Methods Of Optimal Control Theory Research Articles

Application of two classes of inverse problems for optimization and management of the natural hydrocarbon field development process

The article considers two classes of inverse problems in the modeling of hydrocarbon field development: optimization and adaptation of the development history. They jointly form the basis for field development management in a closed loop, based on a constantly updated 3D geological and technological model. The problems of model adaptation (identification) arise due to the point-by-point measurements and the inaccuracy of determining the initial data on porosity and permeability of the host rocks and other parameters of the deposit when building the 3D model. Such problems are regarded as the inverse ones, because the consequence – field measurements – is used to determine their cause. The problems under consideration are related to the identification of both porosity and permeability of the reservoir and other key parameters that determine the dynamics of development. Geologically consistent formulation ensures that the adapted model retains the original geological principles of the distribution of reservoir properties. The second class of problems under consideration is the problems of optimization (regulation) of development. One of the sought-after formulations is related to the optimal redistribution of a given target level of production from the field to all production wells in time. In this case, the technological limitations of surface equipment are taken into account. The maximized criterion reflects the economic effect of the asset development. In this article, without going into details of mathematical realization, the authors consider practical examples of solving these problems using full-scale 3D multiphase filtration models of the real fields. The algorithms based on the current methods of optimal control theory are used, which were developed and implemented by the researchers of Oil and Gas Research Institute of the Russian Academy of Sciences in the SimMatch software package and confirmed their effectiveness on numerous synthetic and real examples. Application and further development of these methods within the framework of a single loop, taking into account continuous information from the entire set of sensors, will significantly increase the efficiency of intelligent wells and field control systems.

Optimal and robust control of population transfer in asymmetric quantum-dot molecules

We present an optimal and robust quantum control method for efficient population transfer in asymmetric double quantum-dot molecules. We derive a long-duration control scheme that allows for highly efficient population transfer by accurately controlling the amplitude of a narrow-bandwidth pulse. To overcome fluctuations in control field parameters, we employ a frequency-domain quantum optimal control theory method to optimize the spectral phase of a single pulse with broad bandwidth while preserving the spectral amplitude. It is shown that this spectral-phase-only optimization approach can successfully identify robust and optimal control fields, leading to efficient population transfer to the target state while concurrently suppressing population transfer to undesired states. The method demonstrates resilience to fluctuations in control field parameters, making it a promising approach for reliable and efficient population transfer in practical applications.

Optimal boundary control of string oscillations by displacement at two ends with specified values of deflection function at intermediate moments of time

We consider the problem of optimal boundary control for the equation of string vibrations with given initial, final conditions and given values of the string deflection function at intermediate moments of time and with a quality criterion specified over the entire time interval. Using the method of separation of variables and methods of optimal control theory with multipoint intermediate conditions, optimal boundary controls are constructed for arbitrary numbers of the first harmonics. As an application of the proposed constructive approach, a boundary optimal control is built with a given string deflection function at an intermediate moment of time.

Optimal step length for the Newton method: case of self-concordant functions

The theoretical foundation of path-following methods is the performance analysis of the (damped) Newton step on the class of self-concordant functions. However, the bounds available in the literature and used in the design of path-following methods are not optimal. In this contribution we use methods of optimal control theory to compute the optimal step length of the Newton method on the class of self-concordant functions, as a function of the initial Newton decrement, and the resulting worst-case decrease of the decrement. The exact bounds are expressed in terms of solutions of ordinary differential equations which cannot be integrated explicitly. We provide approximate numerical and analytic expressions which are accurate enough for use in optimization methods. Consequently, the neighbourhood of the central path in which the iterates of path-following methods are required to stay can be enlarged, enabling faster progress along the central path during each iteration and hence fewer iterations to achieve a given accuracy.

Optimal Dynamic Measurement Method Using Digital Moving Average Filter

The theory of optimal dynamic measurements is based on the use of methods of optimal control theory to find a dynamically distorted input signal from the observed output signal. The model of the measuring system contains a Leontief type system, the initial Showalter–Sidorov condition. Inertia of the measuring system and interference lead to distortion of the input signal. We present a numerical method to use a digital moving average filter for the observed signal with subsequent application of the method of optimal dynamic measurement, as well as the results of computational experiments.

