Three Heated Rooms

Consider three heated rooms depicted in the attached image. The first room can be heated directly by the input $u^1$, the heat transfer into the room. The other two rooms are heated via two boilers with the inputs $u^2$ and $u^3$ respectively. The temperature of each room is described by $x^1$, $x^2$, and $x^3$ respectively, the temperature of each boiler by $x^4$ and $x^5$. The heat emission of each room is considered as a nonlinear function of the room temperature. With $x = (x^1,\ldots, x^5)^{\mathrm T}$ the the nonlinear system has the form

$$\dot{x}=\begin{pmatrix} -c_0(x^2-T_0)-c_1(x^1-T_0)^2\\ -c_0(x^2-T_0)-c1(x^2-T_0)^2 + c_2(x^4-x^2)\\ -c_0(x^3-T_0)-c1(x^3-T_0)^2 + c_2(x^5-x^3)\\ -c_2(x^4-x^2)\\ -c_2(x^5-x^3) \end{pmatrix} + \begin{pmatrix} 1 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} u^1\\ u^2\\ u^3 \end{pmatrix}, $$

where $c_0$, $c_1$, and $c_2$ are parameters for the heat transmission, and $T_0$ is the temperature outside of the rooms. We will choose our parameters so that the time is measured in hours and $x^i$ is measured in Kelvin. The state manifold is $\mathcal{M}= \mathbb{R}^5$, and the coupling conditions are

$y=h(x)=\begin{pmatrix}x^1-x^2\\x^1-x^3\end{pmatrix}=0,$

i.e. the temperature of the three rooms should be equal.