Maximize Frobenius Norm constrained to Linear Dynamics

Question

Dears,

In a Control Systems context, i want to maximize the Frobenius norm of a matrix K (the controller of my system).

Subject to the following constraints:

1. The evolution of the Linear Time Invariant Dynamics: $x_{j+1}=Ax_j+BKx_j$
2. A linear constraint: $Hx_j\leq h$
3. A given time horizon (for example $j$ that goes from 1 to 10).

So i defined it as:

$$\begin{array}{ll} \underset{\mathbf{K}}{\text{maximize}} & \| \mathbf{K} \|_F^2\\ \text{subject to} & x_{j+1} = A x_j + B K x_j\\ & H x_j \leq h, \quad j \in \{0,1,\dots,N-1\}\end{array}$$

For simplicity i have been trying to keep things easy, like using easy matrices (Dimensions used -> A: 2x2 Matrix; B:2x1 Matrix; xj: 2x1 matrix; K: 1x2 Matrix; H: 1x2 Matrix; and h as a scalar).

I believe it has some properties to deal with it as a convex problem because the target function is a Norm, also the inequality constraint is linear, but i'm not fully sure if the equality constraint (LTI Evolution) is a convex constraint or affine. For instance, $X_{10}$ would be too complex to develop and find if its convex.

My first question: How to check if the first constraint (LTI evolution) is convex?
My second question: If it is not convex, is it possible to find a way to convexify it?

Thanks!