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 provide a “robust” approximation result in terms of a linear programming problem with a sufficiently long time horizon and another approximation result in terms of a sequence of dual linear programming problems. We prove that the optimal value of the duals of the truncated linear programming problems with “fixed end-point” converge to the optimal value of the linear optimal control problem with linear constraints beginning from period 1 if and only if a weak transversality condition is satisfied. We prove that if there exists a solution satisfying “interiority condition” for an absolutely convergent linear optimal control problem with exactly one inequality constraint for the control variable in each period, then there is an “implied infinite dual linear programming problem” which has a solution and the optimal value of this dual is equal to the optimal value of the optimal control problem if and only if the same weak transversality condition is satisfied. In the general case of optimal control problems that allow more than one inequality constraint for the control variable in each period, the satisfaction of a boundedness condition for the optimal dual variables leads to the existence of an “implied infinite horizon dual linear programming problem” which has a solution and the optimal value of this dual is equal to the optimal value of the optimal control problem if and only if the weak transversality condition is satisfied.