Lyapunov equation

In control theory, the discrete Lyapunov equation (also known as Stein equation) is of the form where is a Hermitian matrix and is the conjugate transpose of . The continuous Lyapunov equation is of the form 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 , and and are symmetric. The notation means that the matrix is positive definite. Theorem (continuous time version). Given any , there exists a unique satisfying if and only if the linear system is globally asymptotically stable. The quadratic function is a Lyapunov function that can be used to verify stability. Theorem (discrete time version). Given any , there exists a unique satisfying if and only if the linear system is globally asymptotically stable. As before, is a Lyapunov function. The Lyapunov equation is linear, and so if contains entries can be solved in 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. Defining the vectorization operator as stacking the columns of a matrix and as the Kronecker product of and , the continuous time and discrete time Lyapunov equations can be expressed as solutions of a matrix equation. Furthermore, if the matrix 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 , one has where is a conformable identity matrix and is the element-wise complex conjugate of . One may then solve for by inverting or solving the linear equations. To get , one must just reshape appropriately. Moreover, if is stable (in the sense of Schur stability, i.e.