Numerical Solution of Eigenvalue Problems for Alternating Matrix Polynomials and Their Application in Control Problems for Descriptor Systems

Numerical methods for eigenvalue problems associated to alternating matrix pencils and polynomials are discussed. These problems arise in a large number of control applications for differential-algebraic equations ranging from regular and singular linear-quadratic optimal and robust control to dissipativity checking. We present a survey of several of these applications and give a systematic overview over the theory and the numerical solution methods. Our solution concept is based throughout on the computation of eigenvalues and deflating subspaces of even matrix pencils. The unified approach allows to generalize and improve several techniques that are currently in use in systems and control.