Matrix Riccati Equations in Control and Systems Theory

"In summary, this is a very well written and extremely useful book. It should be included in the library of everybody working on Ricatti equations, and in all areas of control, signals, and systems... [It] will be a standard reference book in this area in the future." -SIAM Review

• 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.- 3.2 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.- 3.5 A representation formula for solutions of the discrete time Riccati equation.- 3.6 Global existence results.- 4 Hermitian Riccati differential equations.- 4.1 Comparison results for HRDE.- 4.2 Monotonicity and convexity results: A Fréchet derivative based approach.- 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.- 8 Applications to Robust Control Systems.- 8.1 The Four Block Nehari Problem.- 8.2 Disturbance Attenuation.- 9 Non-symmetric Riccati theory and applications.- 9.1 Non-symmetric Riccati theory.- 9.2 Application to open loop Nash games.- 9.3 Application to open loop Stackelberg games.- A Appendix.- A.1 Basic facts from control theory.- A.2 The implicit function theorem.- References.- List of Figures.