Optimal control models of Einstein’s field equations

Document Type : Mini-review

Authors

1 Mathematics and Computer Science Department, Faculty of Science, Port Said University, Egypt.

2 Department of Mathematical and Natural Sciences Faculty of Engineering Egyptian Russian University Cairo, Egypt

Abstract

It is shown that some problems of Einstein’s field equations of the theory of General Relativity can be modeled as optimal control problems. The advantages of adopting such an approach are explained. As a demonstration of this interdisciplinary research, four examples are reviewed for problems in relativistic astrophysics and cosmology. The first example, in relativistic astrophysics, uses optimal control to prove that the Tolman–Oppenheimer-Volkoff equation of hydrostatic equilibrium is a necessary condition to extremize the mass of a stellar model. The second example, also in relativistic astrophysics, uses optimal control to estimate an upper limit to the mass of a neutron star. The third example, in relativistic cosmology, uses optimal control to construct a closed universe with maximum lifetime. The fourth example, in cosmological inflation, uses optimal control to construct a model of a slow-roll inflationary universe with minimum change of the scalar field. Results show that optimal control is more powerful than classical variational calculus and that optimal control models of Einstein’s field equations add physical significance to their solutions. Extension to other problems is explored, and difficulties in the formulation of optimal control problems in General Relativity are indicated.

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ERU Research Journal
Volume 2, Issue 4
October 2023
Pages 616-630

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