Method for constructing an adaptive suboptimal stationary train traffic controller based on artificial neural networks

The article discusses the modern methodology for performing the synthesis of a suboptimal train controller for the purpose of energy saving. The existing methods of optimal traction control have a number of disadvantages, the main one of which is the lack of direct use in the control program of the data obtained during train operation. Mathematical models used to solve the op- timal problem can be used correctly only in the case of sufficient adequacy. Adequacy check is not part of the known methods of optimal control theory. To eliminate this drawback, it is proposed to use the method of optimal (suboptimal) traction calculations based on artificial neural networks. It improves the accuracy of traction calculations, which is especially important in the aspect of considering energy savings, while reducing the need for computing power. When using this method, it is possible not only to achieve results close to the classical Bellman method, but also to train or verify the network using the recorded data. The article discusses the process of creating and training an artificial neural network based on model data to solve the problem of suboptimal control. The train motion modes obtained by Bellman's method were used as reference data for training the neural network. The presented comparative results of the two methods show the applicability of artificial neural networks for solving applied problems of train traction with the possibility of continuous learning, including the use of trip data, which can be directly included in the training or testing set.

About Identifiability of Oil and Water Relative Permeability Curves and Reservoir Heterogeneity through Integrated Well Test Study

Relative permeabilities (RP) play an important role in reservoir engineering. RP functions together with the permeability anisotropy coefficient predetermine waterflooding efficiency and oil/water ratio in production forecast. Traditionally multiphase flow parameters are estimated from core analysis data. But such measurements suffer from small representative elementary volume (REV) and limited characterization of reservoir properties. Therefore using core data in 3D reservoir modelling could lead to distorted description of actual flow conditions. Despite those functions (RP) could be history matched with assimilated production data, such a procedure would require long history of waterflooding. So the authors’ idea is to apply integrated well test study to estimate the displacement efficiency and/or relative permeability functions at downhole conditions. Well testing is traditionally applied in reservoir engineering for single-phase reservoir parameters estimation. Our approach is extended to multi-phase flow and is based on data collection at well bottomholes and subsequent data assimilation in flow model.In this paper we summarize both features of mathematical problem statement and identifiability of multiphase parameters through inverse problem solution, but also discuss 10+ -year experience in interpretation of two-phase well testing using our technologies. To estimate multiphase parameters, we have to jointly consider complex data of well logging and dynamic data of well testing. Both data sets are used in the inverse problem quality criteria, where the least squares method has been applied. Forward problem corresponds to a 3D multiphase fluid flow model in porous media. The latter consists of continuity equation with multiphase Darcy equation instead of moment balances. For data assimilation modern methods of optimal control theory were successfully applied. For all synthetic test cases real reservoir parameters were accurately recovered through the use of forward simulation data.In order to estimate reservoir heterogeneity, we applied a single-phase model to get the ratio of vertical to horizontal permeability. Data of well self-interference testing were used for vertical permeability estimation. Depending on the depth of reservoir pressure perturbation, reservoir properties could be properly inferred from observations. Two other options of 3D interference testing using vertical and horizontal wells are also presented.In other words, all the problems considered are identifiable, and the level of their correctness is completely predetermined by the amount and quality of observed data. And transition to digital (intelligent) oil-and-gas production and closed-loop reservoir management [1] provides a possibility to remove discussed restrictions and all inverse problems considered should be accounted as fully identifiable.

Control impacts for setting to the set state a wide class of self-oscillations

In the most important oscillation control system, the solution of the problem of determining the necessary control actions for transferring vibrations from one state to another is considered for a wide class of oscillations: harmonic, damped, and forced oscillations obeying the Euler equation. The structure and parameters of control actions are determined based on the methods of optimal control theory.

