The subject of this talk was the development of the Maximum Principle for affine connection control systems. You will recall that the control equations for a general control system, coupled with the so-called adjoint equations, define a time-dependent Hamiltonian system, and that all optimal trajectories are projections of integral curves of this Hamiltonian system. The adjoint equation describing the evolution of the Lagrange multipliers along the optimal trajectory is not generally by itself an intrinsic object. However, we show that for affine connection control systems one can extract the adjoint equation in an intrinsic way, and that the resulting equation, which we call the adjoint Jacobi equation, is related to the Jacobi equation which describes the variation of geodesics.

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