Abstract:
Optimal control techniques have long been applied to different fields: engineering, space science, economics, etc. It is suggested that they be applied to gravitational systems in General Relativity.
As a first example the problem of hydrostatic equilibrium of stellar models is formulated as an optimal control problem. Then the application of Pontryagin's maximum principle leads directly to the Tolman-Oppenheimer-Volkoff equation.
Einstein's interior field equations express the coupling between matter and the gravitational field. They reduce to a system of differential equations. Conventionally, a solution may be sought for a specified equation of state. Alternatively, it is suggested that interior problems be formulated as optimal control problems. The equation of state is thus determined by the optimality conditions. Thus, as a second example the problem of constructing a relativistic cosmological model is considered. The optimal control formulation results in a universe that is optimal with respect to a certain criterion. Specifically, we apply such a formulation to the Friedmann-Robertson-Walker (FRW) closed universe with maximum lifetime. The resulting optimal control problem is reduced to a two-point boundary value problem.