Continuous Maximum Principle Research Articles

Pointwisea posteriorierror analysis of a discontinuous Galerkin method for the elliptic obstacle problem

AbstractWe present a posteriori error analysis in the supremum norm for the symmetric interior penalty discontinuous Galerkin method for the elliptic obstacle problem. We construct discrete barrier functions based on appropriate corrections of the conforming part of the solution obtained via a constrained averaging operator. The corrector function accounts properly for the nonconformity of the approximation and it is estimated by direct use of the Green’s function of the unconstrained elliptic problem. The use of the continuous maximum principle guarantees the validity of the analysis without mesh restrictions but with shape regularity. The proposed residual-type estimators are shown to be reliable and efficient. Numerical results in two dimensions are included to verify the theory and validate the performance of the error estimator.

Multiple-model description and generalised algorithm of ship-building wharf scheduling

We present a new multiple-model description and generalised algorithm of ship-building wharf scheduling. This description is represented as a special case of the job-shop-scheduling problem with dynamically distributed jobs, resources, flows, structures. The approach is based on a natural dynamic decomposition of the problem and its solution with the help of a modified form of continuous maximum principle coupled with combinatorial optimisation.

Blending low-order stabilised finite element methods: A positivity-preserving local projection method for the convection–diffusion equation

In this work we propose a nonlinear blending of two low-order stabilisation mechanisms for the convection–diffusion equation. The motivation for this approach is to preserve monotonicity without sacrificing accuracy for smooth solutions. The approach is to blend a first-order artificial diffusion method, which will be active only in the vicinity of layers and extrema, with an optimal order local projection stabilisation method that will be active on the smooth regions of the solution. We prove existence of discrete solutions, as well as convergence, under appropriate assumptions on the nonlinear terms, and on the exact solution. Numerical examples show that the discrete solution produced by this method remains within the bounds given by the continuous maximum principle, while the layers are not smeared significantly.

Flexible flow shop scheduling for continuous production

Flexible flow shop problem is represented as the dynamic interpretation of scheduling based on a natural dynamic decomposition of the problem and its solution with the help of a modified form of continuous maximum principle blended with combinatorial optimisation. A theoretical analysis of optimal scheduling approach with uniform alternative machines is considered. The scheduling approach is based on a dynamic non-stationary interpretation of the job execution and a temporal decomposition of the scheduling problem. The optimality conditions as well as the structural properties of the model and the algorithm are investigated. Advantages and limitations of the proposed approach are discussed. The results of this study can be used as an extension to the existing mathematical programming models subject to the following issues: dynamics and non-stationarity of the operation execution, non-stationarity of the machine structures and capacity parameters, and representation of continuous flows. In addition, the formulation of the scheduling model in terms of control makes it possible to attract additional tools from mathematics of functional spaces such as stability, robustness, controllability, adaptability, etc. to the schedule analysis and real-time adjustment.

A dynamic model and an algorithm for short-term supply chain scheduling in the smart factory industry 4.0

Smart factories Industry 4.0 on the basis of collaborative cyber-physical systems represents a future form of industrial networks. Supply chains in such networks have dynamic structures which evolve over time. In these settings, short-term supply chain scheduling in smart factories Industry 4.0 is challenged by temporal machine structures, different processing speed at parallel machines and dynamic job arrivals. In this study, for the first time, a dynamic model and algorithm for short-term supply chain scheduling in smart factories Industry 4.0 is presented. The peculiarity of the considered problem is the simultaneous consideration of both machine structure selection and job assignments. The scheduling approach is based on a dynamic non-stationary interpretation of the execution of the jobs and a temporal decomposition of the scheduling problem. The algorithmic realisation is based on a modified form of the continuous maximum principle blended with mathematical optimisation. A detailed theoretical analysis of the temporal decomposition and computational complexity is performed. The optimality conditions as well as the structural properties of the model and the algorithm are investigated. Advantages and limitations of the proposed approach are discussed.

