Optimal Growth with Robust Feedback Control

Abstract

Optimal control is no less important in economics today than welfare economics was twenty years ago. It is used to evaluate different growth paths; and it forms a crucial part of the long-run of the firm. It has been used to derive an aggregate savings function, a demand for real balances function and an optimum dividend policy for a corporation. There is no area in economics which cannot benefit from the use of optimal control theory. As a rule, economists have used the Maximum Principle of Pontryagin to derive optimal control laws. Cass [4] formulated instructions to a central planning board, which attempts to direct the growth of the economy in an optimal manner, on the basis of the maximum principle. In his expository paper, Dorfman [5] used optimal control theory and the maximum principle synonymously. The maximum principle is an open loop type of optimization method in that it yields an entire sequence of controls to be followed from initial conditions. Half of the initial conditions must be obtained from transversality conditions which imply the solution of differential equations. Given the high probability of errors in measurement, formulation and implementation, as well as the likelihood of outside disturbances, the overall system will not be stable unless the open loop optimal policies are converted to a feedback form. This is to be expected since the optimal path to the desired target is unique. It is clearly advantageous in economics to derive policies in feedback form, where the next move depends upon the current state,