On adjoint and complementary systems

A single linear system can represent several different mappings involving states, inputs, and outputs. For each such map, there are associated adjoint and complementary systems that play an important role in inverting the map. The adjoint system determines minimum-norm solutions; it is herein precisely defined, apparently for the first time. The complementary system, a new concept, parametrizes the class of all possible solutions. Applications in system inversion, optimal control, and smoothing are indicated.