Solution Of Optimal Control Problem Research Articles

Analysis and achievement for fractional optimal control of Hepatitis B with Caputo operator

Deciphering the effect of long memory on dynamic behavior is important for controlling disease transmission. Employing fractional order calculus, here, we propose a model with Caputo operator to emphasize the regulation effect of the long memory on the optimal control of Hepatitis B. Considering four different health classes of populations, susceptible, acute infections, chronic infections, and recovered, we try to elucidate the existence of the optimal solution of the fractional optimal control problems (FOCP) by applying a strategy of isolation, treatment, and vaccination. Initially, we discuss the locally asymptotically stability of the equilibrium solutions of the fractional differential model. Sensitivity analysis and numerical simulations are also carried out to verify how parameter changes affect the system dynamics, that is, the threshold effect depending on the basic reproduction number, which provides inspiration for constructing the controlled system. After that, the necessary conditions for the optimality of the Hepatitis B system are derived to focus on the Caputo fractional operator with a singular kernel. An effective numerical simulation algorithm based on trapezoidal approximation is used to validate the control effect in the sense of the transient response by comparing the difference between the fractional-order and integer-order derivatives, and the numeric results can vividly show the attenuation effect of memory on control. This study may provide a powerful rationale for uncovering the importance of the long-memory effects in disease control.

Comparison of metaheuristic optimization algorithms for numerical solutions of optimal control problems

SummaryThe objective in optimal control problems is to determine the control rule to be applied according to the objective function determined for a given system. Theoretically, different solution methods have been developed starting from calculus of variation such as Pontriagin, Hamilton, or Riccati. In this study, apart from analytical methods, it is proposed to apply algorithms inspired by nature to the same problem. Among these algorithms, genetic algorithm, firefly algorithm, Harris Hawk's optimization algorithm, and differential evolution optimization algorithm have been applied to given optimal control problems and their successes were compared in terms of statistical analysis such as minimum, maximum, average, Wilcoxon, and Friedman analysis. The originality of this study is to control the system with discrete control rule by partitioning the control signal, to be applied to the system which is wanted to be optimally controlled. This will allow the control of different techniques, such as MPC or NMPC, which will contribute to the determination of the global optimum control rule, especially in the solution of nonlinear systems. In addition, three different applications have been performed to demonstrate. As a result of these experiments, it has been shown that these algorithms can be applied successfully to such problems.

Эволюционные вычисления для решения задачи терминального оптимального управления

The present article considers the problem of numerical solution of the terminal optimal control problem. The general statement of the terminal optimal control problem and a brief overview of its solving methods are presented. With a direct approach and reduction of the optimal control problem to the finite-dimensional optimization problem, the target functional on the space of desired parameters, regardless of the type of approximation of the control function, may not have the unimodal property. Therefore, it is advisable to use evolutionary algorithms to solve the problem. A general approach to solving the terminal optimal control problem of evolutionary computational algorithms is presented. The paper presents a description of some evolutionary algorithms that were selected as the most effective for solving the optimal control problem. A hybrid evolutionary algorithm based on a combination of several evolutionary algorithms is considered. The computational experiment considers the terminal optimal control problems, for which optimal solutions were found by known classical numerical methods that use the gradient of the target functionality when searching. Comparison of the results obtained by classical and evolutionary methods by functional values and computational costs allows us to conclude that evolutionary algorithms are able to effectively solve the terminal optimal control problems

Scalable Parallel Nonlinear Optimization with PyNumero and Parapint

We describe PyNumero, an open-source, object-oriented programming framework in Python that supports rapid development of performant parallel algorithms for structured nonlinear programming problems (NLP’s) using the Message Passing Interface (MPI). PyNumero provides three fundamental building blocks for developing NLP algorithms: a fast interface for calculating first and second derivatives with the AMPL Solver Library (ASL), a number of interfaces to efficient linear solvers, and block-structured vectors and matrices based on NumPy, SciPy, and MPI that support distributed parallel storage and computation. PyNumero’s design enables efficient, parallel algorithm development using high-level Python syntax while keeping expensive numerical calculations in fast, compiled implementations based on languages like C and Fortran. To demonstrate the utility of PyNumero, we also present Parapint, a Python package built on PyNumero for parallel solution of dynamic optimization problems. Parapint includes a parallel interior-point solver based on Schur-Complement decomposition. We illustrate the effectiveness of PyNumero for developing parallel algorithms with both code examples and scalability analyses for parallel matrix-vector dot products, parallel solution of structured systems of linear equations using Schur-Complement decomposition, and the parallel solution of a two-dimensional PDE optimal control problem. Our numerical results show nearly perfect scaling to more than 1,000 cores for large matrix-vector dot products and structured linear systems. Moreover, we obtain more than 354 times speedup for the optimal control example. History: Accepted by Alice Smith, EIC/Ted Ralphs, Area Editor/Software Tools. Funding: This work was funded in part by the Institute for the Design of Advanced Energy Systems (IDAES) with funding from the Office of Fossil Energy, Cross-Cutting Research, U.S. Department of Energy. This work was also funded by Sandia National Laboratories Laboratory Directed Research and Development (LDRD) program. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.1272 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2021.0285 ) at ( http://dx.doi.org/10.5281/zenodo.7192328 ).

