The Lyapunov equation, named after the Russian mathematician Aleksandr Lyapunov, is a matrix equation used in the stability analysis of linear dynamical systems. In particular, the discrete-time Lyapunov equation (also known as Stein equation) for X {\displaystyle X} is
A X A H − X + Q = 0 {\displaystyle AXA^{H}-X+Q=0}
where Q {\displaystyle Q} is a Hermitian matrix and A H {\displaystyle A^{H}} is the conjugate transpose of A {\displaystyle A} , while the continuous-time Lyapunov equation is
A X + X A H + Q = 0 {\displaystyle AX+XA^{H}+Q=0} .
Application to stability In the following theorems A , P , Q ∈ R n × n {\displaystyle A,P,Q\in \mathbb {R} ^{n\times n}} , and P {\displaystyle P} and Q {\displaystyle Q} are symmetric. The notation P > 0 {\displaystyle P>0} means that the matrix P {\displaystyle P} is positive definite. Theorem (continuous time version). Given any Q > 0 {\displaystyle Q>0} , there exists a unique P > 0 {\displaystyle P>0} satisfying A T P + P A + Q = 0 {\displaystyle A^{T}P+PA+Q=0} if and only if the linear system x ˙ = A x {\displaystyle {\dot {x}}=Ax} is globally asymptotically stable. The quadratic function V ( x ) = x T P x {\displaystyle V(x)=x^{T}Px} is a Lyapunov function that can be used to verify stability. Theorem (discrete time version). Given any Q > 0 {\displaystyle Q>0} , there exists a unique P > 0 {\displaystyle P>0} satisfying A T P A − P + Q = 0 {\displaystyle A^{T}PA-P+Q=0} if and only if the linear system x t + 1 = A x t {\displaystyle x_{t+1}=Ax_{t}} is globally asymptotically stable. As before, x T P x {\displaystyle x^{T}Px} is a Lyapunov function.
Computational aspects of solution The Lyapunov equation is linear; therefore, if X {\displaystyle X} contains n {\displaystyle n} entries, the equation can be solved in O ( n 3 ) {\displaystyle {\mathcal {O}}(n^{3})} time using standard matrix factorization methods. However, specialized algorithms are available which can yield solutions much quicker owing to the specific structure of the Lyapunov equation. For the discrete case, the Schur method of Kitagawa is often used. For the continuous Lyapunov equation the Bartels–Stewart algorithm can be used.
Analytic solution Defining the vectorization operator vec ( A ) {\displaystyle \operatorname {vec} (A)} as stacking the columns of a matrix A {\displaystyle A} and A ⊗ B {\displaystyle A\otimes B} as the Kronecker product of A {\displaystyle A} and B {\displaystyle B} , the continuous time and discrete time Lyapunov equations can be expressed as solutions of a matrix equation. Furthermore, if the matrix A {\displaystyle A} is "stable", the solution can also be expressed as an integral (continuous time case) or as an infinite sum (discrete time case).
Discrete time Using the result that vec ( A B C ) = ( C T ⊗ A ) vec ( B ) {\displaystyle \operatorname {vec} (ABC)=(C^{T}\otimes A)\operatorname {vec} (B)} , one has
( I n 2 − A ¯ ⊗ A ) vec ( X ) = vec ( Q ) {\displaystyle (I_{n^{2}}-{\bar {A}}\otimes A)\operatorname {vec} (X)=\operatorname {vec} (Q)}
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