Abstract 5485: Patient-specific neoadjuvant regimens for breast cancer identified via image-driven mathematical modeling

Abstract Introduction: We show that the combination of quantitative magnetic resonance imaging (MRI) and mathematical modeling can accurately predict tumor response for individual patients, and we demonstrate the selection of personalized therapeutic regimens using our mathematical model to vary, in silico, a range of clinically feasible treatment plans to achieve the greatest tumor control. Methods: Breast cancer patients (N = 18) were scanned by MRI at three time points during neoadjuvant therapy (NAT): 1) prior to NAT, 2) after one cycle of their initial NAT regimen, and 3) after the completion of their initial NAT regimen. Specifically, diffusion-weighted MRI data characterizing tumor cellularity was collected to estimate tumor cell burden, and dynamic contrast-enhanced MRI data was collected to estimate the local drug delivery using pharmacokinetic analysis and population-derived plasma curves of the administered chemotherapies. To predict tumor response, we calibrated our tissue scale, 3D, biophysical mathematical model to each patient's MRI data set using their first two scans. Using the calibrated, patient-specific parameters, the model was projected forward to the third scan time. The predicted total tumor cellularity, volume, and longest axis were compared to the actual values measured from the patient's third scan. Following evaluation of the model's predictive ability, we then employed the model to identify potentially superior, clinically relevant, alternative dosing regimens for each patient. The alternative regimens were defined using the same total amount of drug each patient received during their standard regimen, while varying dosages and frequency between their second and third scans. Statistical analysis was completed with Pearson correlation and Wilcoxon signed rank test. Results: The model's predictions of tumor response are significantly correlated to the measured tumor burden at the time of scan 3 (r2 > 0.88, p < 0.01) for total cellularity, total volume, and longest axis. For the alternative dosing regimens assessed, the model predicted that individual patients could have achieved, on average, an additional 21% (0-46%) reduction in total cellularity. The optimal dosing regimens chosen by the model were predicted to significantly outperform standard regimens for tumor control (p < 0.001). Conclusions: These results suggest that the mathematical model can be predictive of tumor response by MRI in the clinical setting using data at the earliest times of therapy. With in silico studies, we illustrate how therapeutic regimens can be selected for individual patients for better tumor control, revealing that standard regimens may not be the most effective for every patient. These results represent a first step towards mathematically-based, personalized patient regimens. Future work aims to optimize therapy regimens via established optimal control theory methods. Citation Format: Angela M. Jarrett, Ernesto A. Lima, David A. Hormuth, Chengyue Wu, John Virostko, Anna G. Sorace, Julie C. DiCarlo, Debra Patt, Boone Goodgame, Sarah Avery, Thomas E. Yankeelov. Patient-specific neoadjuvant regimens for breast cancer identified via image-driven mathematical modeling [abstract]. In: Proceedings of the Annual Meeting of the American Association for Cancer Research 2020; 2020 Apr 27-28 and Jun 22-24. Philadelphia (PA): AACR; Cancer Res 2020;80(16 Suppl):Abstract nr 5485.

Abstract P2-16-17: Optimizing neoadjuvant regimens for individual breast cancer patients generated by a mathematical model utilizing quantitative magnetic resonance imaging data: Preliminary results

