The Attainable Region and Pontryagin's Maximum Principle

McGregor, Craig; Glasser, David; Hildebrandt, Diane

doi:10.1021/ie980380l

Cited by 13 publications

(9 citation statements)

References 14 publications

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“…160,161 Aydin et al 162 proposed a PMP based quasi-Newton approach for the optimization of semi-batch processes. Their algorithm finds solutions faster than a McGregor et al 136 Reaction and mixing Attainable region analysis 1999 direct method for some case studies. Later, the same authors improved upon their previous results by applying a parsimonious parameterization to the control.…”

Section: -2020mentioning

confidence: 99%

Optimal control and the Pontryagin's principle in chemical engineering: History, theory, and challenges

2022

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In the mid‐1950s, Pontryagin et al. published a principle that became a fundamental concept in optimal control (OC) theory. The principle provides theoretical and practical methods to find the solution of OC problems, in particular, open‐loop control problems. In chemical engineering, the principle has played an important role as a decision making framework for more than 60 years. This study gathers the main contributions on the application of the Pontryagin's principle to the dynamic optimization of chemical processes. A concise overview of the optimality conditions for a wide class of constrained OC problems is provided. Numerical methods to solve the necessary conditions and strategies to address inequality constraints are summarized. The information and illustrative case study presented in this work can be used as a guide to implement the principle in different settings. Opportunities for further application of the principle in relevant chemical engineering problems are also discussed.

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Section: -2020mentioning

confidence: 99%

Optimal control and the Pontryagin's principle in chemical engineering: History, theory, and challenges

2022

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“…Plenty of particular applications was developed: from polymerization [77] to particle breakage in a ball mill [78]. Mathematical methods for study of attainable regions vary from the Pontryagin's maximum principle [79] to linear programming [80], the Shrink-Wrap algorithm [81] and convex analysis.…”

Section: Continuous Time Kineticsmentioning

confidence: 99%

Entropy: The Markov Ordering Approach

Gorban

Judge

2010

Entropy

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The focus of this article is on entropy and Markov processes. We study the properties of functionals which are invariant with respect to monotonic transformations and analyze two invariant “additivity” properties: (i) existence of a monotonic transformation which makes the functional additive with respect to the joining of independent systems and (ii) existence of a monotonic transformation which makes the functional additive with respect to the partitioning of the space of states. All Lyapunov functionals for Markov chains which have properties (i) and (ii) are derived. We describe the most general ordering of the distribution space, with respect to which all continuous-time Markov processes are monotonic (the Markov order). The solution differs significantly from the ordering given by the inequality of entropy growth. For inference, this approach results in a convex compact set of conditionally “most random” distributions

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“…Many particular applications were developed, from polymerization [63] to particle breakage in a ball mill [47] and hydraulic systems [28]. Mathematical methods for the study of attainable regions vary from Pontryagin's maximum principle [46] to linear programming [38], the Shrink-Wrap algorithm [43], and convex analysis. In 1979 it was demonstrated how to utilize the knowledge about partial equilibria of elementary processes to construct the attainable regions [24].…”

Section: Introductionmentioning

confidence: 99%

“…Nevertheless, some of the mathematical backgrounds of this approach were delayed in development and publications. Now, the thermodynamically attainable regions are in extensive use in chemical engineering and beyond [18,19,23,28,34,35,36,37,38,41,43,46,47,60,61,63,69]. In this paper we aim to provide the complete mathematical background for the analysis of the thermodynamically attainable regions.…”

Section: Introductionmentioning

confidence: 99%

Thermodynamic Tree: The Space of Admissible Paths

Gorban¹

2013

SIAM J. Appl. Dyn. Syst.

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Is a spontaneous transition from a state x to a state y allowed by thermodynamics? Such a question arises often in chemical thermodynamics and kinetics. We ask the more formal question: is there a continuous path between these states, along which the conservation laws hold, the concentrations remain non-negative and the relevant thermodynamic potential G (Gibbs energy, for example) monotonically decreases? The obvious necessary condition, G(x)\geq G(y), is not sufficient, and we construct the necessary and sufficient conditions. For example, it is impossible to overstep the equilibrium in 1-dimensional (1D) systems (with n components and n-1 conservation laws). The system cannot come from a state x to a state y if they are on the opposite sides of the equilibrium even if G(x) > G(y). We find the general multidimensional analogue of this 1D rule and constructively solve the problem of the thermodynamically admissible transitions. We study dynamical systems, which are given in a positively invariant convex polyhedron D and have a convex Lyapunov function G. An admissible path is a continuous curve along which $G$ does not increase. For x,y from D, x\geq y (x precedes y) if there exists an admissible path from x to y and x \sim y if x\geq y and y\geq x. The tree of G in D is a quotient space D/~. We provide an algorithm for the construction of this tree. In this algorithm, the restriction of G onto the 1-skeleton of $D$ (the union of edges) is used. The problem of existence of admissible paths between states is solved constructively. The regions attainable by the admissible paths are described.Comment: Extended version, 31 page, 9 figures, 69 cited references, many minor correction

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The Attainable Region and Pontryagin's Maximum Principle

Cited by 13 publications

References 14 publications

Optimal control and the Pontryagin's principle in chemical engineering: History, theory, and challenges

Optimal control and the Pontryagin's principle in chemical engineering: History, theory, and challenges

Entropy: The Markov Ordering Approach

Thermodynamic Tree: The Space of Admissible Paths

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