Optimal Control in Bioprocesses

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 Pontryagin’s Maximum Principle 1Chapter 1 The Classical Calculus of Variations 31.1 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: Euler–Lagrange equations 101.3.1 The problem 101.3.2 The differential of J 111.3.3 Examples 141.4 Hamilton’s equations 201.4.1 Hamilton’s 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 Control 272.1 The vocabulary of optimal 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 Pontryagin’s 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 Applications 653.1 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 Engineering 127Chapter 4 Optimal Filling of a Batch Reactor 1294.1 Reducing the problem 1304.2 Comparison with Bang–Bang 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 Θ+ and Θ− 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 Benyahia et 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