Page 9 - Textos de Matemática Vol. 36

Preface
In many fields of applied mathematics, engineering and economic sciences there appear, e.g., as a consequence of variational problems to be solved, matrix (or operator) Riccati equations. These can be algebraic Riccati equations as well as Riccati difference or differential equations. Riccati equations are in our opinion the simplest but most important class of non-linear equations. They show up in the following domains, just to cite a few:
— linear optimal control and filtering problems with quadratic cost functionals, — linear dynamic games with quadratic cost functionals,
— decoupling of linear systems of differential and difference equations,
— spectral factorization of operators,
— singular perturbation theory,
— boundary value problems for systems of ODEs, — invariant embedding and scattering theory,
— differential geometry.
Besides the well-developed theory of symmetric (or Hermitian) Riccati equations there are more and more demands from applications to develop further the theory for non-symmetric matrix Riccati equations and also for generalized or perturbed Riccati equation.
In the last decades developments of both algebraic and differential matrix Riccati equations theory spread out in the scientific literature. In contrast to the theory of symmetric (or Hermitian) matrix algebraic Riccati equations, which has been presented recently in the book of Lancaster and Rodman [LaRo95], the basic theory of matrix Riccati differential equations is comprised in the mono- graph of Reid [Reid72], along with some applications that were developed before 1972. During the last three decades there was achieved great progress in the mathematical theory of Riccati equations and in its applications, with emphasis on control systems and differential games. Whereas symmetric Riccati equations play a central role in optimal control, non-symmetric matrix Riccati equations show up for instance in the theory of dynamic games and spectral factorization problems, while generalized Riccati equations are common in stochastic control problems or stochastic games.
In a recent book by Abou Kandil, Freiling, Ionescu and Jank [AFIJ03] these problems have been addressed and in this lecture we will present some of these topics. Because of the limited time we confine to the most fundamental properties of matrix Riccati differential equations.
The topics presented in detail in this lecture have been selected according to personal tastes and interests and limited time. Primarily we have focused on the representation formulae of solutions on those parts of the theory that point to applications in linear control theory and dynamic games. For further reading in the theory of Riccati equations and their application, beside [AFIJ03]