Adjoint variable solutions via an auxiliary optimization problem

Abstract

The problem addressed is to obtain an initial guess for the adjoint variables along a trajectory suitable for use in an indirect optimal control method. The proposed method assumes that good estimates of the control and state histories are available. The adjoint estimates are obtained through an auxiliary unconstrained optimization problem of low dimension. The goal of this optimization problem is to find the adjoint values at nodes along the trajectory that cause the control values from the application of the minimum principle to approach the estimated control values as closely as possible in a manner consistent with the adjoint differential equations and transversality conditions. Experience has shown that the method produces adjoint variable estimates sufficiently accurate to cause convergence of a multiple-shooting indirect method. Nomenclature H = Hamiltonian u — p-dimensional control vector x - n -dimensional vector of state variables A = n -dimensional vector of adjoint variables \I> = m -dimensional vector of constraints at terminal time, m < n