| This dissertation is concerned with the controllability properties of so-called bilinear systems of ordinary differential equations (BLS). The most general form of a time-invariant BLS with n-dimensional state vector x and m-dimensional control vector u = (u1, u2,..., um) is x&d2;=Ax+ j=1m ujNjx+b j. * When n = 2 and m = 1 (the planar, scalar control case), (*) becomes x&d2;=Ax+ uNx+b. ** We use a constructive, visualizable approach to derive necessary and sufficient conditions for the complete controllability of (**) in R2. We give these conditions in the form of a table, which determines whether (**) is completely controllable in R2 according to easy computations involving b and the eigenvectors and eigenvalues of A and N.; Unlike other published results on the subject, the visual approach taken here allows some extension of our results to higher-dimensional systems. In every case where (**) is not completely controllable in all of R2 , our approach provides an explicit description of an invariant set (a region I⊆R2 which the state of the system (**) can not be steered out of). Assuming (**) is not controllable in all of R2 , one of our preliminary theorems provides a way to locate regions P⊆R2 in which (**) is completely controllable. Our results also reveal some gaps and mistakes in previously-published, well-known results.; Bilinear systems arise naturally as models for numerous dynamical processes in such fields as engineering, biology, economics, ecology, physiology and physics; we cite several references to this effect. Bilinear control is desirable because it provides much of the power of nonlinearities without being wildly unpredictable. The controllability of bilinear systems is the subject of no less than 19 of our references. In the introduction, we cite specific examples of the application of bilinear control to nuclear and biochemical reactions and electric power transmission. |
