Hamiltonian and self-adjoint control systems (Q1821730)

The paper surveys a remarkable collection of results concerned with Hamiltonian realization of affine analytic control systems. Two results deserve special attention. The first, Theorem 2.3, states a necessary and sufficient condition for an affine system to be Hamiltonian in terms of a special relation between the lifts of the affine system to the tangent and the cotangent bundles the self-adjointness property. Further, Theorem 3.6 characterizes Hamiltonian systems as those affine systems which satisfy the variational criterion \[ \int^{+\infty}_{- \infty}(\delta^ T_ 2y(t)\delta_ 1u(t)\quad -\delta^ T_ 1y(t)\delta_ 2u(t))dt=0, \] for all admissible variations \((\delta_ iu,\delta_ iy)\), \(i=1,2\) of input-output pairs belonging to the external behaviour of the original affine system. The results of the paper generalize to general nonlinear control systems.