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\section{Problem Restatement \& Objectives}
\label{sec:problem}

\subsection{Restatement of the Problem}
\label{subsec:restatement}
The 2026 MCM Problem A concerns continuous-time prediction of smartphone battery drain under time-varying usage.
A smartphone is subject to multiple power-consuming components---most prominently the display, CPU workload, and cellular/Wi-Fi communication---whose intensities evolve over time according to user behavior and network conditions.
Meanwhile, the battery exhibits coupled electro-thermal dynamics and gradual health degradation.
The task is to construct a mechanism-driven, continuous-time model that maps future usage profiles to battery states and terminal voltage, and then to estimate the remaining operating time before the device shuts down.

In particular, given (measured, prescribed, or scenario-generated) time series describing user/device usage and ambient conditions, we aim to:
(i) predict the trajectories of key battery states (e.g., state-of-charge and temperature) and the terminal voltage;
(ii) compute the time-to-empty (TTE) defined by physically meaningful shutdown criteria; and
(iii) interpret sudden shutdown phenomena through explicit feasibility mechanisms rather than black-box regression.

\subsection{Inputs, Outputs, and Prediction Tasks}
\label{subsec:io_tasks}
We represent the battery-in-phone system by a state vector
\[
\mathbf{x}(t)=[z(t),\,v_p(t),\,T_b(t),\,S(t),\,w(t)]^\top,
\]
where \(z\) is state-of-charge (SOC), \(v_p\) is polarization voltage (memory effect of the ECM),
\(T_b\) is battery temperature, \(S\) is state-of-health (SOH, capacity fraction), and \(w\) is the radio tail state.

The exogenous inputs are collected as
\[
\mathbf{u}(t)=[L(t),\,C(t),\,N(t),\,\Psi(t),\,T_a(t)]^\top,
\]
where \(L\in[0,1]\) denotes normalized screen brightness,
\(C\in[0,1]\) the normalized CPU load,
\(N\in[0,1]\) the normalized network activity level (throughput/airtime proxy),
\(\Psi>0\) a signal-quality indicator (larger is better),
and \(T_a\) the ambient temperature.

\paragraph{Primary predicted outputs.}
The model produces the battery terminal voltage and SOC trajectories,
\[
V_{\mathrm{term}}(t), \qquad z(t),
\]
as well as the time-to-empty (TTE), defined as the first time the device becomes inoperable under the specified shutdown criteria:
\[
\mathrm{TTE}=\inf\Big\{t>0:\ V_{\mathrm{term}}(t)\le V_{\mathrm{cut}}\ \text{or}\ z(t)\le 0\Big\}.
\]
Here \(V_{\mathrm{cut}}\) is the cutoff voltage dictated by system protection (BMS/PMIC).

\paragraph{Prediction tasks.}
Given \(\mathbf{x}(0)\) and a future input profile \(\mathbf{u}(t)\) on a horizon \([0,T]\), the prediction tasks are:
\begin{enumerate}
  \item \textbf{State/voltage forecasting:} compute \(\mathbf{x}(t)\) and \(V_{\mathrm{term}}(t)\) for \(t\in[0,T]\);
  \item \textbf{Runtime estimation:} compute \(\mathrm{TTE}\) from the stopping rule above;
  \item \textbf{Mechanistic interpretation:} attribute shutdown to depletion (\(z\to 0\)) or voltage protection (\(V_{\mathrm{term}}\le V_{\mathrm{cut}}\)), and quantify risk of power infeasibility (Section~\ref{subsec:metrics_scenarios}).
\end{enumerate}

\subsection{Performance Metrics and Usage-Scenario Description}
\label{subsec:metrics_scenarios}

