THE CONDITION OF HYDROSTATIC EQUILIBRIUM OF STELLAR MODELS USING OPTIMAL CONTROL SALAH HAGGAG 1,2 and JOHN L. SAFKO 1,3 1 Department of Physics and Astronomy, University of South Carolina, Columbia, South Carolina 29208, USA 2 Permanent address: Mathematics Department, Faculty of Science, Al-Azhar University at Assiut, Assiut 71524, Egypt; 2 Present address: Mathematics Department, College of Science, Qatar University, P.O. Box 2713, Qatar; E-mail: sahaggag@qu.edu.qa. 3 E-mail: safko@mail.psc.sc.edu (Received 6 August 2001; accepted 20 February 2002) Abstract. The condition of hydrostatic equilibrium of relativistic stellar models is formulated as an optimal control problem. Application of Pontryagin’s maximum principle leads directly to the Tolman-Oppenheimer-Volkoff equation. 1. Introduction One of the important applications of general relativity is the study of stellar models: their construction, equilibrium and stability (see for example Misner et al., 1973). The simplest such model is an isolated static sphere of perfect fluid. The vacuum outside the star has the Schwarzschild metric ds 2 =  1 - 2M r  dt 2 -  1 - 2M r  -1 dr 2 - r 2 ( dθ 2 + sin 2 θdφ 2 ) (1) where M is the total mass-energy of the star. In the interior of the star, the spacetime is described by the static spherically symmetric metric ds 2 = e 2ν dt 2 -  1 - 2m r  -1 dr 2 - r 2 ( dθ 2 + sin 2 θdφ 2 ) (2) where ν and m are functions of the coordinate radius r . The energy-momentum tensor for a perfect fluid is given by T ab = (µ + p)U a U b - pg ab (3) where µ and p are, respectively, the fluid density and pressure, and U a is the four- velocity of the matter. The first law of thermodynamics relates the number density of baryons n to µ and p by the equation dn dµ = n p + µ . (4) Astrophysics and Space Science 283: 369–373, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.