Notes on the Riccati operator equation in open quantum systems

Lecture notes on open quantum systems

It is generally impossible to isolate a particular quantum system in which we are interested from its surroundings. Hence, for almost all quantum systems we would like to study, we must consider the influence of the external environment upon the system dynamics, in order to faithfully represent the real dynamical evolution of the physical system. However, we are usually unable to track, either ...

Extended Matrix Pencils for the Delta-Operator Riccati Equation

Modern optimal control techniques such as H2 and H1 control rely on the solution of algebraic Riccati equations for controller synthesis. Reliable numerical techniques for numerical computation of the solution of these equations have been proposed using eigenvector or Schur decompositions of Hamiltonian matrices for continuous-time algebraic Riccati equations (CARE), or symplectic matrices for ...

Conservation law of operator current in open quantum systems

On the existence of solutions to the operator Riccati equation and the tan Θ theorem

Let A and C be self-adjoint operators such that the spectrum of A lies in a gap of the spectrum of C and let d > 0 be the distance between the spectra of A and C . We prove that under these assumptions the sharp value of the constant c in the condition ‖B‖ < cd implying the solvability of the operator Riccati equation XA−CX+XBX = B∗ is equal to √ 2. We also prove an extension of the Davis-Kahan...

A Survey of Riccati Equation Results in Negative Imaginary Systems Theory and Quantum Control Theory

This paper presents a survey of some new applications of algebraic Riccati equations. In particular, the paper surveys some recent results on the use of algebraic Riccati equations in testing whether a system is negative imaginary and in synthesizing state feedback controllers which make the closed loop system negative imaginary. The paper also surveys the use of Riccati equation methods in the...