Optimal Control from Theory to Computer Programs

ISBN-13: 9789048164981 / Angielski / Miękka / 2010 / 330 str.

Viorel Arnăutu;Pekka Neittaanmäki

Opis

The aim of this book is to present the mathematical theory and the know-how to make computer programs for the numerical approximation of Optimal Control of PDE's. The computer programs are presented in a straightforward generic language. As a consequence they are well structured, clearly explained and can be translated easily into any high level programming language. Applications and corresponding numerical tests are also given and discussed. To our knowledge, this is the first book to put together mathematics and computer programs for Optimal Control in order to bridge the gap between mathematical abstract algorithms and concrete numerical ones.
The text is addressed to students and graduates in Mathematics, Mechanics, Applied Mathematics, Numerical Software, Information Technology and Engineering. It can also be used for Master and Ph.D. programs.

Recenzje

From the reviews:

"This text provides a comprehensive introduction to several techniques for the numerical computation of control problems governed by (mainly elliptic) partial differential equations (PDE). ... The pseudo-code is clear and well documented ... . This book is most suited for graduate students in applied mathematics, numerical analysis or control engineering." (Andrew C. Eberhard, Zentralblatt MATH, Vol. 1056 (7), 2005)

Spis treści

1: Selected Topics From Functional And Convex Analysis. 1. Differentiability topics. 2. Convex functionals. 2: Optimization Problems. 1. Existence and uniqueness results for minimization problems. 2. The Azimuth Mark method. 3. The steepest descent method. 4.Projection operators in Hilbert spaces. 5. Projected gradient methods. 3: Numerical Approximation of Elliptic Variational Problems. 1. An Example from Fluid Mechanics problems in media with semi-permeable boundaries. 2. The general form of Elliptic Variational Inequalities. 3. The internal approximation of Elliptic Variational Inequalities. 4.The Finite Element Method. Error Estimates. 5. Optimization Methods. 5.1. The successive approximations method. 5.2. The penalization method. 5.3. The Lagrange multipliers method. 6. Computer realization of the optimization methods. 6.1. The seepage flow of water through a homogeneous rectangular dam. 6.2. The successive approximations method. 6.3. The penalization method. 6.4. The Lagrange multipliers method. 4: Indirect Methods For Optimal Control Problems. 1. The elimination of the state. 2. An optimal control problem related to the inverse Stefan problem. 2.1. The one-phase Stefan problem. 2.2. The inverse Stefan problem and the related optimal control problem. 2.3. The numerical realization of the Algorithm ALG-R. 3. Optimal control for a two-phase Stefan Problem. 3.1. The two-phase Stefan Problem. 3.2. The optimal control problem. 3.3. A numerical algorithm. 5: A Control Problem For A Class Of Epidemics. 1. Statement of the control problem. 2. The numerical realization of the algorithm. 6: Optimal Control For Plate Problems. 1. Decomposition of fourth order elliptic equations. 2. The clamped plate problem. 3. Optimization of plates. 4. A Fourth-Order Variational Inequality. 7: Direct Numerical Methods for Optimal Control Problems. 1. The abstract optimal control problem. 2. The quadratic programming problem. 3. Interior-point methods for the solution of problem (QP). 4. Numerical solution of the linear system. 4.1. Krylov subspace algorithms. 4.2. Convergence properties. 4.3. The implementation of the algorithms. 5. Preconditioning. 5.1. Preconditioning for MINRES and SYMMLQ. 5.2. Preconditioning for the KKT system. 8: Stochastic Control Problems. 1. Stochastic processes. 2. Stochastic control problems. An introduction. 3. The Hamilton-Jacobi-Bellman equations. 4.The Markov chain approximation. 5. Numerical algorithms. 5.1. Approximation in Policy Space. 5.2. Approximation in Value Space. 5.3. Computational Problems. References. Topic Index. Author Index.