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Time minimal control of batch reactors. (English) Zbl 0914.93043
Necessary mathematical tools for the time-optimal control of batch irreversible chain reaction are presented. The study is based on classical optimization theory and Pontryagin’s maximum principle. The mathematical complexity of the general design technique is structured into problems, subproblems, and side problems with explanations. Each problem is treated in good old mathematical fashion by the use of definitions, conditions, propositions, lemmas, and theorems.
The main objective of the paper is threefold: 1) to compute the closed-loop time-optimal controls near the terminal constraint, 2) to find the estimate of the number of switchings, to determine the singular extremals and switching functions, and 3) to introdue the concept of focal points and give an algorithm to compute conjugate and focal points along the singular extremal.
It is shown that the results of the analysis in the domain of set objectives combined with numerical simulation allow to find the time-optimal control law in many situations. Also, some results are extended to closed-loop suboptimal control of more complicated reactions.
The paper is organized into sections. In §1 the chemical kinetics under study is introduced, the control problems are stated, and the main results of the paper overviewed. §2 recalls the extremality results for time-optimal control. In §3 the concepts of projected and reduced control problems are introduced. In §4 the global bounds on the number of switches are found. §5 and §6 expose application examples and simulation results. §7 deals with the concept of conjugate and focal points. And §8 exposes the commented numerical experiments in terms of the statements.
The found time-optimal and suboptimal control laws are implemented on a batch reactor located at Caen, France.

MSC:
93C95 Application models in control theory
49N05 Linear optimal control problems
93C83 Control/observation systems involving computers (process control, etc.)
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