(pp. 36-39)

Amol J. Sasane

Problem: Let$R\; \in \;\mathbb{C}{[{\eta _1},\; \ldots ,\;{\eta _{\text{m}}},\;\xi ]^{{\text{g}}\; \times \;{\text{w}}}}$and let$\mathfrak{B}$be the behavior given by the kernel representation corresponding to$R$. Find an algebraic test on$R$characterizing the time-controllability of$\mathfrak{B}$.

In the above, we assume$\mathfrak{B}$to comprise of only smooth trajectories, that is,

$\mathfrak{B} = \{ w\; \in \;{C^\infty }\;({\mathbb{R}^{{\text{m + 1}}}},\;{\mathbb{C}^{\text{w}}})\;|\;{D_R}w\; = \;0\} $,

where${D_R}:\;{C^\infty }\;({\mathbb{R}^{{\text{m}} + 1}},\;{\mathbb{C}^{\text{w}}})\; \to \;{{\text{C}}^\infty }\;({\mathbb{R}^{{\text{m}} + 1}},\;{\mathbb{C}^{\text{g}}})$is the differential map that acts as follows: if$R\; = \;{[{r_{{\text{ij}}}}]_{{\text{g}}\; \times \;{\text{w}}}}$, then

${D_R}\;\left[ {\begin{array}{*{20}{c}} {{w_1}} \\ \vdots \\ {{w_{\text{w}}}} \\ \end{array} } \right] = \;\left[ {\begin{array}{*{20}{c}} {\sum\nolimits_{{\text{k}} = 1}^{\text{w}} {{r_{1{\text{k}}}}\;\left( {\frac{\partial } {{\partial {x_1}}},\; \ldots ,\;\frac{\partial } {{\partial {x_{\text{m}}}}},\;\frac{\partial } {{\partial t}}} \right)\;{w_{\text{k}}}} } \\ \vdots \\ {\sum\nolimits_{{\text{k}} = 1}^{\text{w}} {{r_{{\text{gk}}}}\;\left( {\frac{\partial } {{\partial {x_1}}},\; \ldots ,\;\frac{\partial } {{\partial {x_{\text{m}}}}},\;\frac{\partial } {{\partial t}}} \right)\;{w_{\text{k}}}} } \\ \end{array} } \right]\;$.

Time-controllability is a property of the behavior, defined as follows. The behavior$\mathfrak{B}$is said to be*time-controllable*if for any${w_1}$and${w_2}$in$\mathfrak{B}$, there exits a$w\; \in \;\mathfrak{B}$and a$\tau \geqslant 0$such that

$w( \bullet ,\;t)\; = \;\left\{ {\begin{array}{*{20}{c}} {{w_1}( \bullet ,\;t)} & {{\text{for}}\;{\text{all}}\;t\; \leqslant \;0} \\ {{w_2}( \bullet ,\;t - \tau )} & {{\text{for}}\;{\text{all}}\;t\; \geqslant \;\tau } \\ \end{array} } \right.$.

The behavioral theory for systems described...