Quadratic Functionals in Variational Analysis and Control Theory

This monograph is intended to provide classical and recent results on quadratic functionals. On the one hand it contains the well-known facts on the second variation from the calculus of variations, including Jacobi-type conditions, and on the other hand new aspects in control theory, in particular the notion of strong observability and its relation to the optimal linear regulator. The main object described is a general theory of self-adjoint eigenvalue problems for linear Hamiltonian systems, which includes Morse's oscillation theory and his extensions of Sturmian theory. The dependence on the eigenvalue parameter may be nonlinear. The treatment is based on a novel approach via field theory, in particular Picone's identity. The central features needed for the method are recent results on Riccati matrix differential equations and on monotone matrix-valued functions. Applications of the theory yield classical and new results in such areas as, for example, linear control theory, variational analysis (Rayleigh's principle), or Sturm-Liouville eigenvalue problems. The book is self-contained, and accessible to mathematics or science students entering the graduate level.