# On Lagrangian Dynamics and Its Control Formulation

S. H. Lam

## Abstract

This paper exploits the equivalence between constrained dynamics problems and a class of tracking control problems. An alternative derivation of the modern formulation of Parczewski and Blajer (1989) is presented, including some new analysis on the almost singular'' case, followed by a critique of the classical Lagrangian formulation. In addition, a new formulation which allows the constraints to be satisfied approximately is presented. A general closed-loop control law which depends on a user-specified threshold of constraint errors, \epsilon, is proposed, transforming the default DAE problem into a stiff ODE initial-value problem which can be routinely solved on computers. It is shown that certain explosively unstable open-loop systems may be stabilized by the use of closed-loop control using a small but finite $\epsilon$. An example is worked out using both the open-loop and the closed-loop control formulations.

Revised February, 1998.
Published in Applied Mathematics and Computations, 91, pp. 259-284, Elsevier Science, 1998.