Dear Bachir,

Yes we can absolutely do this,  there are many references that prove the feasibility of this. I suggest to you links and attached files in topics.

-Design LQG Tracker Using Control System Designer - MATLAB ...

https://www.mathworks.com › ... › LQG Design

-Form Linear-Quadratic-Gaussian (LQG) servo controller - MATLAB ...

https://www.mathworks.com/help/control/ref/lqgtrack.html

-Example: LQG Design for Set Point Tracking - rohan.sdsu.edu

www-rohan.sdsu.edu/doc/matlab/toolbox/control/.../desig33b.html

-Aircraft Control System Using LQG and LQR Controller with Optimal ...

www.sciencedirect.com/science/article/pii/S1877705814011771

Best regards

I'd say that as long as you guarantee such stability with Lyapunov theory, you are done! Just write the desired performance and look for it the right Lypunov candidate. In linear systems, a quadratic matrix function does the job (i.e., an energy -like function used in your system). Further steps are straightforward. Nowadays, you could include the LINEAR MATRIX INEQUALITY (LMI) THEORY which is more powerful, general and elegant than working with Lypunov only. Indeed, Lypunov stability theory is a particular case of LMIs:

https://web.stanford.edu/~boyd/lmibook/lmibook.pdf

I used that book in my PhD and another "local" one:

http://www.dcsc.tudelft.nl/~cscherer/lmi/notes99.pdf

I found quite useful to combine both of them.

Do not separate the question of stability and controllability.  Understanding in right perspective these two concepts are very important because it is a chicken and egg problem. Take the example of an aircraft control to explain this. By structural construction of aircraft using L/D ratio one can achieve complete stability. But one should make it unstable but controllable to achieve the goal of manoeuvrability of the aircraft. So to stay it is a balance of stability and controllability needed rather looking at it as isolated issues.