%AEide, Joseph%AHager, William%ARao, Anil%Anull Ed.%BJournal Name: Journal of Optimization Theory and Applications
%D2021%I
%JJournal Name: Journal of Optimization Theory and Applications
%K
%MOSTI ID: 10282141
%PMedium: X
%TModified Legendre–Gauss–Radau Collocation Method for Optimal Control Problems with Nonsmooth Solutions
%XA new method is developed for solving optimal control problems whose solutions
are nonsmooth. The method developed in this paper employs a modified form of
the Legendre–Gauss–Radau orthogonal direct collocation method. This modified
Legendre–Gauss–Radau method adds two variables and two constraints at the end
of a mesh interval when compared with a previously developed standard Legendre–
Gauss–Radau collocation method. The two additional variables are the time at the
interface between two mesh intervals and the control at the end of each mesh inter-
val. The two additional constraints are a collocation condition for those differential
equations that depend upon the control and an inequality constraint on the control
at the endpoint of each mesh interval. The additional constraints modify the search
space of the nonlinear programming problem such that an accurate approximation to
the location of the nonsmoothness is obtained. The transformed adjoint system of the
modified Legendre–Gauss–Radau method is then developed. Using this transformed
adjoint system, a method is developed to transform the Lagrange multipliers of the
nonlinear programming problem to the costate of the optimal control problem. Fur-
thermore, it is shown that the costate estimate satisfies one of the Weierstrass–Erdmann
optimality conditions. Finally, the method developed in this paper is demonstrated on
an example whose solution is nonsmooth.
%0Journal Article