Optimal Control Theory for Infinite Dimensional Systems (Hardcover)

1. Control Problems in Infinite Dimensions.- 1. Diffusion Problems.- 2. Vibration Problems.- 3. Population Dynamics.- 4. Fluid Dynamics.- 5. Free Boundary Problems.- Remarks.- 2. Mathematical Preliminaries.- 1. Elements in Functional Analysis.- 1.1. Spaces.- 1.2. Linear operators.- 1.3. Linear functional and dual spaces.- 1.4. Adjoint operators.- 1.5. Spectral theory.- 1.6. Compact operators.- 2. Some Geometric Aspects of Banach Spaces.- 2.1. Convex sets.- 2.2. Convexity of Banach spaces.- 3. Banach Space Valued Functions.- 3.1. Measurability and integrability.- 3.2. Continuity and differentiability.- 4. Theory of Co Semigroups.- 4.1. Unbounded operators.- 4.2. Co semigroups.- 4.3. Special types of Co semigroups.- 4.4. Examples.- 5. Evolution Equations.- 5.1. Solutions.- 5.2. Semilinear equations.- 5.3. Variation of constants formula.- 6. Elliptic Partial Differential Equations.- 6.1. Sobolev spaces.- 6.2. Linear elliptic equations.- 6.3. Semilinear elliptic equations.- Remarks.- 3. Existence Theory of Optimal Controls.- 1. Souslin Space.- 1.1. Polish space.- 1.2. Souslin space.- 1.3. Capacity and capacitability.- 2. Multifunctions and Selection Theorems.- 2.1. Continuity.- 2.2. Measurability.- 2.3. Measurable selection theorems.- 3. Evolution Systems with Compact Semigroups.- 4. Existence of Feasible Pairs and Optimal Pairs.- 4.1. Cesari property.- 4.2. Existence theorems.- 5. Second Order Evolution Systems.- 5.1. Formulation of the problem.- 5.2. Existence of optimal controls.- 6. Elliptic Partial Differential Equations and Variational Inequalities.- Remarks.- 4. Necessary Conditions for Optimal Controls - Abstract Evolution Equations.- 1. Formulation of the Problem.- 2. Ekeland Variational Principle.- 3. Other Preliminary Results.- 3.1. Finite codimensionality.- 3.2. Preliminaries for spike perturbation.- 3.3. The distance function.- 4. Proof of the Maximum Principle.- 5. Applications.- Remarks.- 5. Necessary Conditions for Optimal Controls - Elliptic Partial Differential Equations.- 1. Semilinear Elliptic Equations.- 1.1. Optimal control problem and the maximum principle.- 1.2. The state coastraints.- 2. Variation along Feasible Pairs.- 3. Proof of the Maximum Principle.- 4. Variational Inequalities.- 4.1. Stability of the optimal cost.- 4.2. Approximate control problems.- 4.3. Maximum principle and its proof.- 5. Quasilinear Equations.- 5.1. The state equation and the optimal control problem.- 5.2. The maximum principle.- 6. Minimax Control Problem.- 6.1. Statement of the problem.- 6.2. Regularization of the cost functional.- 6.3. Necessary conditions for optimal controls.- 7. Bounary Control Problems.- 7.1. Formulation of the problem.- 7.2. Strong stability and the qualified maximum principle.- 7.3. Neumann problem with measure data.- 7.4. Exact penalization and a proof of the maximum principle.- Remarks.- 6. Dynamic Programming Method for Evolution Systems.- 1. Optimality Principle and Hamilton-Jacobi-Bellman Equations.- 2. Properties of the Value Functions.- 2.1. Continuity.- 2.2. B-continuity.- 2.3. Semi-concavity.- 3. Viscosity Solutions.- 4. Uniqueness of Viscosity Solutions.- 4.1. A perturbed optimization lemma.- 4.2. The Hilbert space X?.- 4.3. A uniqueness theorem.- 5. Relation to Maximum Principle and Optimal Synthesis.- 6. Infinite Horizon Problems.- Remarks.- 7. Controllability and Time Optimal Control.- 1. Definitions of Controllability.- 2. Controllability for linear systems.- 2.1. Approximate controllability.- 2.2. Exact controllability.- 3. Approximate controllability for semilinear systems.- 4. Time Optimal Control - Semilinear Systems.- 4.1. Necessary conditions for time optimal pairs.- 4.2. The minimum time function.- 5. Time Optimal Control - Linear Systems.- 5.1. Convexity of the reachable set.- 5.2. Encounter of moving sets.- 5.3. Time optimal control.- Remarks.- 8. Optimal Switching and Impulse Control