NUMERICAL APPROXIMATION OF OPTIMAL CONTROL PROBLEMS CONSTRAINED BY DYNAMIC EQUATIONS VIA GALERKIN METHOD

Abstract

The research investigates the application of the Galerkin method to optimal control problems constrained by coupled dynamic equations. These constrained problems are reformulated into unconstrained ones using the Hamiltonian approach, which facilitates the determination of boundary conditions for both the state and costate variables. By assuming a polynomial solution, the weighted and residual functions were derived. The Orthogonality of the product of these functions leads to the formation of a system of linear equations. Solving these equations provides the solution for the boundary conditions through direct substitution. This scheme was developed for the Lagrange form of optimal control problems to assess its accuracy in approximating exact solutions. Several optimal control problems with known exact solutions were solved using the proposed scheme, and the results were compared to evaluate its effectiveness.