= Behavioral systems theory
{title2=$\mathcal B$}
<Behavioral systems theory> describes a <dynamical system> by the complete set of time histories its variables are allowed to follow. With time set $\mathbb T$ and signal-value set $\mathbb W$, its behavior is a subset
$$
\mathcal B\subseteq\mathbb W^{\mathbb T}.
$$
An equation specifies which functions belong to $\mathcal B$. It need not initially designate some variables as <control inputs> and others as <plant outputs>. <Jan C. Willems> emphasized physical relations and interconnection by shared variables as a motivation for this viewpoint.<The Behavioral Approach to Open and Interconnected Systems by Jan C. Willems>[Printed pp46–49; section What Is a Dynamical System?]{ref}
An original <electrical circuit> example uses two series <resistors>, $R_1=1\,\Omega$ and $R_2=2\,\Omega$, followed by a <capacitor> of <capacitance> $C=0.5\,\mathrm F$. Let $i$ be the common <electric current>, $V$ the total <electric potential difference>, and $v_C$ the <capacitor>'s <electric potential difference>. The component laws and connection give
$$
V=(R_1+R_2)i+v_C=3i+v_C,\qquad \dot v_C=i/C=2i,
$$
with the numerical equations understood in SI units. These equations define allowed histories $(V,i,v_C)$ without choosing which external variable is imposed.
Eliminating the internal variable gives, for differentiable histories,
$$
\dot V=3\dot i+2i.
$$
Conversely, a pair $(V,i)$ satisfying this relation can be extended by $v_C=V-3i$ to satisfy the component equations. Elimination therefore describes exactly the allowed external histories under these regularity assumptions. A prescribed initial <electric charge> would impose an additional initial constraint.
If the apparatus holds $i=1\,\mathrm A$ and starts with $v_C=0$, it must supply $V=(3+2t)\,\mathrm V$ with $t$ in seconds. If it instead holds $V=3\,\mathrm V$ with the same initially uncharged <capacitor>, the allowed <electric current> is $i=e^{-2t/3}\,\mathrm A$. These are different experiments on the same component relation. A chosen input–output description remains useful, but its choice follows the connection and experiment rather than being built into the definition of the physical device.
In the general framework, interconnection restricts the allowed histories by requiring designated variables to agree or satisfy connection laws. Keeping internal variables gives a component model; eliminating them gives an external description. This supports reusable models of interconnected <electrical circuits>, mechanical systems and other <dynamical systems> without assuming that every connection transmits information in only one direction.
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