This paper presents new solutions to certain non-standard non-linear H-infinity control problems. We consider non-linear affine plants whose measurement output is of dimension larger than the dimension of the external input. This problem is, under proper assumptions. transformed to the problem of stabilization by means of output injection and solution of a Hamilton-Jacobi partial differential inequality arising in singular H-infinity state-feedback control. General sufficient solvability conditions are given, Explicit solutions are available in the local and semilocal cases. The former concerns a certain neighbourhood of the origin in the closed loop state-space. whereas the latter assumes that the trajectories are restricted to a neighbourhood of an invariant manifold. The issue of the controller order is addressed and a reduced-order controller is obtained in the local case. A new generalization of the chain-scattering formalism provides a very useful framework for solving this problem.