Author | Li, Xunjing. author |
---|---|
Title | Optimal Control Theory for Infinite Dimensional Systems [electronic resource] / by Xunjing Li, Jiongmin Yong |
Imprint | Boston, MA : Birkhรคuser Boston, 1995 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-4260-4 |
Descript | XII, 450 p. online resource |
1. Control Problems in Infinite Dimensions -- ยง1. Diffusion Problems -- ยง2. Vibration Problems -- ยง3. Population Dynamics -- ยง4. Fluid Dynamics -- ยง5. Free Boundary Problems -- Remarks -- 2. Mathematical Preliminaries -- ยง1. Elements in Functional Analysis -- ยง1.1. Spaces -- ยง1.2. Linear operators -- ยง1.3. Linear functional and dual spaces -- ยง1.4. Adjoint operators -- ยง1.5. Spectral theory -- ยง1.6. Compact operators -- ยง2. Some Geometric Aspects of Banach Spaces -- ยง2.1. Convex sets -- ยง2.2. Convexity of Banach spaces -- ยง3. Banach Space Valued Functions -- ยง3.1. Measurability and integrability -- ยง3.2. Continuity and differentiability -- ยง4. Theory of Co Semigroups -- ยง4.1. Unbounded operators -- ยง4.2. Co semigroups -- ยง4.3. Special types of Co semigroups -- ยง4.4. Examples -- ยง5. Evolution Equations -- ยง5.1. Solutions -- ยง5.2. Semilinear equations -- ยง5.3. Variation of constants formula -- ยง6. Elliptic Partial Differential Equations -- ยง6.1. Sobolev spaces -- ยง6.2. Linear elliptic equations -- ยง6.3. Semilinear elliptic equations -- Remarks -- 3. Existence Theory of Optimal Controls -- ยง1. Souslin Space -- ยง1.1. Polish space -- ยง1.2. Souslin space -- ยง1.3. Capacity and capacitability -- ยง2. Multifunctions and Selection Theorems -- ยง2.1. Continuity -- ยง2.2. Measurability -- ยง2.3. Measurable selection theorems -- ยง3. Evolution Systems with Compact Semigroups -- ยง4. Existence of Feasible Pairs and Optimal Pairs -- ยง4.1. Cesari property -- ยง4.2. Existence theorems -- ยง5. Second Order Evolution Systems -- ยง5.1. Formulation of the problem -- ยง5.2. Existence of optimal controls -- ยง6. Elliptic Partial Differential Equations and Variational Inequalities -- Remarks -- 4. Necessary Conditions for Optimal Controls โ Abstract Evolution Equations -- ยง1. Formulation of the Problem -- ยง2. Ekeland Variational Principle -- ยง3. Other Preliminary Results -- ยง3.1. Finite codimensionality -- ยง3.2. Preliminaries for spike perturbation -- ยง3.3. The distance function -- ยง4. Proof of the Maximum Principle -- ยง5. Applications -- Remarks -- 5. Necessary Conditions for Optimal Controls โ Elliptic Partial Differential Equations -- ยง1. Semilinear Elliptic Equations -- ยง1.1. Optimal control problem and the maximum principle -- ยง1.2. The state coastraints -- ยง2. Variation along Feasible Pairs -- ยง3. Proof of the Maximum Principle -- ยง4. Variational Inequalities -- ยง4.1. Stability of the optimal cost -- ยง4.2. Approximate control problems -- ยง4.3. Maximum principle and its proof -- ยง5. Quasilinear Equations -- ยง5.1. The state equation and the optimal control problem -- ยง5.2. The maximum principle -- ยง6. Minimax Control Problem -- ยง6.1. Statement of the problem -- ยง6.2. Regularization of the cost functional -- ยง6.3. Necessary conditions for optimal controls -- ยง7. Bounary Control Problems -- ยง7.1. Formulation of the problem -- ยง7.2. Strong stability and the qualified maximum principle -- ยง7.3. Neumann problem with measure data -- ยง7.4. Exact penalization and a proof of the maximum principle -- Remarks -- 6. Dynamic Programming Method for Evolution Systems -- ยง1. Optimality Principle and Hamilton-Jacobi-Bellman Equations -- ยง2. Properties of the Value Functions -- ยง2.1. Continuity -- ยง2.2. B-continuity -- ยง2.3. Semi-concavity -- ยง3. Viscosity Solutions -- ยง4. Uniqueness of Viscosity Solutions -- ยง4.1. A perturbed optimization lemma -- ยง4.2. The Hilbert space X? -- ยง4.3. A uniqueness theorem -- ยง5. Relation to Maximum Principle and Optimal Synthesis -- ยง6. Infinite Horizon Problems -- Remarks -- 7. Controllability and Time Optimal Control -- ยง1. Definitions of Controllability -- ยง2. Controllability for linear systems -- ยง2.1. Approximate controllability -- ยง2.2. Exact controllability -- ยง3. Approximate controllability for semilinear systems -- ยง4. Time Optimal Control โ Semilinear Systems -- ยง4.1. Necessary conditions for time optimal pairs -- ยง4.2. The minimum time function -- ยง5. Time Optimal Control โ Linear Systems -- ยง5.1. Convexity of the reachable set -- ยง5.2. Encounter of moving sets -- ยง5.3. Time optimal control -- Remarks -- 8. Optimal Switching and Impulse Controls -- ยง1. Switching and Impulse Controls -- ยง2. Preliminary Results -- ยง3. Properties of the Value Function -- ยง4. Optimality Principle and the HJB Equation -- ยง5. Construction of an Optimal Control -- ยง6. Approximation of the Control Problem -- ยง7. Viscosity Solutions -- ยง8. Problem in Finite Horizon -- Remarks -- 9. Linear Quadratic Optimal Control Problems -- ยง1. Formulation of the Problem -- ยง1.1. Examples of unbounded control problems -- ยง1.2. The LQ problem -- ยง2. Well-posedness and Solvability -- ยง3. State Feedback Control -- ยง3.1. Two-point boundary value problem -- ยง3.2. The Problem (LQ)t -- ยง3.3. A Fredholm integral equation -- ยง3.4. State feedback representation of optimal controls -- ยง4. Riccati Integral Equation -- ยง5. Problem in Infinite Horizon -- ยง5.1. Reduction of the problem -- ยง5.2. Well-posedness and solvability -- ยง5.3. Algebraic Riccati equation -- ยง5.4. The positive real lemma -- ยง5.5. Feedback stabilization -- ยง5.6. Fredholm integral equation and Riccati integral equation -- Remarks -- References