On the variational discretization of optimal control problems for Lagrangian dynamics

<title>Abstract</title> <p>We derive the necessary optimality conditions of an optimal control problem with dynamical constraints described by forced Euler-Lagrange equations, using a recently proposed new Lagrangian approach [1] and use variational integrators to solve them. We show that for a family of low-order discretizations the resulting numerical schemes are ‘doubly symplectic’, meaning they provide forced symplectic integrators for the underlying controlled mechanical system and symplectic integrators in the state-adjoint space. This paves the way for variational error analysis to be used in the optimal control setting to derive the order of convergence of the resulting numerical schemes and further the possibility to apply discrete Noether’s theorem for the calculation of conserved quantities there. The schemes derived behave well numerically, benefiting from this ’double-symplecticity’. A multi-body example is solved and the benefits of this ‘double-symplecticity’ illustrated. The example demonstrates the convergence and also the ability to preserve first integrals associated to symmetries in the considered optimal control problem. <bold>Mathematics Subject Classification:</bold> 37M15, 49K15, 49M05, 49M25, 65K10, 65P10, 70H15, 70G45, 70G65, 70Q05.</p>