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Anant Agarwal is a computer scientist and electrical engineer whose work includes computer architecture, multicore processors and circuit education. MIT's departmental profile lists computer architecture among his research areas, while his CSAIL profile describes work involving multicore computing and authorship of a circuits textbook.^[ref]

His identified MIT faculty page links an external portrait of Anant Agarwal.

The MIT lecture State and Memory by Anant Agarwal and Jeffrey Lang, jointly credited with Jeffrey Lang, connects analog stored state to computer memory. A capacitor can summarize prior input by one state quantity, such as stored electric charge $q=C{v}_C$; digital storage then adds a rule for distinguishing logical values. The slides explicitly introduce capacitor state, zero-input response, zero-state response and a one-bit memory abstraction.^[ref]

For an original physical-memory example, let a $1\mathrm{nF}$ capacitor initially hold $3\mathrm{V}$ and leak through a $1\mathrm{M\Omega }$ resistor. With no driving input,If a stipulated readout interprets values above $2\mathrm{V}$ as the stored high state, the ideal decay reaches that threshold afterThe initial stored electric charge is $3\mathrm{nC}$, but the logical interpretation also depends on leakage, the elapsed time and the decision threshold. This is a teaching model, not a specification for a particular memory chip.

The example exposes why merely retaining some electric charge is different from reliably retaining a bit. A practical memory circuit must preserve distinguishable states over the required interval and allow the state to be read. Agarwal and Lang's lecture uses the state concept to connect the circuit's dynamics with that information-storage task.

Jeffrey Lang is an electrical engineer whose work concerns electromechanical conversion power balance, control and small-scale devices. MIT's Research Laboratory of Electronics identifies the same professor as Jeffrey H. Lang and describes research on electric motors, electrical generators and microelectromechanical systems, together with teaching in circuits and electromagnetism.^[ref]

The identified MIT profile provides an externally hosted portrait of Jeffrey H. Lang.

For an original motor-scale illustration, a shaft delivering torque $\tau =2\mathrm{mN}\mathrm{m}$ at angular velocity $\omega =1000\mathrm{rad}{\mathrm{s}}^{-1}$ delivers $P_m=\tau \omega =2\mathrm{W}$. If electrical input is $3\mathrm{W}$ and the operating point is steady, the remaining $1\mathrm{W}$ must be accounted for by losses or another specified output. During acceleration some input can instead increase rotational kinetic energy. Reporting only shaft speed would not establish either output power or conversion efficiency.

With Anant Agarwal, Lang is credited on the MIT course material State and Memory by Anant Agarwal and Jeffrey Lang. It explains how a capacitor's state summarizes the part of past input needed to predict its future response.^[ref] For a series RC circuit with a constant later input $V_f$,Two different earlier input histories that leave the same initial capacitor electric potential difference give the same subsequent response under the same future input. For stipulated $RC=1\mathrm{ms}$, $v_C(0)=1\mathrm{V}$ and $V_f=3\mathrm{V}$, the value after one time constant is $3-2/e\approx 2.2642\mathrm{V}$. This original numerical example shows what it means for the state to retain enough of the past without retaining the entire waveform.

The zero-input response is the output due to a system's initial internal state when its externally applied input is set to zero. It describes memory left by earlier excitation. For a linear time-invariant system, linearity separates this from the zero-state response due to the subsequent input.

For constant matrices in $\dot{x}=Ax+Bu$, $y=Cx+Du$, setting $u=0$ and $x(0)=x_0$ givesThe matrix exponential solves the homogeneous state equation. With an arbitrary input, the complete solution adds the zero-state response:A zero initial output $C{x}_0=0$ alone does not establish that the whole state is zero.

For an RC circuit with capacitor difference $v(0)=V_0$ and no applied source, $RC\dot{v}+v=0$, so $v(t)=V_0{e}^{-t/(RC)}$. The capacitor energy falls as $C{V}_0^2{e}^{-2t/(RC)}/2$; the resistor receives the dissipated energy. The primary MIT lecture explicitly distinguishes the capacitor's initial state, zero external input and zero initial state.^[ref]

At a time constant $RC=2\mathrm{s}$ and $V_0=3\mathrm{V}$, the modeled difference after two seconds is $3/e\approx 1.104\mathrm{V}$. Zero input has not meant zero output or zero stored energy. A direct experiment charges the same capacitor to different known initial differences, removes the charging source and records the decay across a fixed resistor. The amplitudes should scale with $V_0$ while the normalized decay remains the same in the linear model.

A zero-input response need not decay in every system: an ideal undamped oscillator retains its initial motion, and an unstable homogeneous system can amplify a nonzero initial state. The transfer function describes the zero-state input-output map, so its product with a zero input cannot represent a separate nonzero initial-state contribution.

The zero-state response is the output produced by an applied input when the system starts with zero internal state. For a linear time-invariant system it separates the effect of the input from the zero-input response caused by an initially stored state. A zero initial output alone need not mean zero internal state.

For a causal linear time-invariant system with impulse response $h$, the zero-state response is the convolutionAn instantaneous feedthrough term, if present, is represented by a Dirac delta term in $h$. Equivalently, for constant matrices in $\dot{x}=Ax+Bu$, $y=Cx+Du$ with $x(0)=0$,For nonzero $x(0)=x_0$, linearity adds the zero-input response $C{e}^{At}{x}_0$. Here the matrix exponential propagates the state left by the earlier input.

For example, $\tau \dot{y}+y=u$ with $\tau >0$, $y(0)=0$ and a constant input $u(t)=A$ for $t\ge 0$ has the step response $y_{\mathrm{zs}}=A(1-e^{-t/\tau })$. At one time constant it has reached about $63.2\%$ of its final value. If the initial value instead equals $y_0$, the total output is $A(1-e^{-t/\tau })+y_0{e}^{-t/\tau }$. A charged capacitor in an RC circuit provides a direct physical example: discharge it before repeating the same input step to isolate the zero-state response, and compare with a run from a known initial electric potential difference.^[ref]

With compatible transforms, the zero-state input-output relation is $Y=HU$, where $H$ is the transfer function. For a nonzero initial state, a unilateral Laplace transform includes additional initial-condition terms. Those terms cannot be absorbed into an input-independent transfer function multiplying $U$.