Nonlinear Dynamics

, Volume 82, Issue 1–2, pp 547–562 | Cite as

Optimal stable control for nonlinear dynamical systems: an analytical dynamics based approach

Original Paper


This paper presents a method for obtaining optimal stable control for general nonlinear nonautonomous dynamical systems. The approach is inspired by recent developments in analytical dynamics and the observation that the Lyapunov criterion for stability of dynamical systems can be recast as a constraint to be imposed on the system. A closed-form expression for control is obtained that minimizes a user-defined control cost at each instant of time and enforces the Lyapunov constraint simultaneously. The derivation of this expression closely mirrors the development of the fundamental equation of motion used in the study of constrained motion. For this control method to work, the positive definite functions used in the Lyapunov constraint should satisfy a consistency condition. A class of positive definite functions has been provided for mechanical systems that meet this criterion. To illustrate the broad scope of the method, for linear systems it is shown that a proper choice of these positive definite functions results in conventional LQR control. Control of the Lorenz system and a multi-degree of freedom nonlinear mechanical system are considered. Numerical examples demonstrating the efficacy and simplicity of the method are provided.


Nonlinear dynamical systems  Minimization of control cost Lyapunov constraint Analytical dynamics approach Global asymptotic stability Consistent constraint Control of Lorenz and mechanical systems 


Conflict of interest



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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of Aerospace and Mechanical EngineeringUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.Department of Civil EngineeringUniversity of Southern CaliforniaLos AngelesUSA
  3. 3.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA
  4. 4.Department of Information and Operations ManagementUniversity of Southern CaliforniaLos AngelesUSA
  5. 5.Department of Civil EngineeringUniversity of Southern CaliforniaLos AngelesUSA

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