# Value of $\gamma$ in the H-infinity norm

Suppose I have the system:

$$\dot{x} = Ax+Bu\\ y=Cx+Du$$

and the following Hamiltonian matrix:

$$H=\begin{pmatrix} A & \frac{1}{2}B^TB\\ -CC^T&-A \end{pmatrix}$$

I want to find the value of $$\gamma$$ which is the bound of the $$H_{\infty}$$ norm, so it is the value such that $$\left |T(j\omega) \right |_{\infty }<\gamma$$, where $$\left |T(j\omega) \right |_{\infty }$$ is the H-infinity of the transfer function $$T(j\omega)$$.

I know that for the bounded real lemma, if the eigenvalues of $$A$$ have a negative real part, and $$I\gamma^2-DD^T>0$$, then the Hamiltonian have no eigenvalues on the imaginary axis. I also know that if the eigenvalues of $$A$$ have a negative real part, then $$\left |T(j\omega) \right |_{\infty }<\gamma$$ holds.

But my question is: how do I find the value of $$\gamma$$?

I have been told the result is $$\gamma=0.5$$ but I really can't get to this result. I have tried using the Shur's complement to see if this matrix is negative definite (so before doing that I switched sign to the Hamiltonian). In this way, I thought that if the A matrix is negative definite, it has all eigenvalues with a negative real part, so the resulting value of $$\gamma$$ form the computation would have been the searched value. But I don't find the desired result. Maybe I am missing a point and doing something wrong, or maybe I am completely on the wrong path.

Finding the $$H_\infty$$ norm of a linear system is not trivial. There are many numerical methods. A classic paper is: http://stanford.edu/~boyd/papers/bisection_hinfty.html