Note that one way to solve the algebraic Riccati equation is by iterating the dynamic Riccati equation of the finite-horizon case until it converges.
nˈoʊt ðˈætwˌʌn wˈeɪ tə sˈɑːlv ðɪ ˌældʒɪbɹˈeɪɪk ɹɪkˈɑːɾi ᵻkwˈeɪʒən ɪz baɪ ˈɪɾɚɹˌeɪɾɪŋ ðə daɪnˈæmɪk ɹɪkˈɑːɾi ᵻkwˈeɪʒən ʌvðə fˈaɪnaɪthɚɹˈaɪzən kˈeɪs ʌntˈɪl ɪt kənvˈɜːdʒᵻz.