Bilinear Control: Rank-one Inputs

indeed has $D=b¥cdot c^{*}$ for $b^{*}=(0,¥ldots,0,1)$ and $c^{*}$ with entries $¥frac{1}{2}(¥beta_{k}-a_{k})$ . (As Jan Willems once pointed out, all the entries are constants; however, the zeros and ones are stiff structure constants, while only the last row has “soft” parameters, to be encompassed by a rank-one control matrix.) Section 1 presents a canonic decomposition of the state space of (1) into linear subspaces (which are controllable or observable, or not, in a suitable sense); a somewhat surprising analogue of the Kalman decomposition that applies to linear control systems. What makes this possible is the rather technical observation (Lemma 1) that, in the Taylor expansion of $Dx(t)$ , the first nonvanishing term does not depend on the controls. The basic result of Section 2 is that, for any initial point $p$ and small times