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AuthorLi, Xunjing. author
TitleOptimal Control Theory for Infinite Dimensional Systems [electronic resource] / by Xunjing Li, Jiongmin Yong
ImprintBoston, MA : Birkhรคuser Boston, 1995
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Descript XII, 450 p. online resource


Infinite dimensional systems can be used to describe many phenomena in the real world. As is well known, heat conduction, properties of elasticยญ plastic material, fluid dynamics, diffusion-reaction processes, etc., all lie within this area. The object that we are studying (temperature, displaceยญ ment, concentration, velocity, etc.) is usually referred to as the state. We are interested in the case where the state satisfies proper differential equaยญ tions that are derived from certain physical laws, such as Newton's law, Fourier's law etc. The space in which the state exists is called the state space, and the equation that the state satisfies is called the state equation. By an infinite dimensional system we mean one whose corresponding state space is infinite dimensional. In particular, we are interested in the case where the state equation is one of the following types: partial differential equation, functional differential equation, integro-differential equation, or abstract evolution equation. The case in which the state equation is being a stochastic differential equation is also an infinite dimensional problem, but we will not discuss such a case in this book


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{128}{148} 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{128}{148} 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{128}{148} Semilinear Systems -- ยง4.1. Necessary conditions for time optimal pairs -- ยง4.2. The minimum time function -- ยง5. Time Optimal Control โ{128}{148} 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

Mathematics Applied mathematics Engineering mathematics System theory Mathematics Applications of Mathematics Systems Theory Control


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