Boundary value problem

About: Boundary value problem is a research topic. Over the lifetime, 145355 publications have been published within this topic receiving 2731135 citations. The topic is also known as: boundary value problems.

Spectral Methods in MATLAB

Abstract: Preface 1 Differentiation matrices 2 Unbounded grids: the semidiscrete Fourier transform 3 Periodic grids: the DFT and FFT 4 Smoothness and spectral accuracy 5 Polynomial interpolation and clustered grids 6 Chebyshev differentiation matrices 7 Boundary value problems 8 Chebyshev series and the FFT 9 Eigenvalues and pseudospectra 10 Time-stepping and stability regions 11 Polar coordinates 12 Integrals and quadrature formulas 13 More about boundary conditions 14 Fourth-order problems Afterword Bibliography Index

Optimal Control of Systems Governed by Partial Differential Equations

Abstract: Principal Notations.- I Minimization of Functions and Unilateral Boundary Value Problems.- 1. Minimization of Coercive Forms.- 1.1. Notation.- 1.2. The Case when ?: is Coercive.- 1.3. Characterization of the Minimizing Element. Variational Inequalities.- 1.4. Alternative Form of Variational Inequalities.- 1.5. Function J being the Sum of a Differentiable and Non-Differentiable Function.- 1.6. The Convexity Hypothesis on $$ {U_{ad}} $$.- 1.7. Orientation.- 2. A Direct Solution of Certain Variational Inequalities.- 2.1. Problem Statement.- 2.2. An Existence and Uniqueness Theorem.- 3. Examples.- 3.1. Function Spaces on ?.- 3.2. Function Spaces on ?.- 3.3. Subspaces of Hm(?).- 3.4. Examples of Boundary Value Problems.- 3.5. Unilateral Boundary Value Problems (I).- 3.6. Unilateral Boundary Value Problems (II).- 3.7. Unilateral Boundary Value Problems (III).- 3.8. Unilateral Boundary Value Problems Case of Systems.- 3.9. Elliptic Operators of Order Greater than Two.- 3.10. Non-differentiable Functionals.- 4. A Comparison Theorem.- 4.1. General Results.- 4.2. An Application.- 5. Non Coercive Forms.- 5.1. Convexity of the Set of Solutions.- 5.2. Approximation Theorem.- Notes.- II Control of Systems Governed by Elliptic Partial Differential Equations.- 1. Control of Elliptic Variational Problems.- 1.1. Problem Statement.- 1.2. First Remarks on the Control Problem.- 1.3. The Set of Inequalities Defining the Optimal Control.- 2. First Applications.- 2.1. System Governed by the Dirichlet Problem Distributed Control.- 2.2. The Case with No Constraints.- 2.3. System Governed by a Neumann Problem Distributed Control.- 2.4. System Governed by a Neumann Problem Boundary Control.- 2.5. Local and Global Constraints.- 2.6. System Governed by a Differential System.- 2.7. System Governed by a 4th Order Differential Operator.- 2.8. Orientation.- 3. A Family of Examples with N = 0 and $$ {U_{ad}} $$ Arbitrary.- 3.1. General Case.- 3.2. Application (I).- 3.3. Application (II).- 4. Observation on the Boundary.- 4.1. System Governed by a Dirichlet Problem (I).- 4.2. Some Results on Non-homogeneous Dirichlet Problems.- 4.3. System Governed by a Dirichlet Problem (II).- 4.4. System Governed by a Neumann Problem.- 5. Control and Observation on the Boundary. Case of the Dirichlet Problem.- 5.1. Orientation.- 5.2. Boundary Control in L2(?).- 5.3. A "Controllability-Like" Problem.- 5.4. Pointwise Control and Observation.- 6. Constraints on the State.- 6.1. Orientation.- 6.2. Control and Constraints on the Boundary.- 7. Existence Results for Optimal Controls.- 7.1. Orientation.- 7.2. Distributed Control.- 7.3. Singular Perturbation of the System.- 7.4. Boundary Control.- 7.5. Control of Systems Governed by Unilateral Problems.- 8. First Order Necessary Conditions.- 8.1. Statement of the Theorem.- 8.2. Proof of the Theorem.- 8.2.1. "Algebraic" Transformation.- 8.2.2. General Remarks on the Utilization of (8.13.).- 8.2.3. Proof that dj,?0.- Notes.- III Control of Systems Governed by Parabolic Partial Differential Equations.- 1. Equations of Evolution.- 1.1. Data.- 1.2. Evolution Problems.- 1.3. Proof of Uniqueness.- 1.4. Proof of Existence.- 1.5. Some Examples.- 1.6. Semi-groups.- 2. Problems of Control.- 2.1. Notation. Immediate Properties.- 2.2. Set of Inequalities Characterizing the Optimal Control.- 2.3. Case (i). Set of Inequalities.- 2.4. Case (ii). Set of Inequalities.- 2.5. Orientation.- 3. Examples.- 3.1. Mixed Dirichlet Problem for a Second Order Parabolic Equation.- 3.1.1. C = Injection Map of L2(0, T V)?L2(Q).- 3.1.2. C = Identity Map of L2(0, T V) into itself.- 3.1.3. Observation of the Final State.- 3.2. Mixed Neumann Problem for a Parabolic Equation of Second Order.- 3.2.1. Case (i).- 3.2.2. Case (ii).- 3.3. System of Equations and Equations of Higher Order.- 3.3.1. System of Equations.- 3.3.2. Higher Order Equations.- 3.4. Additional Results.- 3.5. Orientation.- 4. Decoupling and Integro-Differential Equation of Riccati Type (I).- 4.1. Notation and Assumptions.- 4.2. Operator P(t), Function r(t).- 4.3. Formal Calculations.- 4.4. The Finite Dimensional Case Approximation.- 4.5. Passage to the Limit.- 4.6. Integro-Differential Equation of Riccati Type.- 4.7. Connections with the Hamilton-Jacobi Theory.- 4.8. The Case where Constraints are Present.- 4.9. Various Remarks.- 4.9.1. Direct Study of the "Riccati Equation".- 4.9.2. Another Approach to the Direct Study of the "Riccati Equation".