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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 subsetAn 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.^[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 givewith 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,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.

Dissipativity expresses an input–output balance for a dynamical system: the increase of a nonnegative storage function cannot exceed the accumulated supply rate. For state $x$, control input $u$, plant output $y$ and chosen supply rate $w(u,y)$, the requirement isfor every admissible trajectory and time interval, with $S\ge 0$. The storage function and supply rate need not be literal physical energy and power; choosing them makes a particular performance statement precise. Jan C. Willems's1972paper develops this general storage–supply formulation.^[ref]

An original mechanical example is a mass with viscous damping, driven by an applied force $u$:Choose the kinetic energy $S(v)=m{v}^2/2$ and entering power $w(u,y)=uy$. ThenIntegration gives the required inequality. The nonnegative integral $\int b{v}^{2}dt$ accounts for the mechanical energy dissipated by the damper. With no applied force, the stored kinetic energy cannot increase. A dashpot driven at constant velocity provides an experimental interpretation: supplied power is dissipated even when the mass's stored kinetic energy remains constant.

A different choice tests an input–output gain. If a storage function establishes the inequality forthen $S(x(0))=0$ and nonnegative final storage implyThe physical power balance and this squared-signal bound use the same logical structure. Neither inequality, by itself, proves that every internal state converges to zero: a hidden lossless motion can retain storage.

Mechanical available storage is the supremum of net work that can be extracted from a system's current state through its allowed mechanical terminals. If $Fv$ is entering power and the initial state is $x_0$, defineThe zero-duration choice gives $S_a\ge 0$. Any nonnegative mechanical energy storage function satisfying the mechanical dissipation inequality bounds the extracted work, so $S_a(x_0)\le E(x_0)$. Available storage is thus an operational recoverability quantity; it need not equal every form of internal stored energy. The general supply/storage construction was developed by Jan C. Willems.^[ref]

Consider a spring and dashpot in parallel, with $F=kx+c\dot{x}$ and initially $x=x_0$. Unload to $x=0$ at constant speed over duration $T$, so $\dot{x}=-x_0/T$. The outgoing work isThe first term is released elastic potential energy; the second is lost in the dashpot. For $k=1000N/m$, $c=10\mathrm{N}s/m$ and $x_0=0.10\mathrm{m}$, the initial spring energy is $5\mathrm{J}$. A one-second unloading recovers $4.90\mathrm{J}$ and a ten-second unloading recovers $4.99\mathrm{J}$. With unrestricted duration and suitable terminal control, these paths approach the energy upper bound, proving $S_a=5\mathrm{J}$ for this ideal example. A supremum need not be attained in finite time.

The allowed operations are essential. If the same example must finish at zero extension within one second, the Cauchy-Schwarz inequality gives ${\int }_0^T{\dot{x}}^{2}dt\ge x_0^2/T$, so the best recoverable work under that extra requirement is $4.90\mathrm{J}$. A physically inaccessible internal motion may store energy without allowing its recovery at the chosen terminals. Thus stating the initial state, usable ports, time and actuator limits is part of an available-storage claim, not an optional qualification.

A storage function assigns a nonnegative quantity to the state of a dynamical system so that its increase is bounded by the integrated supply rate. In dissipativity, it must satisfyfor every allowed history and interval. A storage function may represent physical energy, but it can also certify an abstract input–output bound. Jan C. Willems's theory relates possible storage functions to recoverable supply, rather than selecting an arbitrary positive expression.^[ref]

For a differentiable storage function and state equation $\dot{x}=f(x,u)$, the pointwise inequalityis sufficient: integrating it along a solution proves the interval statement. Positivity alone is insufficient; the derivative must be checked against the specified supply rate.

An original dimensionless example makes this check exact. For $\dot{x}=-x+u$, output $y=x$, and supply rate $w=uy=ux$, try $S_a(x)=a{x}^2/2$ with $a>0$. Its derivative isThe difference from supply is ${\dot{S}}_a-ux=-a{x}^2+(a-1)xu$. Requiring it to be nonpositive for every real $u$ and fixed nonzero $x$ forces $a=1$. That value works because ${\dot{S}}_1=ux-x^2\le ux$. Thus even within this simple positive quadratic family, the supply law fixes the admissible scale.

A storage function also limits extraction. Rearranging the interval inequality givesStarting in a given state, one cannot withdraw more net supply than the initial storage certificate permits. A chosen storage function need not make that upper bound attainable.

When unforced operation makes the supply rate zero, storage cannot increase. This resembles a Lyapunov function, but nonincrease alone does not prove asymptotic stability: the function may fail to distinguish some states, or its derivative may vanish along persistent motion. The storage balance and a convergence proof are separate claims.