Discrete Pseudocontrol Sets for Optimal Control Problems

A method of optimal control problem solutions based on a new concept of pseudocontrol sets is presented. This approach combines large-scale linear programming algorithms with the well-known discretization of the continuous system dynamics on small segments and uses discrete pseudocontrol sets, which are considered independently for each segment. Every set is expressed as a mesh approximation of an admissible control space. The method is associated with significant increases in the number of decision variables and requires introducing artificial variables or pseudovariables. Terminal conditions are presented as a linear matrix equation. An extension of the matrix equation for the sums of the pseudocontrols is used to transform the problem into a linear programming form. Interior-point inequality constraints are represented as a linear matrix inequality. The resulting linear programming form is characterized by matrices that are very large and sparse. The number of decision variables is on the order of tens of thousands. In modern linear programming, there are effective interior-point algorithms to solve such problems. A minimum path-planning problem with nonlinear constraints, reentry trajectory optimization with maximum cross range, and maximum-radius orbit transfer are considered as application examples. The results of the last example are almost coincident with known solutions using other methods.