LQR with derivative-dependent performance?

Question

Given a standard LTI system with

$$ \dot{x} = A x + B u $$

The standard LQR finds the control gain $K$ of the state feedback $u = -Kx$ such that

$$ J_1 = \int_0^\infty \big( x^T Q x + u^T R u \big) dt \tag{1} $$

is minimized, given some positive (semi) definite $Q$ and $R$.

However, what if one is also interested in penalizing the derivatives of the controlled system? With that I mean to extend $(1)$ like

$$ J_2 = \int_0^\infty \big( x^T Q x + u^T R u + \dot{x}^T S \dot{x} \big) dt \tag{2} $$

with some positive (semi) definite $S$.

Questions:

1. Has something like $(2)$ been considered before? If so, I am very interested in references.
2. Is it possible to minimize $(2)$ in a similar way like $(1)$? Will the optimal controller be linear after all?

Answer 1

You have $\dot{x}^TS\dot{x} = (Ax+Bu)^TS(Ax+Bu)$ so it's just a standard LQ problem with a mixed term (the general LQ penalty is $x^TQx+u^TRu + x^TNu$)