Lyapunov equation

In control theory, the discrete Lyapunov equation is of the form In control theory, the discrete Lyapunov equation is of the form where Q {displaystyle Q} is a Hermitian matrix and A H {displaystyle A^{H}} is the conjugate transpose of A {displaystyle A} . The continuous Lyapunov equation is of form: A X + X A H + Q = 0 {displaystyle AX+XA^{H}+Q=0} . The Lyapunov equation occurs in many branches of control theory, such as stability analysis and optimal control. This and related equations are named after the Russian mathematician Aleksandr Lyapunov. In the following theorems A , P , Q ∈ R n × n {displaystyle A,P,Qin mathbb {R} ^{n imes 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, z T P z {displaystyle z^{T}Pz} is a Lyapunov function. Specialized software is available for solving Lyapunov equations. For the discrete case, the Schur method of Kitagawa is often used. For the continuous Lyapunov equation the method of Bartels and Stewart can be used. Defining the vec ⁡ ( A ) {displaystyle operatorname {vec} (A)} operator as stacking the columns of a matrix A {displaystyle A} and A ⊗ B {displaystyle Aotimes 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). 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