Optimal Control in Bioprocesses von Alain/Sari Rapaport

<p>Optimal control is a branch of applied mathematics that engineers need in order to optimize the operation of systems and production processes. Its application to concrete examples is often considered to be difficult because it requires a large investment to master its subtleties.<br /><br /> The purpose of Optimal Control in Bioprocesses is to provide a pedagogical perspective on the foundations of the theory and to support the reader in its application, first by using academic examples and then by using concrete examples in biotechnology. The book is thus divided into two parts, the first of which outlines the essential definitions and concepts necessary for the understanding of Pontryagins maximum principle or PMP while the second exposes applications specific to the world of bioprocesses.<br /><br /> This book is unique in that it focuses on the arguments and geometric interpretations of the trajectories provided by the application of PMP.</p>

Jérôme Harmand is a researcher at the LBE laboratory of INRA in Narbonne, France.Claude Lobry is a former Professor of both the University of Bordeaux and University of Nice in France.Alain Rapaport is a researcher at the Applied Mathematics and Informatics Department of INRA in Montpellier, France.Tewfik Sari is Research Director at the National Research Institute of Science and Technology for Environment and Agriculture (IRSTEA) in Montpellier, France.

Introduction ixPart 1 Learning to use Pontryagins Maximum Principle1Chapter 1 The Classical Calculus of Variations31.1 Introduction: notations 31.2 Minimizing a function 41.2.1 Minimum of a function of one variable 41.2.2 Minimum of a function of two variables 61.3 Minimization of a functional: EulerLagrange equations 101.3.1 The problem 101.3.2 The differential ofJ 111.3.3 Examples 141.4 Hamiltons equations 201.4.1 Hamiltons classical equations 201.4.2 The limitations of classical calculus of variations and small steps toward the control theory 231.5 Historical and bibliographic observations 25Chapter 2 Optimal Control272.1 The vocabulary ofoptimal control theory 272.1.1 Controls and responses 272.1.2 Class of regular controls 282.1.3 Reachable states 312.1.4 Controllability 342.1.5 Optimal control 372.1.6 Existence of a minimum 382.1.7 Optimization and reachable states 422.2 Statement of Pontryagins maximum principle 442.2.1 PMP for the canonical problem 442.2.2 PMP for an integral cost 472.2.3 The PMP for the minimum-time problem 502.2.4 PMP in fixed terminal time and integral cost 522.2.5 PMP for a non-punctual target 562.3 PMP without terminal constraint 572.3.1 Statement 572.3.2 Corollary 592.3.3 Dynamic programming and interpretation of the adjoint vector 59Chapter 3 Applications653.1 Academic examples (to facilitate understanding) 653.1.1 The driver in a hurry 653.1.2 Profile of a road 673.1.3 Controlling the harmonic oscillator: the swing (problem) 703.1.4 The Fuller phenomenon 753.2 Regular problems 773.2.1 A regular Hamiltonian and the associated shooting method 773.2.2 The geodesic problem (seen as a minimum-time problem) 803.2.3 Regularization of the problem of the driver in a hurry 903.3 Non-regular problems and singular arcs 923.3.1 Optimal fishing problem 923.3.2 Discontinuous value function: the Zermelo navigation problem 1023.4 Synthesis of the optimal control, discontinuity of the value function, singular arcs and feedback 1183.5 Historical and bibliographic observations 125Part 2 Applications in Process Engineering127Chapter 4 Optimal Filling of a Batch Reactor1294.1 Reducing the problem 1304.2 Comparison with BangBang policies 1314.3 Optimal synthesis in the case of Monod 1354.4 Optimal synthesis in the case of Haldane 1354.4.1 Existence of an arc that (partially) separates + and 1364.4.2 Using PMP 1384.5 Historical and bibliographic observations 141Chapter 5 Optimizing Biogas Production 1435.1 The problem 1435.2 Optimal solution in a well-dimensioned case 1465.3 The Hamiltonian system 1485.4 Optimal solutions in the underdimensioned case 1565.5 Optimal solutions in the overdimensioned case 1635.6 Inhibition by the substrate 1675.7 Singular arcs 1705.8 Historical and bibliographic observations 176Chapter 6 Optimization of a Membrane Bioreactor (MBR)1776.1 Overview of the problem 1776.2 The model proposed by Benyahiaet al 1856.3 The model proposed by Cogan and Chellamb 1866.4 Historical and bibliographic observations 188Appendices 191Appendix 1 Notations and Terminology 193Appendix 2 Differential Equations and Vector Fields 197Appendix 3 Outline of the PMP Demonstration 205Appendix 4 Demonstration of PMP without a Terminal Target 215Appendix 5 Problems that are Linear in the Control 221Appendix 6 Calculating Singular Arcs 231References 237Index 243

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