Direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using a Radau pseudospectral method

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A unified framework for the numerical solution of optimal control problems using pseudospectral methods (2010, methodology): “…methods using these sets of collocation points are the Legendre–Gauss–Lobatto pseudospectral method (Elnagar et al., 1995; Fahroo & Ross, 2001), the Legendre–Gauss pseudospectral method (Benson et al., 2006; Rao et al., 2010), and the Legendre–Gauss–Radau pseudospectral method (Garg et al., 2009).”
AN OVERVIEW OF THREE PSEUDOSPECTRAL METHODS FOR THE NUMERICAL SOLUTION OF OPTIMAL CONTROL PROBLEMS (2017, neutral): “LetΛ denote theN by n matrix of Lagrange multipliers associated with the system dynamics (8), let ΛN+1 denote the1 by n vector of multipliers associated with the equation (11) for XN+1, and letμ be the 1 by n row vector of Lagrange multipliers associated with the initial condition.”
Pseudospectral methods for solving infinite-horizon optimal control problems (2011, background): “Moreover, from the definition of λN+1, the following identity can be derived (see Garg et al., 2009, for the details): µ = λN+1 + N− j=1 wjTj∇xH(Xi,Ui, λi).”