I have the heat equation :
\begin{equation} \frac{\partial u\left(x,y,t\right)}{\partial t}=\alpha\nabla^2 u\left(x,y,t\right)+\beta I\left(x,y\right) \label{eq:HE} \end{equation}
as $u$ to be the temperature value at time $t$ correspond to the point $\left(x,y\right)$ and
\begin{equation} \nonumber \nabla^2 u\left(x,y,t\right):=\left(\frac{\partial^2 u\left(x,y,t\right)}{\partial x^2}\right)+ \left(\frac{\partial^2 u\left(x,y,t\right)}{\partial y^2}\right). \label{eq:Nabla-2} \end{equation}
In addition
\begin{equation} \nonumber \alpha:= \frac{\kappa}{c \rho} \label{eq:alpha} \end{equation}
and
\begin{equation} \nonumber \beta:= \frac{1}{c \rho} \label{eq:beta} \end{equation}
with $\kappa$, $c$ and $\rho$ all in $\mathbb{R}^{+}$ to be thermal conductivity, the specific heat capacity and the mass density, respectively. We let $x\in\left[-1,+1\right]$, $y\in\left[-1,+1\right]$, $t\in\mathbb{R}^{+}$ and also $u\left(x,y,0\right)=0$.
Here, $I$ to simulate an external source of heat that is a Gaussian function$\colon$
\begin{equation} I\left(x,y\right) = I_0 \cdot \exp{ \left[-2 \left[ \left(\frac{x-x_c}{\omega}\right)^2 + \left(\frac{y-y_c}{\omega}\right)^2 \right] \right]} \label{eq:Heat} \end{equation}
with an initial value
\begin{equation} I_0= \frac{2\mathcal{P}}{\pi\omega^2} \label{eq:Heat-I} \end{equation}
by knowing $\omega$ and $\mathcal{P}$ to be the radius of the Gaussian waist and the power of the heat power of the Gaussian distribution, respectively. We set $\mathcal{P}=10^6$ (Watt) and $\omega=3$ pixels while letting $\left(x_c,y_c\right)$ to take all possible trajectory points that the Gaussian distribution can move over.
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My question is now, how the heat equation explained above along with its external source of heat can be rewritten in terms of a state-space model?
The final goal is to solve the state-space model of heat equation using linear quadratic regulator (LQR) approach.
Many thanks