A nonlinear economic control problem with a linear feedback solution

This paper discusses a special optimal control problem having a solution expressible in linear feedback form. The problem structure is characteristic of certain economic investment problems where the control influences the rate of deterioration of a capital good. An example of an economic problem of this type is the determination of the optimal maintenance policy for a housing unit when the effectiveness of maintenance is subject to decreasing returns to scale. Mathematically, the problem is characterized by a dynamic system that is nonlinear with respect to both the state and the control and by a linear objective function. Both the control and the state are restricted to be nonnegative and to lie below given upper bounds. The problem possesses a unique global optimal solution expressible in linear feedback form. The associated control law is derived from the solution of an auxiliary ordinary differential equation that can be solved backward in time. In addition to the linear control law, the problem possesses other important linearity properties. The closed-loop system is linear; that is, although the system itself is nonlinear, when the optimal control law is substituted for the control, the resulting system is linear with respect to the state. And, the optimal return function is affine; that is, the optimal value that can be attained from a given point is an affine function of the value of the state at that time.