Method for automatic costate calculation

Abstract

A method for the automatic calculation of costates using only the results obtained from direct optimization techniques is presented. The approach exploits the relation between the time-varying costates and certain sensitivities of the variational cost function, a relation that also exists between the Lagrangian multipliers obtained from a direct optimization approach and the sensitivities of the associated nonlinear-programming cost function. The complete theory for treating free, control-constrained, interior-point-constrained, and state-constrained optimal control problems is presented. As a numerical example, a state-constrained version of the brachistochrone problem is solved and the results are compared to the optimal solution obtained from Pontryagin's minimum principle. The agreement is found to be excellent. Nomenclature / = right-hand side of state equations ge = control equality constraints gi = control inequality constraints he = state equality constraints hi = state inequality constraints J = cost function M = interior-point constraints m = dimension of control vector u N = total number of nodes minus 1 = total number of subintervals n — dimension of state vector x PWC = set of piecewise continuous functions t = time tf = final time ti = nodes along the time axis to = initial time u = control vector x = state vector Xf = final state Xi = state vector at node £/ Xo = initial state \(t) = costate A/ = Lagrangian multiplier associated with differential constraints along subinterval / Hi - Lagrangian multiplier associated with state constraints at node i (Ti = Lagrangian multiplier associated with control constraints along subinterval / <£ = cost function if} j. = boundary conditions at final time •00 = boundary conditions at initial time