Coordination of the supply chain schedules with re-scheduling considerations

We consider two synchronized schedules in the supply chain (e.g., an assembly line schedule at a producing company and a supply schedule for a module). This is a multi-objective dynamic scheduling problem with constrained machine capacities. The optimization criteria include total lateness minimization and throughput maximization. Due to some random re-scheduling activities (i.e., new rush customer orders) at one of the companies, the schedule coordination should be performed again. This problem is a dynamic scheduling problem where machine capacities are constrained. If a machine is assigned to a new introduced job that came during the re-scheduling, it cannot be used for processing the initially planned jobs at the same time. This conflict should be resolved on the basis of new schedule coordination to execute both new and initial jobs. For such problem statement, a new dynamic model for coordinated scheduling of interlinked processes in a supply chain under partial re-scheduling is presented. The peculiarity of the proposed approach is the dynamic interpretation of scheduling based on a natural dynamic decomposition of the problem and its solution with the help of a modified form of continuous maximum principle blended with combinatorial optimization. The special properties of the developed model allow using methods of discrete optimization for the schedule calculation. Optimality and sufficiency conditions as well as structural properties of the model are investigated. Advantages and limitations of the proposed approach are discussed. With the developed approach, an explicit inclusion of a schedule changes in the SC coordinated decisions for a wide ranges of possible applications as well as a dynamic model and a tractable algorithm for optimal discrete time scheduling on the basis of continuous maximum principle have been obtained.

Dynamic co-ordinated scheduling in the supply chain under a process modernisation

A new dynamic model for co-ordinated scheduling of interlinked processes in a supply chain under a process modernisation is presented. Such a problem is vital in many of the supply chain management domains. This problem is represented as a special case of the scheduling problem with dynamically distributed jobs. The peculiarity of the proposed approach is the dynamic interpretation of scheduling based on a natural dynamic decomposition of the problem and its solution with the help of a modified form of continuous maximum principle blended with combinatorial optimisation. The special properties of the developed model allow using methods of discrete optimisation for the schedule calculation. Optimality and sufficiency conditions as well as structural properties of the model are investigated. Advantages and limitations of the proposed approach are discussed. With the developed approach, an explicit inclusion of a process modernisation in the SC co-ordinated decisions for a wide ranges of possible applications as well as a dynamic model and a tractable algorithm for optimal discrete time scheduling on the basis of continuous maximum principle have been obtained.

Discrete and continuous maximum principles for parabolic and elliptic operators

The major qualitative properties of linear parabolic and elliptic operators/PDEs are the different maximum principles (MPs). Another important property is the stabilization property (SP), which connects these two types of operators/PDEs. This means that under some assumptions the solution of the parabolic PDE tends to an equilibrium state when t → ∞ , which is the solution of the corresponding elliptic PDE. To solve PDEs we need to use some numerical methods, and it is a natural requirement that these qualitative properties are preserved on the discrete level. In this work we investigate this question when a two-level discrete mesh operator is used as the discrete model of the parabolic operator (which is a one-step numerical procedure for solving the parabolic PDE) and a matrix as a discrete elliptic operator (which is a linear algebraic system of equations for solving the elliptic PDE). We clarify the relation between the discrete parabolic maximum principle (DPMP), the discrete elliptic maximum principle (DEMP) and the discrete stabilization property (DSP). The main result is that the DPMP implies the DSP and the DEMP.

Pointwise a posteriori error estimates for monotone semi-linear equations

We derive upper and lower a posteriori estimates for the maximum norm error in finite element solutions of monotone semi-linear equations. The estimates hold for Lagrange elements of any fixed order, non-smooth nonlinearities, and take numerical integration into account. The proof hinges on constructing continuous barrier functions by correcting the discrete solution appropriately, and then applying the continuous maximum principle; no geometric mesh constraints are thus required. Numerical experiments illustrate reliability and efficiency properties of the corresponding estimators and investigate the performance of the resulting adaptive algorithms in terms of the polynomial order and quadrature.

