Infinite Horizon Linear Optimal Control with Linear Constraints

We define infinite horizon linear optimal control problem with linear constraints. We provide a necessary condition for an optimal trajectory in terms of an infinite sequence of linear programming problems. We also provide a similar sufficient condition for optimality in terms of a related infinite sequence of linear programming problems. We define a “bang-bang sequence of decision rules”, and provide sufficient conditions for the existence of a unique optimal trajectory that is generated by such a sequence of decision rules. We also provide a “robust” approximation result in terms of a linear programming problem with a sufficiently long time horizon. We use the strong duality theorem and complementary slackness condition of linear programming to obtain necessary conditions for an optimal trajectory. These necessary conditions lead to a very general “transversality condition”, the satisfaction of which is a characteristic feature of optimal trajectories in infinite horizon optimization. A more compact transversality condition is realized when along an optimal trajectory the control variable is “eventually” strictly positive. Under suitable assumptions we prove that there is a infinite horizon “implied dual linear programming problem” which has a solution and which along with the optimal trajectory satisfies the complementary slackness conditions. Further, the optimal value of the implied dual linear programming problem is equal to the optimal value of the maximization problem that gives rise to it. We obtain sufficient conditions for a trajectory to be an optimal trajectory by using the strong duality theorem and complementary slackness condition of linear programming.