- 4.9.3. Yet Another Approach to the Direct Study of the "Riccati Equation".- 5. Decoupling and Integro-Differential Equation of Riccati Type (II).- 5.1. Application of the Schwartz-Kernel Theorem.- 5.2. Example of a Mixed Neumann Problem with Boundary Control.- 5.3. Example of a Mixed Neumann Problem with Observation of the Final State.- 5.4. Mixed Neumann Problem, Observation of the Final State and Constraints in a Vector Space.- 5.5. Remarks on Decoupling in the Presence of Constraints.- 6. Behaviour as T ? + ?.- 6.1. Orientation and Hypotheses.- 6.2. The Case T = ?.- 6.3. Passage to the Limit as T ? + ?.- 7. Problems which are not Necessarily Coercive.- 7.1. Distributed Observation.- 7.2. Observation of the Final State.- 7.3. Examples where N = 0 and $$ {U_{ad}} $$ is not Bounded.- 8. Other Observations of the State and other Types of Control.- 8.1. Pointwise Observation of the State.- 8.2. Pointwise Control.- 8.3. Control and Observation on the Boundary.- 9. Boundary Control and Observation on the Boundary or of the Final State for a System Governed by a Mixed Dirichlet Problem.- 9.1. Orientation and Problem Statement.- 9.2. Non Homogeneous Mixed Dirichlet Problem.- 9.3. Definition of $$ \frac{{\partial y}}{{\partial {v_A}}} $$ Observation.- 9.4. Cost Function Equations of Optimal Control.- 9.5. Regular Control.- 9.6. Observation of the Final State.- 9.7. Observation of the Final State, Second Order Parabolic Operator.- 10. Controllability.- 10.1. Problem Statement.- 10.2. Controllability and Uniqueness.- 10.3. Super-Controllability and Super-Uniqueness.- 11. Control via Initial Conditions Estimation.- 11.1. Problem Statement. General Results.- 11.2. Examples.- 11.3. Controllability.- 11.4. An Estimation Problem.- 12. Duality.- 12.1. General Remarks.- 12.2. Example.- 13. Constraints on the Control and the State.- 13.1. A General Result.- 13.2. Applications (I).- 13.3. Applications (II).- 14. Non Quadratic Cost Functions.- 14.1. Orientation.- 14.2. An Example.- 14.3. Remarks on Decoupling.- 15. Existence Results for Optimal Controls.- 15.1. Orientation.- 15.2. Non-linear Problem with Distributed Control (I).- 15.3. Non-linear Problem with Distributed Control. Singular Perturbation.- 15.4. Non-linear Problem. Boundary Control.- 15.5. Utilization of Convexity and the Maximum Principle for Second Order Parabolic Equations.- 15.6. Control of Systems Governed by Evolution Inequalities.- 16. First Order Necessary Conditions.- 16.1. Statement of the Theorem.- 16.2. Proof of Theorem 16.1.- 16.2.1. "Algebraic" Transformation.- 16.2.2. Utilization of (16.11.).- 16.2.3. Proof of (16.12.).- 16.3. Remarks.- 17. Time Optimal Control.- 17.1. Problem Statement.- 17.2. Existence Theorem.- 17.3. Bang-Bang Theorem.- 18. Miscellaneous.- 18.1. Equations with Delay.- 18.1.1. Definition of the State.- 18.1.2. Control Problem.- 18.2. Spaces which are not Normable.- Notes.- IV Control of Systems Governed by Hyperbolic Equations or by Equations which are well Posed in the Petrowsky Sense.- 1. Second Order Evolution Equations.- 1.1. Notation and Hypotheses.- 1.2. Problem Statement. An Existence and Uniqueness Result.- 1.3. Proof of Uniqueness.- 1.4. Proof of Existence.- 1.5. Examples (I).- 1.6. Examples (II).- 1.7. Orientation.- 2. Control Problems.- 2.1. Notation. Immediate Properties.- 2.2. Case (2.5.).- 2.3. Case (2.6.).- 2.4. Case (2.7.).- 2.5. Case (2.8.).- 3. Transposition and Applications to Control.- 3.1. Transposition of Theorem 1.1.- 3.2. Application (I).- 3.3. Application (II).- 3.4. Application (III).- 4. Examples.- 4.1. Examples of Hyperbolic Problems. Distributed Control, Distributed Observation.- 4.2. Examples of Hyperbolic Systems. Distributed Control, Observation of the Final State.- 4.3. Petrowsky Type Equation. Distributed Control. Distributed Observation.- 4.4. Petrowsky Type Equation. Distributed Control. Observation of the Final State.- 4.5. Orientation.- 5. Decoupling.- 5.1. Problem Statement. Rewriting as a System of First Order Equations.- 5.2. Rewriting of the Set of Equations Determining the Optimal Control.- 5.3. Decoupling.- 5.4. Riccati Integro-differential Equation.- 5.5. Another Optimal Control Problem. Decoupling.- 6. Control via Initial Conditions. Estimation.- 6.1. Problem Statement.- 6.2. Coercivity of J(?).- 6.3. System of Equations Determining the Optimal Control.- 7. Boundary Control (I).- 7.1. Problem Statement.- 7.2. Definition of the State of the System.- 7.3. Distributed Observation.- 7.4. Boundary Observation.- 8. Boundary Control (II).- 8.1. Problem Statement.- 8.2. Control ? Regular.- 8.3. Examples.- 9. Parabolic-Hyperbolic Systems.- 9.1. Recapitulation of Some General Results.- 9.2. Complement.- 9.3. Control Problems.- 9.4. Example (I).- 9.5. Example (II).- 9.6. Decoupling.- 10. Existence Theorems.- 10.1. Orientation.- 10.2. Example. Introduction of a "Viscosity" Term.- 10.3. Time Optimal Control.- Notes.- V Regularization, Approximation and Penalization.- 1. Regularization.- 1.1. Parabolic Regularization.- 1.2. Application to Optimal Control.- 1.3. Application to Decoupling.- 1.4. Various Remarks.- 1.5. Regularization of the Control.- 2. Approximation in Terms of Systems of Cauchy-Kowaleska Type.- 2.1. Evolution Equation on a Variety.- 2.2. Approximation by a System of Cauchy-Kowaleska Type.- 2.3. Linearized Navier-Stokes Equation.- 3. Penalization.- Notes.

