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\title{\bf The GCAT Framework: A Control-Theoretic Model of Governed Autonomy}
\author{[Authors anonymized for submission]}

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\begin{document}
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\begin{abstract}
We present a control-theoretic realization of the GCAT framework. A four-dimensional state space over governance capacity, control authority, autonomy, and trust is coupled to a legitimacy-capacity function that induces an admissible set. We show that this admissible set can be enforced through a barrier condition, derive the relevant directional derivative, and construct a minimally invasive control law that preserves admissibility under control-affine dynamics. This version of GCAT is intentionally dynamical and serves as the control-oriented counterpart to the geometric foundation.
\end{abstract}
\section{Introduction}
The geometric GCAT paper defines what states are admissible. This paper asks a different question: if the system evolves dynamically, can admissibility be maintained by control? The answer is yes under standard control-affine assumptions, provided the admissibility margin is treated as a barrier quantity.
\section{State, Capacity, and Admissibility}
Let
\[
\x=(g,c,a,t)\in[0,1]^4
\]
and
\[
\Lam(\x)=K g^\alpha c^\beta t^\gamma,
\qquad
m(\x)=\Lam(\x)-a.
\]
The admissible set is
\[
\A=\{\x:m(\x)\ge 0\}.
\]
\section{Dynamics}
We consider control-affine dynamics
\[
\dot{\x}=f(\x)+G(\x)u,
\]
with locally Lipschitz $f$ and $G$. For analysis we instantiate
\begin{align}
\dot g &= k_g(g_0-g)-\eta_g a,\\
\dot c &= k_c(c_0-c),\\
\dot a &= k_a a(1-a)+\eta_a(1-g)+u_a,\\
\dot t &= k_t(t_0-t),
\end{align}
where $u_a$ is the control channel used to suppress autonomy growth near the boundary.
\section{Barrier Formulation}
The admissibility margin acts as the barrier quantity. Its derivative is
\[
\dot m(\x)=\nabla \Lam(\x)\cdot \dot{\x}-\dot a.
\]
A sufficient condition for forward invariance of $\A$ is
\[
\dot m(\x,u)\ge -\kappa m(\x),\qquad \kappa>0.
\]
When this holds, trajectories starting in $\A$ remain in $\A$.
\section{Control Synthesis}
The barrier inequality can be solved for admissible inputs. A practical choice is a quadratic program:
\begin{align}
u^\star=\arg\min_u \|u-u_{nom}\|^2\\
\text{s.t.}\quad \dot m(\x,u)\ge -\kappa m(\x).
\end{align}
For the scalar autonomy channel, a simple fallback law is
\[
u_a^\star=\min\left(0,\phi(\x)+\kappa m(\x)\right),
\]
where $\phi(\x)$ collects the noncontrol terms in the barrier derivative.
\section{Boundary Behavior}
At the admissible boundary $m(\x)=0$, the condition reduces to
\[
\dot m(\x,u)\ge 0.
\]
This is the tangency condition preventing outward crossing.
\section{Conclusion}
We presented a control-oriented version of GCAT in which the admissibility margin functions as a barrier quantity. Under control-affine dynamics, this yields an enforcement mechanism that preserves capacity-bounded autonomy over time.
\end{document}
