Representation and Control of Infinite Dimensional Systems

I Qualitative Properties of Linear Dynamical Systems.- 1 Control of Linear Finite Dimensional Differential Systems Revisited.- 1 Introduction.- 2 Controllability and observability of finite dimensional linear systems.- 2.1 Controllability.- 2.2 Observability.- 2.3 Duality.- 2.4 Canonical structure for linear systems.- 2.5 The pole-assigment theorem.- 2.6 Stabilizability and detectability.- 2.7 Applications of controllability and observability.- 3 Optimal control.- 3.1 Optimal control in a finite time interval.- 3.2 Optimal control over an infinite time interval.- 4 A glimpse into H?-theory: state feedback case.- 4.1 Introduction.- 4.2 Main results.- 5 Final remarks.- Notes.- 2 Controllability and Observability for a Class of Infinite Dimensional Systems.- 1 Introduction.- 2 Main definitions.- 2.1 Notation.- 2.2 Definitions.- 3 Criteria for approximate and exact controllability.- 3.1 Criterion for approximate controllability.- 3.2 Criteria for exact controllability and continuous observability.- 3.3 Approximation.- 4 Finite dimensional control space.- 4.1 Finite dimensional case.- 4.2 General state space.- 5 Controllability for the heat equation.- 5.1 Distributed control.- 5.2 Boundary control.- 5.3 Neumann boundary control.- 5.4 Pointwise control.- 6 Controllability for skew-symmetric operators.- 6.1 Notation and general comments.- 6.2 Dynamical system.- 6.3 Approximation.- 6.4 Exact controllability for T arbitrarily small.- 7 General framework: skew-symmetric operators.- 7.1 Operator A.- 7.2 Operator B.- 7.3 Dynamical system.- 7.4 Exact controllability.- 8 Exact controllability of hyperbolic equations.- 8.1 Wave equation with Dirichlet boundary control.- 8.2 Wave equation with Neumann boundary control.- 8.3 Maxwell equations.- 8.4 Plate equation.- References to Part I.- II Quadratic Optimal Control: Finite Time Horizon.- 1 Systems with Bounded Control Operators: Control Inside the Domain.- 1 Introduction and setting of the problem.- 2 Solution of Riccati equation.- 2.1 Notation and preliminaries.- 2.2 Riccati equation.- 2.3 Representation formulas for the solution of the Riccati equation.- 3 Strict and classical solutions of the Riccati equation.- 3.1 The general case.- 3.2 The analytic case.- 3.3 The variational case.- 4 The case of the unbounded observation.- 4.1 The analytic case.- 4.2 The variational case.- 5 The case when A generates a group.- 6 The linear quadratic control problem with finite horizon.- 6.1 The main result.- 6.2 The case of unbounded observation.- 6.3 Regularity properties of the optimal control.- 6.4 Hamiltonian systems.- 7 Some generalizations and complements.- 7.1 Non homogeneous state equation.- 7.2 Time dependent state equation and cost function.- 7.3 Dual Riccati equation.- 8 Examples of controlled systems.- 8.1 Parabolic equations.- 8.2 Wave equation.- 8.3 Delay equations.- 8.4 Evolution equations in noncylindrical domains.- 2 Systems with Unbounded Control Operators: Parabolic Equations with Control on the Boundary.- 1 Introduction.- 2 Riccati equation.- 2.1 Notation.- 2.2 Riccati equation for ? >1/2.- 2.3 Solution of the Riccati equation for ? >1/2.- 3 Dynamic Programming.- 3 Systems with Unbounded Control Operators: Hyperbolic Equations with Control on the Boundary.- 1 Introduction.- 2 Riccati equation.- 3 Dynamic Programming.- 4 Examples of controlled hyperbolic systems.- 5 Some result for general semigroups.- References to Part II.- III Quadratic Optimal Control: Infinite Time Horizon.- 1 Systems with Bounded Control Operators: Control Inside the Domain.- 1 Introduction and setting of the problem.- 2 The algebraic Riccati equation.- 3 Solution of the control problem.- 3.1 Feedback operator and detectability.- 3.2 Stabilizability and stability of the closed loop operator F in the point spectrum case.- 3.3 Stabilizability.- 3.4 Exponential stability of F.- 4 Qualitative properties of the solutions of the Riccati equation.- 4.1 Local stability results.- 4.2 Attractivity properties of a stationary solution.- 4.3 Maximal solutions.- 4.4 Continuous dependence of stationary solutions with respect to the data.- 4.5 Periodic solutions of the Riccati equation.- 5 Some generalizations and complements.- 5.1 Non homogeneous state equation.- 5.2 Time dependent state equation and cost function.- 5.3 Periodic control problems.- 6 Examples of controlled systems.- 6.1 Parabolic equations.- 6.2 Wave equation.- 6.3 Strongly damped wave equation.- 2 Systems with Unbounded Control Operators: Parabolic Equations with Control on the Boundary.- 1 Introduction and setting of the problem.- 2 The algebraic Riccati equation.- 3 Dynamic programming.- 3.1 Existence and uniqueness of the optimal control.- 3.2 Feedback operator and detect ability.- 3.3 Stabilizability and stability of F in the point spectrum case.- 3 Systems with Unbounded Control Operators: Hyperbolic Equations with Control on the Boundary.- 1 Introduction and setting of the problem.- 2 Main results.- 3 Some result for general semigroups.- References to Part III.- Appendix A.- An Isomorphism Result.- Index to Volume II.- Corrections to Volume I.