Abstract Introduction: Tumor forecasting methods for predicting treatment response of individual breast cancer patients to neoadjuvant therapy (NAT) have shown promise in clinical application. Our framework for predicting tumor response integrates quantitative magnetic resonance imaging (MRI) data acquired early in the course of NAT into a mechanism-based, biophysical model that predicts the eventual treatment response of breast tumors. Being able to predict which patient will respond effectively to NAT would have a fundamental and lasting impact on healthcare. However, the ultimate goal is to optimize therapy given the unique characteristics of each patient. Here we show that the detailed combination of advanced image analysis and rigorous mathematical modeling can accurately predict response for the individual patient. Further, we use the model to demonstrate the potential selection of personalized therapeutic regimens. This is accomplished by initializing the mathematical model with patient-specific characteristics and then varying, in silico, a range of treatment plans to achieve the greatest tumor control. Methods: Quantitative MRI was acquired from breast cancer patients (N = 11) at three time points during the course of NAT: 1) prior to NAT, 2) after 1 cycle of their initial chemotherapy, and 3) after the completion of the initial chemotherapy regimen. With these data, we implemented our recently established mechanically coupled, reaction-diffusion model at the tissue scale for predicting breast tumor response to therapy. The 3D model is initialized with patient-specific, diffusion-weighted MRI data characterizing tumor cellularity. Additionally, the model includes a tumor cell reduction term for local drug delivery as estimated from pharmacokinetic analysis of dynamic contrast-enhanced MRI data and population-derived plasma curves of therapeutic concentrations. The model’s predictive ability was assessed using three different measures. Using the first two scans, the model is calibrated and simulated forward to the third scan time to compare the predicted total tumor cellularity, volume, and longest axis to the actual values measured from the patient’s third scan. We then simulate alternate regimens using the same total dose each patient received during their standard regimen, while varying dosages and frequency between their second and third scans. Results: After calibrating the model using the first two imaging time points, the model’s predictions are significantly correlated to the measured tumor burden at scan three with Pearson Correlation Coefficients of 0.93, 0.89, 0.96 (p < 0.01) for total cellularity, total volume, and longest axis, respectively. The model predicts that for the alternative dosing regimens assessed, individual patients could have achieved an additional 0-43% reduction in total cellularity compared to the therapeutic regimens patients actually received. This indicates that standard regimens may not be the most effective for every patient. Discussion and future directions: These results demonstrate that the mathematical model can be predictive of tumor response using data at the earliest times of therapy regimens. The in silico results illustrate how for individual patients (depending on their unique tumor characteristics and vasculature—captured by the calibrated parameters of the model), therapy regimens can be tailored and even optimized (via established optimal control theory methods) to each patient using a mathematical model and simulation studies. The present investigation represents a significant first step towards personalizing patient regimens through quantitative imaging and mathematical modeling. NCI U01 CA174706, NCI U01 CA154602, CPRIT RR160005, ACS-RSG-18-006-01-CCE Citation Format: Angela M Jarrett, David A Hormuth II, Chengyue Wu, John Virostko, Anna G Sorace, Julie C DiCarlo, Debra Patt, Boone Goodgame, Sarah Avery, Thomas E Yankeelov. Optimizing neoadjuvant regimens for individual breast cancer patients generated by a mathematical model utilizing quantitative magnetic resonance imaging data: Preliminary results [abstract]. In: Proceedings of the 2019 San Antonio Breast Cancer Symposium; 2019 Dec 10-14; San Antonio, TX. Philadelphia (PA): AACR; Cancer Res 2020;80(4 Suppl):Abstract nr P2-16-17.

О разрешимости задачи нелинейной оптимизации колебательных процессов при появлении особых управлений

The article deals with the cases when special controls appear in nonlinear optimization of oscillatory processes, when the function of external influence depends non-linearly on the control parameter. Quality-managed process is to minimize the integral of the functional, that is, in the final moment of time square deviation of a controlled process from a given desired state was minimal. The study was conducted using a generalized solution of the boundary value problem, which more or less adequately describes the actual process. According to the well-known method of optimal control theory, the increment of the functional is calculated and the Pontryagin function is written out, which is studied for the maximum in the range of acceptable values of the control function. The optimal control conditions are written out in the form of equality and differential inequality, which must be fulfilled simultaneously. The case when the maximum principle degenerates is studied. It is established that the desired control satisfies an infinite-dimensional system of linear integral Fredholm equations of the first kind, the solvability of which is studied by operator methods. It is proved that the operator equation has infinitely many solutions and an algorithm for their construction is developed. Then the functional is minimized on the set of solutions of the operator equation. This problem has at least one solution that is the desired special optimal control.

