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OPTIMAL CONTROL THEORY
TIME OPTIMAL CONTROL OF SECOND ORDER SYSTEMS
TIME OPTIMAL CONTROL OF A DUFFING PLANT
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adjoint equations algebraic approximate switching surface Assume the existence bang-bang control boundary conditions calculus of variations constant control control constraint control history coordinates different approximations Duffing equation e2 approximations equal to zero equation II.2 equations III.38 error in excess evaluate the constants exact switching surface excess of true expansion for q Figure function Hamiltonian higher order segments initial conditions isochrone known point x,y magnitude method n/gj n/gq negative time integration nonlinear system optimal control problem optimal switching surface optimal trajectory order e approximation order e1 equation ordinary differential equations partial differential equations Percentage error phase plane point on W2 Pol plants Pontryagin Pontryagin's maximum principle series solutions set equal set of initial Setting the coefficient solution for xq transformation of derivatives true optimal uniformly valid expansion variable expansion procedure vector W1 segments x-axis approximation zero gives zeroes of q