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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.

Jan C. Willems was an electrical engineer and systems theorist whose work helped make dissipativity a general tool for studying dynamical systems. His university-hosted curriculum records an electrical-engineering doctorate at MIT in1968, a systems-and-control professorship at Groningen from1973, and later work as a guest professor at KU Leuven. It records his death on31August2013.^[ref]

His1972general theory expresses a system's balance using a storage function and supply rate. The formulation compares a change in stored quantity with integrated input supply, allowing physical energy accounting and input–output performance bounds to share one mathematical language.^[ref]

An original example illustrates the scientific idea. A2kilogram mass with viscous coefficient3newton-seconds per metre obeys $2\dot{v}=u-3v$. Its kinetic energy is $S=v^2$ joules when $v$ is expressed in metres per second, andAt constant velocity1metre per second, a3newton force supplies3watts, all dissipated while the stored kinetic energy stays1joule. The dissipativity article develops the general inequality; this simple case shows why its supply and storage terms must be distinguished.

Willems also developed behavioral systems theory, which describes a dynamical system by a family of admissible time histories. His curriculum explains the motivation: an interconnected system need not come with a uniquely privileged division into control inputs and plant outputs. Interconnection can instead impose shared variables and compatibility between trajectories.^[ref] The two contributions address complementary questions: which histories are possible, and what balance law must each history obey?

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.

A supply rate specifies the instantaneous quantity whose time integral is compared with a change of storage function in dissipativity. With control input $u$, plant output $y$ and state $x$, a chosen function $w(u,y)$ enters the balanceThe supply rate can be negative: it then represents net withdrawal in the chosen accounting. Its physical interpretation depends on the variables and sign convention, rather than on the word supply alone.^[ref]

For a mechanical terminal with applied force $F$ and inward-convention velocity $v$, choose $w=Fv$, the entering power. For an electrical terminal with electric current entering its positive-potential terminal, the corresponding choice is $w=Vi$, with $V$ an electric potential difference. Both have units of watts when $S$ is energy. Reversing one terminal sign without changing the other reverses the apparent power flow.

An original capacitor example illustrates the sign. Let $C=2\mathrm{F}$, terminal electric potential difference $V=3\mathrm{V}$, and charging electric current $i=1\mathrm{A}$. Its stored electrostatic energy is $S=C{V}^2/2=9\mathrm{J}$, and at that instant $\dot{S}=Vi=3\mathrm{W}$. With the same potential and $i=-1\mathrm{A}$, the supply rate is $-3\mathrm{W}$ and storage decreases. This is consistent with nonnegative stored energy; negative supply does not mean negative storage.

A different supply rate can express a signal-performance requirement:Nonnegative final storage and zero initial storage then bound accumulated output-square by ${\gamma }^2$ times accumulated input-square. These squares need not be physical power. Scaling or changing the units of the signals requires corresponding changes in the coefficients. A general real quadratic choice can be written as $w=y^{T}Qy+2y^{T}Lu+u^{T}Ru$, with symmetric $Q,R$; the selected quadratic form states which input–output relation is being tested.