ОПТИМІЗАЦІЯ ТРАЄКТОРІЙ КРИВОЛІНІЙНИХ СВЕРДЛОВИН МЕТОДАМИ ЛІНІЙНОГО ТА НЕЛІНІЙНОГО ПРОГРАМУВАННЯ

Andrusenko E.N. Minimization of the cost of oil and gas borehole by the methods of linear and nonlinear programming. Visnyk National Transport University. Series «Economical sciences». Scientific and Technical Collection. - Kyiv: National Transport University, 2020. - Issue 2 (47). The task is posed of constructing optimal trajectories of oil and gas wells by linear and nonlinear programming methods, as well as the theory of optimal control. The object of study is the geometry of the axial line of deep directional, horizontal and branching trajectories of oil and gas wells. The purpose of the work is to build the optimal trajectory of an oil or gas well, ensuring its minimum cost, minimum length and minimum values of its construction parameters. A technique was developed for optimizing the geometry of the trajectory of an oil or gas well using the methods of optimal control theory. Cases of the formulation of the Lagrange, Mayer, and Boltz problems are considered under various (linear and nonlinear) constraints on the values of phase variables and control functions.

Global Stability for a Class of Nonlinear PDE with Non-local Term

This paper is concerned with establishing global asymptotic stability results for a class of non-linear PDE, in which the nonlinearity enters as a scalar multiple which is non-local. The method of proof does not involve a Lyapounov function. It is shown that stability for the PDE is equivalent to stability for a differential delay equation. Stability for the delay equation is proven by exploiting certain monotonicity properties. These are established by using the methods of optimal control theory.

Effective methods for optimal design of induction coils on example of surface hardening

Purpose This paper aims to describe main ideas and demonstrates results of the research activities carried out by the authors in the field of optimal design concepts for induction heater for surface hardening. The main goal of the research studies is the application of different optimization methods and numerical finite element method (FEM) codes for field analysis to solve the optimal design problem that is mathematically formulated in terms of the one of the most important optimization criteria for surface hardening technology, e.g. maximum temperature uniformity within the hardening surface layer. Design/methodology/approach Evolutionary algorithm based on Adaptive Gaussian Process-Assisted Differential Evolution for MEMS Design Optimization (AGDEMO) and alternance method of parametric optimization based on optimal control theory are applied as effective tools for the practice-oriented problem for optimization of induction heater design based on non-linear coupled electromagnetic and temperature field analysis. Different approaches are used for combining FEM codes for interconnected field analysis and optimization algorithms into automated optimization procedure. Findings Optimization procedures are tested and investigated for optimal design problem solution on the examples of induction hardening of steel cylindrical billet. Practical implications Solved problems are based on the design of practical industrial applications. The developed optimization procedures are planned to be applied to the wide range of real-life problems of the optimal design of different electromagnetic devices and systems. Originality/value This paper describes main ideas and results of the research activities carried out by the authors in the field of optimal design of induction heaters for hardening based on numerical coupled electromagnetic and temperature field analysis. The implementation of the automated procedure that combines a numerical FEM code for coupled field analysis with an optimization algorithm and its subsequent application for designing induction heaters makes the proposed approach specific and original. This paper also demonstrates that different optimization strategies used (evolutionary algorithm based on AGDEMO and alternance method of optimal control theory) are effective for real-life industrial applications for optimization of induction heaters design.

Design and control of electrotechnological systems

Purpose This paper aims to investigate different multi-objective optimization (MOO) approaches for design and control of electromagnetic devices. The main goal of MOO is to find the set of design variables or control parameters which will provide the best possible values of typical conflicting objective functions. Design/methodology/approach In the research studies, standard genetic algorithm (GA), non-dominated sorting GA (NSGA-II), migration NSGA algorithm and alternance method of optimal control theory are discussed and compared. Findings The test practical problems of multi-criteria optimization of induction heating processes with respect to chosen quality criteria confirm the effectiveness of application of considered MOO approaches both for the problems of design and control. Originality/value This paper represents and investigates different MOO approaches for design and control of electrotechnological systems.

