンサード リンク
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