Recent work (Rantzer, 2022) formulated a class of optimal control problems involving positive linear systems, linear stage costs, and elementwise constraints on control. It was shown that the problem admits linear optimal cost and the associated Bellman's equation can be characterized by a finite-dimensional nonlinear equation, which is solved by linear programming. In this work, we report exact dynamic programming (DP) theories for the same class of problems. Moreover, we extend the results to a related class of problems where the norms of control are bounded while the optimal costs remain linear. In both cases, we provide conditions under which the solutions are unique, investigate properties of the optimal policies, study the convergence of value iteration, policy iteration, and optimistic policy iteration applied to such problems, and analyze the boundedness of the solution to the associated optimization programs. Apart from a form of the Frobenius-Perron theorem, the majority of our results are built upon generic DP theory applicable to problems involving nonnegative stage costs.