Matrix Riccati Equations in Control and Systems Theory

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 Fréchet 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.