You can also solve the LQR problem for discrete systems instead of continuous. One does needs to solve the DARE instead of CARE. But this does require having a discretized state space model. A simple but bad discetization method would be forward Euler. But there are a lot of other methods out there, such as zero-order-hold, first-order-hold and bilinear-transform/Tustin. In certain situations one method would be preferable over another and a decent overview is given in this YouTube series by Brian Douglas. But at low frequencies they all yield the same response, so if the sample rate is significantly faster than the dynamics of the system then it won't matter much which method you pick.
It can also be noted that it is very likely that the continuous LQR gain you calculated for the continuous system will also work on a digital controller if the absolute values of the closed loop poles lie significantly below the sample frequency. Namely most often zero-order-hold would be a good representation of what the digital controller does. Namely it calculates the controller output and holds it for one sample time. This roughly adds half a sample time delay, which might make the system unstable if the closed loop poles lie close to the sample frequency. But a delay only reduces the phase margin of the controller, but LQR has a guaranteed phase margin of 60°, so should always be able to handle some decent amount of delay. For example when use would use poles placement you would not have this 60° phase margin guarantee. But with LQR the performance might not be as optimal as the LQR designed for the discretized system.