The authors present the theory of symmetric (Hermitian) matrix Riccati equations and contribute to the development of the theory of non-symmetric Riccati equations as well as to certain classes of coupled and generalized Riccati equations occurring in differential games and stochastic control. The volume offers a complete treatment of generalized and coupled Riccati equations. It deals with differential, discrete-time, algebraic or periodic symmetric and non-symmetric equations, with special emphasis on those equations appearing in control and systems theory. Extensions to Riccati theory allow to tackle robust control problems in a unified approach.The book makes available classical and recent results to engineers and mathematicians alike. It is accessible to graduate students in mathematics, applied mathematics, control engineering, physics or economics. Researchers working in any of the fields where Riccati equations are used can find the main results with the proper mathematical background.
1 Basic results for linear equations.- 1.1 Linear differential equations and linear algebraic equations.- 1.2 Exponential dichotomy and L2evolutions.- 2 Hamiltonian Matrices and Algebraic Riccati equations.- 2.1 Solutions of algebraic Riccati equations and graph subspaces.- 2.2 Indefinite scalar products and a canonical form of Hamiltonian matrices.- 2.3 Hermitian algebraic Riccati equations.- 2.4 Positive semi-definite solutions of standard algebraic Riccati equations.- 2.5 Hermitian discrete-time algebraic Riccati equations.- 3 Global aspects of Riccati differential and difference equations.- 3.1 Riccati differential equations and associated linear systems.- 3.1.1 Riccati differential equations, Riccati-transformation and spectral factorization.- 3.1.2 Riccati differential equations and linear boundary value problems.- 3.2 A representation formula.- 3.3 Flows on Grassmann manifolds: The extended Riccati differential equation.- 3.4 General representation formulae for solutions of RDE and PRDE, the time-continuous and periodic Riccati differential equation, and dichotomy.- 3.4.1 A general representation formula for solutions of RDE.- 3.4.2 A representation formula for solutions of the periodic Riccati differential equation PRDE.- 3.5 A representation formula for solutions of the discrete time Riccati equation.- 3.5.1 Properties of the solutions to DARE.- 3.5.2 Properties of the solutions to DRDE.- 3.6 Global existence results.- 4 Hermitian Riccati differential equations.- 4.1 Comparison results for HRDE.- 4.1.1 Arbitrary coefficients.- 4.1.2 Periodic coefficients.- 4.1.3 Constant coefficients.- 4.1.4 Riccati inequalities.- 4.2 Monotonicity and convexity results: A Frechet derivative based approach.- 4.2.1 Notation and preliminaries.- 4.2.2 Results for HARE.- 4.2.3 Results for HDARE.- 4.2.4 Results for HRDE.- 4.3 Convergence to the semi-stabilizing solution.- 4.4 Dependence of HRDE on a parameter.- 4.5 An existence theorem for general HRDE.- 4.6 A special property of HRDE.- 5 The periodic Riccati equation.- 5.1 Linear periodic differential equations.- 5.2 Preliminary notation and results for linear periodic systems.- 5.3 Existence results for periodic Hermitian Riccati equations.- 5.4 Positive semi-definite periodic equilibria of PRDE.- 6 Coupled and generalized Riccati equations.- 6.1 Some basic concepts in dynamic games.- 6.2 Non-symmetric Riccati equations in open loop Nash differential games.- 6.3 Discrete-time open loop Nash Riccati equations.- 6.4 Non-symmetric Riccati equations in open loop Stackelberg differential games.- 6.5 Discrete-time open loop Stackelberg equations.- 6.6 Coupled Riccati equations in closed loop Nash differential games.- 6.7 Rational matrix differential equations arising in stochastic control.- 6.8 Rational matrix difference equations arising in stochastic control.- 6.9 Coupled Riccati equations in Markovian jump systems.- 7 Symmetric differential Riccati equations: an operator based approach.- 7.1 Popov triplets: definition and equivalence.- 7.2 Associated objects.- 7.3 Associated operators.- 7.4 Existence of the stabilizing solution.- 7.5 Positivity theory and applications.- 7.6 Differential Riccati inequalities.- 7.7 The signature condition.- 7.8 Differential Riccati theory: A Hamiltonian descriptor operator approach.- 7.8.1 Descriptors and dichotomy.- 7.8.2 Hamiltonian descriptors.- 7.8.3 The stabilizing (anti-stabilizing) solution.- 8 Applications to Robust Control Systems.- 8.1 The Four Block Nehari Problem.- 8.1.1 Problem Statement.- 8.1.2 A characterization of all solutions.- 8.1.3 Main Result.- 8.2 Disturbance Attenuation.- 8.2.1 Problem statement.- 8.2.2 A necessary condition.- 8.2.3 The Disturbance Feedforward Problem.- 8.2.4 The least achievable tolerance of the DF problem.- 9 Non-symmetric Riccati theory and applications.- 9.1 Non-symmetric Riccati theory.- 9.1.1 Basic notions and preliminary results.- 9.1.2 Toeplitz operators and Riccati equations.- 9.2 Application to open loop Nash games.- 9.2.1 Definitions and Hilbert space.- 9.2.2 Unique Nash equilibria.- 9.2.3 The general case.- 9.2.4 If any, then one or infinitely many.- 9.3 Application to open loop Stackelberg games.- 9.3.1 Characterization in Hilbert space.- 9.3.2 Unique Stackelberg equilibria.- 9.3.3 A value function type approach.- A Appendix.- A.1 Basic facts from control theory.- A.2 The implicit function theorem.- References.- List of Figures.