Trajectory Optimization of Chance-Constrained Nonlinear Stochastic Systems for Motion Planning Under Uncertainty

In this article, we present generalized polynomial chaos-based sequential convex programming (gPC-SCP) to compute a suboptimal solution for a continuous-time chance-constrained stochastic nonlinear optimal control (SNOC) problem. The approach enables motion planning for robotic systems under uncertainty. The gPC-SCP method involves two steps. The first step is to derive a surrogate problem of deterministic nonlinear optimal control (DNOC) with convex constraints by using gPC expansion and the distributionally robust convex subset of the chance constraints. The second step is to solve the DNOC problem using sequential convex programming for trajectory generation and control. We prove that in the unconstrained case, the optimal value of the DNOC converges to that of SNOC asymptotically and that any feasible solution of the constrained DNOC is a feasible solution of the chance-constrained SNOC. We also present the predictor–corrector extension (gPC-SCP <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^\text{PC}$</tex-math></inline-formula> ) for real-time motion trajectory generation in the presence of stochastic uncertainty. In the gPC-SCP <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^\text{PC}$</tex-math></inline-formula> method, we first predict the uncertainty using the gPC method and then optimize the motion plan to accommodate the uncertainty. We empirically demonstrate the efficacy of the gPC-SCP and the gPC-SCP <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^\text{PC}$</tex-math></inline-formula> methods for the following two test cases: first, collision checking under uncertainty in actuation and physical parameters and second, collision checking with stochastic obstacle model for 3DOF and 6DOF robotic systems. We validate the effectiveness of the gPC-SCP method on the 3DOF robotic spacecraft testbed.

Solving distributed-order fractional optimal control problems via the Fibonacci wavelet method

A new approach to finding the approximate solution of distributed-order fractional optimal control problems (D-O FOCPs) is proposed. This method is based on Fibonacci wavelets (FWs). We present a new Riemann–Liouville operational matrix for FWs using the hypergeometric function. Using this, an operational matrix of the distributed-order fractional derivative is presented. Implementing the mentioned operational matrix with the help of the Gauss–Legendre numerical integration, the problem converts to a system of algebraic equations. Error analysis is proposed. Finally, the validation of the present technique is checked by solving some numerical examples.

Two-grid methods of finite element approximation for parabolic integro-differential optimal control problems

&lt;abstract&gt;&lt;p&gt;In this paper, we present a two-grid scheme of fully discrete finite element approximation for optimal control problems governed by parabolic integro-differential equations. The state and co-state variables are approximated by a piecewise linear function and the control variable is discretized by a piecewise constant function. First, we derive the optimal a priori error estimates for all variables. Second, we prove the global superconvergence by using the recovery techniques. Third, we construct a two-grid algorithm and discuss its convergence. In the proposed two-grid scheme, the solution of the parabolic optimal control problem on a fine grid is reduced to the solution of the parabolic optimal control problem on a much coarser grid; additionally, the solution of a linear algebraic system on the fine grid and the resulting solution maintain an asymptotically optimal accuracy. Finally, we present a numerical example to verify the theoretical results.&lt;/p&gt;&lt;/abstract&gt;

Optimal control of the Navier— Stokes system with a space variable in a network-like domain

