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adjoint application approximation boundary conditions boundary terms calculus of variations Chapter coefficients control problem cost functional diffusion equation distributed distributed-parameter systems dynamic programming equa Erzberger and Kim estimate Evaluation Example 4.3 extremum feedback gain formulation ft ft ft func Gateaux derivative given gives Green's function grid points Helmholtz Equation initial integral interval iteration Lagrange multiplier linear lumped-parameter systems maximum principle minimization modal representation Mode Number obtained operator optimal control optimal control strategies optimal solution Optimal Trajectory optimization problem polynomial procedure quadratic quadratic approximation reciprocal methods rH rH Riccati equation scalar product Section sensor locations shown in equation shown in Figure side of equation subroutine Substitution from equation system dynamics Taylor series techniques thesis tion transformation trial function trial variable UJ UJ vector