 Award ID(s):
 1819002
 NSFPAR ID:
 10282084
 Date Published:
 Journal Name:
 ACM Transactions on Mathematical Software
 Volume:
 46
 Issue:
 3
 ISSN:
 00983500
 Page Range / eLocation ID:
 1 to 38
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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A new method is developed for solving optimal control problems whose solutions contain a nonsmooth optimal control. The method developed in this paper employs a modified form of the LegendreGaussRadau (LGR) orthogonal direct collocation method in which an additional variable and two additional constraints are included at the end of a mesh interval. The additional variable is the switch time where a discontinuity occurs. The two additional constraints are a collocation condition on each differential equation that is a function of control along with a control constraint at the endpoint of the mesh interval that defines the location of the nonsmoothness. These additional constraints modify the search space of the NLP in a manner such that an accurate approximation to the location of the nonsmoothness is obtained. An example with a nonsmooth solution is used throughout the paper to illustrate the improvement of the method over the standard LegendreGaussRadau collocation method.more » « less

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Abstract A mesh refinement method is described for solving optimal control problems using Legendre‐Gauss‐Radau collocation. The method detects discontinuities in the control solution by employing an edge detection scheme based on jump function approximations. When discontinuities are identified, the mesh is refined with a targeted
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null (Ed.)A 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.more » « less

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