Optimal shape design as a material distribution problem

Abstract: Shape optimization in a general setting requires the determination of the optimal spatial material distribution for given loads and boundary conditions. Every point in space is thus a material point or a void and the optimization problem is a discrete variable one. This paper describes various ways of removing this discrete nature of the problem by the introduction of a density function that is a continuous design variable. Domains of high density then define the shape of the mechanical element. For intermediate densities, material parameters given by an artificial material law can be used. Alternatively, the density can arise naturally through the introduction of periodically distributed, microscopic voids, so that effective material parameters for intermediate density values can be computed through homogenization. Several examples in two-dimensional elasticity illustrate that these methods allow a determination of the topology of a mechanical element, as required for a boundary variations shape optimization technique.

Non-Homogeneous Media and Vibration Theory

Abstract: Distributions and Sobolev spaces.- Operators in Banach spaces.- Examples of boundary value problems.- Semigroups and laplace transform.- Homogenization of second order equations.- Homogenization in elasticity and electromagnetism.- Fluid flow in porous media.- Vibration of mixtures of solids and fluids.- Examples of perturbations for elliptic problems.- The Trotter-Kato theorem and related topics.- Spectral perturbation. Case of isolated eigenvalues.- Perturbation of spectral families and applications to selfadjoint eigenvalue problems.- Stiff problems in constant and varialbe domains.- Averaging and two-scale methods.- Generalities and potential method.- Functional methods.- Scattering problems depending on a parameter.