A nonlinear optimal control approach to stabilization of a macroeconomic development model

Abstract

A nonlinear optimal (H-infinity) control approach is proposed for the problem ofstabilization of the dynamics of a macroeconomic development model that is known as the Grossman-Helpman model of endogenous product cycles. The dynamics of the macroeconomic developmentmodel is divided in two parts. The first one describes economic activities in a developed country andthe second part describes variation of economic activities in a country under development which tries tomodify its production so as to serve the needs of the developed country. The article shows that throughcontrol of the macroeconomic model of the developed country, one can finally control the dynamicsof the economy in the country under development. The control method through which this is achievedis the nonlinear H-infinity control. The macroeconomic model for the country under developmentundergoes approximate linearization round a temporary operating point. This is defined at each timeinstant by the present value of the system’s state vector and the last value of the control input vectorthat was exerted on it. The linearization is based on Taylor series expansion and the computation of theassociated Jacobian matrices. For the linearized model an H-infinity feedback controller is computed.The controller’s gain is calculated by solving an algebraic Riccati equation at each iteration of thecontrol method. The asymptotic stability of the control approach is proven through Lyapunov analysis.This assures that the state variables of the macroeconomic model of the country under developmentwill finally converge to the designated reference values.

Keywords

• macroeconomic development models; Grossman-Helpman model; endogenous growth;nonlinear optimal control| H-infinity control| approximate linearization| Jacobian matrices| Riccatiequation| asymptotic stability