Departmental Papers (ESE)


It is well known that unconstrained infinite-horizon optimal control may be used to construct a stabilizing controller for a nonlinear system. In this note, we show that similar stabilization results may be achieved using unconstrained finite horizon optimal control. The key idea is to approximate the tail of the infinite horizon cost-to-go using, as terminal cost, an appropriate control Lyapunov function. Roughly speaking, the terminal control Lyapunov function (CLF) should provide an (incremental) upper bound on the cost. In this fashion, important stability characteristics may be retained without the use of terminal constraints such as those employed by a number of other researchers. The absence of constraints allows a significant speedup in computation. Furthermore, it is shown that in order to guarantee stability, it suffices to satisfy an improvement property, thereby relaxing the requirement that truly optimal trajectories be found. We provide a complete analysis of the stability and region of attraction/operation properties of receding horizon control strategies that utilize finite horizon approximations in the proposed class. It is shown that the guaranteed region of operation contains that of the CLF controller and may be made as large as desired by increasing the optimization horizon (restricted, of course, to the infinite horizon domain). Moreover, it is easily seen that both CLF and infinite-horizon optimal control approaches are limiting cases of our receding horizon strategy. The key results are illustrated using a familiar example, the inverted pendulum, where significant improvements in guaranteed region of operation and cost are noted.

Document Type

Journal Article

Subject Area


Date of this Version

May 2001


Copyright 2001 IEEE. Reprinted from IEEE Transactions on Automatic Control, Volume 46, Number 5, May 2001, pages 776-783.
Publisher URL:

This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to By choosing to view this document, you agree to all provisions of the copyright laws protecting it.

NOTE: At the time of publication, author Ali Jadbabaie was affiliated with the California Institute of Technology. Currently (March 2005), he is a faculty member in the Department of Electrical and Systems Engineering at the University of Pennsylvania.


Lyapunov methods, control system synthesis, nonlinear control systems, optimal control, stability, control Lyapunov function, guaranteed region of operation, infinite horizon cost-to-go, inverted pendulum, region of attraction, stability characteristics, terminal cost, unconstrained finite horizon optimal control, unconstrained receding-horizon control, control Lyapunov functions (CLFs), model predictive control, nonlinear control design, receding horizon control



Date Posted: 30 April 2005

This document has been peer reviewed.