The minimal realization theory for input-output map8 that arise from finitedimensional, continuous time, bilinear systems is discussed. It is shown that an observed bilinear system (i.e. a bilinear system together with an observation functional, but without a Jixed initial state) is completely determined by its input-output correspondence, i.e. its multivalued inpu.&-output map. A precise formulation and proof of the result are given that continuous canonical fornw for bilinear systems do not exist. Thie is done by adapting
to the bilinear case an idea due to Hazewinkel and Kalman for the case of linear systema.
In addition, the proof presented here has the advantage of not involving any algebraic geometry, which makes it considerably Gnpler than the original proof of the linear systems result.
I. Zntroduction
This paper has two parts. In the first part we give a brief presentation of the theory of minimal realizations of input-output maps which arise from continuous time, bilinear systems, in an abstract setting. In particular, we prove a new result (Theorem I) which says that an observed bilinear system, without a fixed initial state, is uniquely characterized by its input-output correspondence, i.e. by the set of input-output pairs which is the union of the sets that correspond to all the initial states. We also present the duality between reachability and observability. For completeness, the proof of the uniqueness theorem of Brockett (1) and D'Alessandro et al. ( 2) is also included.
The second part is devoted to the proof that continuous canonical forms do not exist. This is done by adapting to the bilinear case the idea which Hazewinkel and Kalman (3) used for the linear case. However, the proof given here involves a considerable simplification, in that no algebraic geometry whatsoever is used, and no explicit calculations are needed.
ZZ. Minimal Realiaations
Let m be a positive integer. A bilinear system with m inputs is an m + 2-tuple B = (7; A,, . . . . A,,J, where V is a finite-dimensional real vector space (called