Optimal reflux rate policy determination for multicomponent batch distillation columns

This paper describes a procedure to obtain optimal reflux or optimal distillate rate policy for multicomponent batch distillation columns using the Pontryagin's continuous maximum principle. For application to multicomponent systems, a reduced order model is used using short-cut procedure developed specifically for analysis and design of batch columns.

Development of an optimal linear controler for antibiotic fermentation

The optimization method based on the Pontryagin's continuous maximum principle and was used to predict the optimal specific growth rate profile for a typical antibiotic fermentation model with a given nominal set of start values and parameters. If this set is unknown or known only to be in some domains for the process, then it is necessary to use an o ptimal l inear c ontroler (OLC). The behaviour is demonstrated by simulation runs.

Maximum cell productivity by repeated fed‐batch culture for constant yield case

AbstractOptimal operation of repeatedly fed‐batch was determined by the continuous maximum principle for the constant yield case. The objective of maximum cell productivity for a fixed cell concentration was achieved by finding the substrate feeding policy that minimized the processing time. Analytical criteria for the optimal filling policy show that an exponential policy is optimum when the specific growth rate has a maximum, and also that operation in the simple repeated batch mode is optimum when the specific growth rate is monotonic increasing. Comparisons between optimal repeated fed‐batch culture and other modes of operation were made for the case of substrate‐inhibited growth. Cell productivity by repeated fed‐batch exceeds both batch and continuous operation for the case of low residual substrate concentration.

Optimal aeration control in a post-aeration basin

Abstract The optimal aeration policy in a post-aeration basin is investigated. The physical problem is formulated using a dispersion model to describe the dissolved oxygen and the pollutant concentrations. The mathematical model is amenable to the Pontryagin's continuous maximum principle which reduces the optimization problem to a boundary value problem. The optimal aeration policy is determined by numerically integrating the boundary value problem. Simulations are made to show the important effects of the inlet waste strength, the flow pattern and the flow retention on the optimal aeration policy.

Optimal operation of a batch cooling crystallizer

A theory of programmed cooling crystallization is presented based on the moment transformation of the population balance. Numerical predictions for the behaviour of a batch crystallizer agree closely with those from an alternative theory based on the discrete-supersaturation balance. A potential advantage of the present approach is shown to lie in the application of the Continuous Maximum Principle in optimal control theory to facilitate the numerical computation of optimal cooling curves. The transient behavior of a computed “size-optimal” operating policy which maximises the terminal size of the largest crystals is shown to be significantly different from either natural of linear cooling or from crontrolled cooling at a constant nucleation rate. A peak in the nucleation rate towards the end of the operation and an increased terminal crystal size is predicted for the conditions considered and this is supported by preliminary experimental work with potassium sulphate solutions in a laboratory-scale crystallizer.

Optimization of batch fermentation processes. II. Optimum temperature profiles for batch penicillin fermentations.

AbstractOptimization methods based on the continuous maximum principle and the calculus of variations were used to calculate optimum temperature profiles for batch penicillin fermentations. These methods were first applied to several general models to develop effective techniques for the numerical solution of the equations. Subsequently, these methods were applied to two particular models, derived from experimental data, and the optimum temperature profiles were determined. The results indicated that an improvement, in penicillin yield of about 15% was possible if the optimum temperature profiles were followed.

Statistical optimal control†

In an attempt to modify Pontryagin's method in such a way that more complicated problems if optimization can be brought within the capabilities of present-day computers, the analogy between hamiltonian dynamics and the continuous maximum principle has been extended. This paper gives some basic theorems.

The Continuous Maximum Principle

The Continuous Maximum Principle

The Continuous Maximum Principle

The Continuous Maximum Principle

The Continuous Maximum Principle, L. T. Fan, John Wiley and Sons, New York (1966). 411 pages, $16.00

The Continuous Maximum Principle, L. T. Fan, John Wiley and Sons, New York (1966). 411 pages, $16.00