A semi-infinite programming approach to continuously constrained LQ optimal control problems

Consider the class of LQ optimal control problems with continuous linear state constraints, that is, constraints imposed on every instant of the time horizon. This class of problems are known to be difficult to solve numerically. In this paper, a computational method based on a semi-infinite programming approach is given. The LQ optimal control problem is formulated as a positive quadratic infinite programming problem. This can be done by considering the control as the decision variable while taking the state as a function of the control. After parameterizing the decision variable, an approximate quadratic semi-infinite programming problem is obtained. It is shown that as we refine the parameterization, the solution sequence of the approximate problems converge to the solution of the infinite programming problem (hence to the solution of the original optimal control problem). Numerically, the semi-infinite programming problems obtained above can be efficiently solved using an algorithm based on a dual parameterization method.