[proofplan] Substitute the controller equations into the plant equation and write the resulting autonomous system for the augmented state $(x,x_c)$. Internal stability for this finite-dimensional autonomous interconnection is exactly exponential stability of the augmented homogeneous system, and the standard finite-dimensional criterion says this is equivalent to the augmented matrix being Hurwitz. [/proofplan] [step:Compute the augmented closed-loop dynamics] Substitute $y(t)=Cx(t)$ into the controller output equation: \begin{align*} u(t)=C_cx_c(t)+D_cCx(t). \end{align*} Then the plant state equation becomes \begin{align*} \dot{x}(t)=(A+BD_cC)x(t)+BC_cx_c(t). \end{align*} Similarly, substituting $y(t)=Cx(t)$ into the controller state equation gives \begin{align*} \dot{x}_c(t)=B_cCx(t)+A_cx_c(t). \end{align*} Thus the augmented state $z(t):=(x(t),x_c(t))\in\mathbb R^n\times\mathbb R^q$ satisfies \begin{align*} \dot{z}(t)=A_{\mathrm{cl}}z(t). \end{align*} [/step] [step:Apply the finite-dimensional Hurwitz stability criterion] By definition, internal stability of this autonomous finite-dimensional closed-loop interconnection means that every component of the augmented homogeneous response decays exponentially to zero for every initial state $(x(0),x_c(0))$. This is the same as exponential stability of the origin for \begin{align*} \dot{z}(t)=A_{\mathrm{cl}}z(t). \end{align*} For a finite-dimensional real linear autonomous system, exponential stability of the origin is equivalent to the system matrix being Hurwitz. Applying this criterion to the matrix of $A_{\mathrm{cl}}$ gives that the interconnection is internally stable if and only if $A_{\mathrm{cl}}$ is Hurwitz. [guided] The only calculation is to form the augmented state equation. The measured output is $y=Cx$, so the controller output is $u=C_cx_c+D_cCx$. Substituting this into $\dot{x}=Ax+Bu$ gives $\dot{x}=(A+BD_cC)x+BC_cx_c$. The controller state equation becomes $\dot{x}_c=B_cCx+A_cx_c$. Therefore the pair $z=(x,x_c)$ evolves by $\dot{z}=A_{\mathrm{cl}}z$ with exactly the displayed operator $A_{\mathrm{cl}}$. Internal stability for this finite-dimensional closed-loop system means exponential decay of the augmented homogeneous response from every initial state. The standard finite-dimensional linear stability criterion says that $\dot{z}=Mz$ is exponentially stable exactly when every eigenvalue of $M$ has negative real part, that is, exactly when $M$ is Hurwitz. Taking $M=A_{\mathrm{cl}}$ proves both directions. [/guided] [/step]