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Named after I. Michael Ross and F. Fahroo, the Ross–Fahroo lemma is a fundamental result in optimal control theory.〔 I. M. Ross and F. Fahroo, A Pseudospectral Transformation of the Covectors of Optimal Control Systems, Proceedings of the First IFAC Symposium on System Structure and Control, Prague, Czech Republic, 29–31 August 2001.〕 〔 I. M. Ross and F. Fahroo, Discrete Verification of Necessary Conditions for Switched Nonlinear Optimal Control Systems, ''Proceedings of the American Control Conference, Invited Paper'', June 2004, Boston, MA.〕〔N. Bedrossian, M. Karpenko, and S. Bhatt, "Overclock My Satellite: Sophisticated Algorithms Boost Satellite Performance on the Cheap", ''IEEE Spectrum'', November 2012.〕 It states that dualization and discretization are, in general, non-commutative operations. The operations can be made commutative by an application of the covector mapping principle. ==Description of the theory== A continuous-time optimal control problem is information rich. A number of interesting properties of a given problem can be derived by applying the Pontryagin's minimum principle or the Hamilton–Jacobi–Bellman equations. These theories implicitly use the continuity of time in their derivation.〔B. S. Mordukhovich, Variational Analysis and Generalized Differentiation: Basic Theory, Vol.330 of Grundlehren der Mathematischen Wissenschaften (Principles of Mathematical Sciences ) Series, Springer, Berlin, 2005.〕 When an optimal control problem is discretized, the Ross–Fahroo lemma asserts that there is a fundamental loss of information. This loss of information can be in the primal variables as in the value of the control at one or both of the boundary points〔 F. Fahroo and I. M. Ross, Pseudospectral Methods for Infinite Horizon Nonlinear Optimal Control Problems, AIAA Guidance, Navigation and Control Conference, August 15–18, 2005, San Francisco, CA.〕 or in the dual variables as in the value of the Hamiltonian over the time horizon. To address the information loss, Ross and Fahroo introduced the concept of closure conditions which allow the known information loss to be put back in. This is done by an application of the covector mapping principle.〔 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「Ross–Fahroo lemma」の詳細全文を読む スポンサード リンク
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