If all the matrices in the problem formulation are time-invariant and if the horizon  tends to infinity the discrete-time LQG controller becomes time-invariant.
ɪf ˈɔːl ðə mˈeɪtɹɪsˌiːz ɪnðə pɹˈɑːbləm fˌɔːɹmjʊlˈeɪʃən ɑːɹ tˈaɪmɪnvˈɛɹiənt ænd ɪf ðə hɚɹˈaɪzən tˈɛndz tʊ ɪnfˈɪnᵻɾi ðə dɪskɹˈiːttˈaɪm ˌɛlkjˌuːdʒˈiː kəntɹˈoʊlɚ bɪkˌʌmz tˈaɪmɪnvˈɛɹiənt.