\paragraph{Operational termination and reliability-oriented metrics.}
The principal performance metric is the operating time before shutdown, \(\mathrm{TTE}\).
For evaluation and comparison across scenarios, we also report:
\begin{itemize}
  \item \textbf{Terminal-voltage margin:} \(\min_{t\in[0,\mathrm{TTE}]}(V_{\mathrm{term}}(t)-V_{\mathrm{cut}})\), which indicates how close the device operates to the cutoff boundary;
  \item \textbf{Delivered-energy proxy:} \(E_{\mathrm{del}}=\int_{0}^{\mathrm{TTE}} V_{\mathrm{term}}(t)I(t)\,dt\) (when current \(I(t)\) is available from the closure), which supports sanity checks against SOC depletion;
  \item \textbf{Thermal exposure:} \(\max_{t\in[0,\mathrm{TTE}]} T_b(t)\), reflecting potential thermal throttling or safety constraints.
\end{itemize}

\paragraph{Risk event: CPL feasibility (voltage-collapse risk).}
Because the load is modeled as a constant-power demand (CPL) coupled to the electrochemical model, a feasibility condition naturally arises.
Let
\[
\Delta(t)=\big(V_{\mathrm{oc}}(z(t)) - v_p(t)\big)^2 - 4R_0(T_b(t),S(t))\,P_{\mathrm{tot}}(t),
\]
where \(P_{\mathrm{tot}}(t)\) is the demanded total power and \(R_0\) the ohmic resistance.
When \(\Delta(t)<0\), the CPL algebraic closure admits no real current solution, indicating that the demanded power is infeasible given the instantaneous battery capability and may lead to abrupt voltage collapse.
We therefore introduce the \emph{first risk time}
\[
t_{\Delta}=\inf\{t>0:\ \Delta(t)\le 0\},
\]
as an auxiliary diagnostic.
In later sections, we use \(t_\Delta\) to distinguish \emph{infeasibility-driven} shutdown risk from ordinary energy depletion.

\paragraph{Representative usage scenarios.}
To ensure that conclusions are interpretable and reproducible, we evaluate the model under a small set of canonical usage scenarios, each defined by a characteristic input pattern \(\mathbf{u}(t)\).
Table~\ref{tab:scenarios} summarizes the scenarios used throughout the paper.

\begin{table}[t]
\centering
\caption{Representative usage scenarios and their qualitative input characteristics.}
\label{tab:scenarios}
\begin{tabular}{p{2.4cm}p{10.6cm}}
\hline
Scenario & Input characteristics \(\mathbf{u}(t)=[L(t),C(t),N(t),\Psi(t),T_a(t)]^\top\) \\
\hline
Standby/Idle &
Low brightness \(L\approx 0\) (screen off), low CPU \(C\ll 1\), sporadic network \(N\approx 0\) with residual tail \(w\), typical \(\Psi\), moderate \(T_a\). \\
Browsing/Social &
Moderate \(L\), moderate CPU \(C\), intermittent network bursts \(N(t)\) with tail effects, typical-to-good \(\Psi\), moderate \(T_a\). \\
Video Streaming &
High \(L\), sustained moderate CPU \(C\), sustained network activity \(N\) (downlink), sensitivity to \(\Psi\); moderate \(T_a\). \\
Gaming/High Compute &
High \(L\), high CPU \(C\) (near saturation), moderate network \(N\), typical \(\Psi\); emphasizes thermal rise and possible throttling. \\
Weak Signal (Stress) &
Moderate-to-high \(L\), moderate CPU \(C\), nontrivial \(N\) under poor signal \(\Psi\downarrow\); stresses the signal-quality penalty in \(P_{\mathrm{net}}(N,\Psi,w)\) and increases collapse risk. \\
Cold Ambient (Stress) &
Any of the above with low \(T_a\); highlights increased \(R_0\) and reduced \(Q_{\mathrm{eff}}\), potentially shortening TTE and increasing \(t_\Delta\) likelihood. \\
\hline
\end{tabular}
\end{table}

The above scenarios are not tied to a specific dataset; they can be instantiated using recorded traces or generated synthetically (e.g., piecewise-smooth profiles or stochastic processes) while keeping the same physical meaning of each input channel.
This design supports both deterministic simulations and uncertainty quantification (Monte Carlo) in later sections.
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