A Linear Model of Optimal Control with One-dimensional Control and State Variables

We consider discrete-time infinite-horizon optimal control problems with a linear objective function. For absolutely convergent linear optimal control problems, we prove the existence of a solution, the necessity of a Euler and transversality conditions for a solution and the sufficiency of competitive condition and a different transversality for a solution. We show that the optimal value functions satisfy a “functional equation of dynamic programming” and that the satisfaction of this functional equation is necessary and sufficient for a trajectory to solve the optimization problem. Under the additional assumption, namely, “concave in pay-offs from the control variable”, being satisfied by absolutely convergent linear optimal control problems, we show that the optimal value functions are concave and continuous. We obtain closed form solutions for such problems under the assumption that there is a state transition function that is strictly increasing and strictly concave in the gap variable and satisfy mild interiority conditions.