A new numerical solution of<tex>X = A₁X + XA₂ + D, X(0) = C</tex>

Another numerical solution of the general matrix differential equation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∘{X}=A₁X+XA₂+D, X(0)=C</tex> for X is considered without any stability condition for A <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> and A <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> . Like Davison's method, the proposed algorithm requires only some n <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> words of memory and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n_{3</tex> multiplications where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n=max(n₁,n₂)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A ∈ R^{n₁ × n₁},A₂ ∈ R^{n₂ × n₂}</tex> . This new approach is well suited to solve large and possibly unstable systems. We take the opportunity to run the differential equation for various D. A very efficient technique follows to design the so-called receding horizon control problem.