Two-stroke optimization scheme for mesoscopic refrigerators

Refrigerators use a thermodynamic cycle to move thermal energy from a cold reservoir to a hot one. Implementing this operation principle with mesoscopic components has recently emerged as a promising strategy to control heat currents in micro and nanosystems for quantum technological applications. Here we combine concepts from stochastic and quantum thermodynamics with advanced methods of optimal control theory to develop a universal optimization scheme for such small-scale refrigerators. Covering both the classical and the quantum regime, our theoretical framework provides a rigorous procedure to determine the periodic driving protocols that maximize either cooling power or efficiency. As a main technical tool, we decompose the cooling cycle into two strokes, which can be optimized one by one. In the regimes of slow or fast driving, we show how this procedure can be simplified significantly by invoking suitable approximations. To demonstrate the practical viability of our scheme, we determine the exact optimal driving protocols for a quantum microcooler, which can be realized experimentally with current technology. Our work provides a powerful tool to develop optimal design strategies for engineered cooling devices and it creates a versatile framework for theoretical investigations exploring the fundamental performance limits of mesoscopic thermal machines.

Sub-Finsler structures on the Engel group

A one-parameter family of left-invariant sub-Finsler problems on a four-dimensional nilpotent Lie group of depth 3 with two generators is considered. The indicatrix of sub-Finsler structures is a square rotated by an arbitrary angle in the distribution. Methods of optimal control theory are applied. Abnormal and singular normal trajectories are described, and their optimality is proved. Singular trajectories arriving at the boundary of the reachable set in fixed time are characterized. A bang-bang phase flow is constructed, and estimates for the number of switchings on bang-bang trajectories are obtained. The structure of all normal extremals is described. Mixed trajectories are studied.

Infinite Geodesics of Sub-Finsler Distances in Heisenberg Groups

Abstract We consider Heisenberg groups equipped with a sub-Finsler metric. Using methods of optimal control theory we prove that in this geometric setting the infinite geodesics are horizontal lines under the assumption that the sub-Finsler metric is defined by a strictly convex norm. This answers a question posed in [8] and has applications in the characterization of isometric embeddings into Heisenberg groups.

Sub-Finsler Structures on the Engel Group

A one-parameter family of left-invariant sub-Finsler problems on a four-dimensional nilpotent Lie group of depth 3 with two generators is considered. The indicatrix of sub-Finsler structures is a square rotated by an arbitrary angle in the distribution. Methods of optimal control theory are applied. Abnormal and singular normal trajectories are described, and their optimality is proved. Singular trajectories arriving at the boundary of the reachable set in fixed time are characterized. A bang-bang phase flow is constructed, and estimates for the number of switchings on bang-bang trajectories are obtained. The structure of all normal extremals is described. Mixed trajectories are studied.

Positive Nonlinear Systems Synthesis Based on Optimal Control Methods

This paper considers a nonlinear positive control system of autonomous intellectual agent’s motions. Single or multiple intellectual agents are capable of independent decision-making to achieve an odor source or radioactive source. Various biologically inspired behaviour-based approaches, such as chemotaxis, the moth inspired casting algorithm, flocking behavior and foraging, population development or species interaction are used for creation of control system of an intellectual agent. The agent is a tracking system, for example, unmanned aerial vehicle or a multicopter. The intellectual agent searches hazardous pollutants in dangerous environments. The task of a flying robot is to find the source of invisible pollutions. The aim of this work is to synthesize the control law, that provides a predetermined degree of exponential stability in a closed-loop positive system based on Lotka—Volterra equations. The methods of optimal control theory are used in the synthesis of the system. Asymptotic stability is achieved by solving the Riccati equation. The stability of the control system is an important criterion of quality to be ensured. Therefore, the intellectual agent is able to control the movements to the right and to the left, reaching the source. With the model proposed we provide the simulation and experimental results, which correspond to the quality metrics required. In this work, the control system has a given degree of exponential stability and the transient response, which confirms the possibility of using the model proposed in terrestrial mobile robots, unmanned aerial vehicles, autonomous underwater vehicles and other robots in searching a large area.