The research of the problem of optimal control of the Navier—Stokes evolutionary differential system, considered in Sobolev spaces, the elements of which are functions with carriers in an n-dimensional network-like domain, is presented. Such domain consists of a finite number of subdomains, mutually adjacent to certain parts of the surfaces of their boundaries according to the graph type. For functions that are elements of these spaces, the conditions for the existence of traces on the surfaces of the joining are presented and the conditions of adjacency subdomains to which these functions satisfy are described. In applied questions of the analysis of the processes of transport of continuous media, the conditions of adjacency describe the regularities of the flow of fluid through the boundaries of the adjacent domains. The paper presents the results of following main research questions: 1) weak solvability of the initial boundary value problem for the Navier—Stokes system and obtaining the conditions for the existence of a weak solution to this problem; 2) the formation and solution of optimal control problems of various types of Navier—Stokes system. The fundamental approach to the analysis of the weak solvability of the initial boundary value problem is its reduction to the differential-difference problem (semi-digitization of the original system by a time variable) and subsequent use of a priori estimates for weak solutions of the obtained boundary value problems. The obtained a priori estimates are used to prove the theorem of the existence of a weak solution of the original differential system and indicate the way of the actual construction of this solution. A universal approach to solving the problems of optimal distributed and starting control of the Navier—Stokes evolutionary system is presented. The latter essentially expands the possibilities of analyzing non-stationary network-like processes of applied hydrodynamics (for example, processes of transporting various types of liquids through network or main line pipelines) and optimal control of these processes.

Optimal control problems of nonlocal interaction equations

In the present work we deal with the existence of solutions for optimal control problems associated to transport equations. The behaviour of a population of individuals will be influenced by the presence of a population of control agents whose role is to lead the dynamics of the individuals towards a specific goal. The dynamics of the population of individuals is described by a suitable nonlocal transport equation, while the role of the population of agents is designed by the optimal control problem. This model has been first studied in [12] for a class of continuous nonlocal potentials, while in the present project we consider the case of mildly singular potentials in a gradient flow formulation of the target transport equation.

A Note on the Realization of Nonlinear ODEs in Optimal Control Problems

When in the differential equation describing the behaviour of a dynamical system the time derivative of the input is involved, a naive realization may mislead the application of the Pontryagin Maximum Principle for the solution of optimal control problems. We show that a suitable procedure to eliminate the time derivative of the input leads to a realization that provides more information on the existence and on the form of the solution.

Operational matrix-based solution of optimal control problems using hybrid Chelyshkov polynomials

Operational matrix-based solution of optimal control problems using hybrid Chelyshkov polynomials

Nonlinear Model Predictive Control: an Optimal Search Domain Reduction

Nonlinear Model Predictive Control (NMPC) is a powerful control method, used in many industrial contexts. NMPC is based on the online solution of a suitable Optimal Control Problem (OCP) but this operation may require high computational costs, which may compromise its implementation in “fast” real-time applications. In this paper, we propose a novel NMPC approach, aiming to improve the numerical efficiency of the underlying optimization process. In particular, a Set Membership approximation method is applied to derive from data tight bounds on the optimal NMPC control law. These bounds are used to restrict the search domain of the OCP, allowing a significant reduction of the computation time. The effectiveness of the proposed NMPC strategy is demonstrated in simulation, considering an overtaking maneuver in a realistic autonomous vehicle scenario.

An Adaptive Multi-Level Max-Plus Method for Deterministic Optimal Control Problems

We introduce a new numerical method to approximate the solution of a finite horizon deterministic optimal control problem. We exploit two Hamilton-Jacobi-Bellman PDE, arising by considering the dynamics in forward and backward time. This allows us to compute a neighborhood of the set of optimal trajectories, in order to reduce the search space. The solutions of both PDE are successively approximated by max-plus linear combinations of appropriate basis functions, using a hierarchy of finer and finer grids. We show that the sequence of approximate value functions obtained in this way does converge to the viscosity solution of the HJB equation in a neighborhood of optimal trajectories. Then, under certain regularity assumptions, we show that the number of arithmetic operations needed to compute an approximate optimal solution of a d-dimensional problem, up to a precision ε, is bounded by O(Cd|logε|), for some constant C > 1, whereas ordinary grid-based methods have a complexity in O(1/εad) for some constant a > 0.

Fitness Dependent Optimizer Based Computational Technique for Solving Optimal Control Problems of Nonlinear Dynamical Systems

This paper presents a pragmatic approach established on the hybridization of nature-inspired optimization algorithms and Bernstein Polynomials (BPs), achieving the optimum numeric solution for Nonlinear Optimal Control Problems (NOCPs) of dynamical systems. The approximated solution for NOCPs is obtained by the linear combination of BPs with unknown parameters. The unknown parameters are evaluated by the conversion of NOCP to an error minimization problem and the formulation of an objective function. The Fitness Dependent Optimizer (FDO) and Genetic Algorithm (GA) are used to solve the objective function, and subsequently the optimal values of unknown parameters and the optimum solution of NOCP are attained. The efficacy of the proposed technique is assessed on three real-world NOCPs, including Van der Pol (VDP) oscillator problem, Chemical Reactor Problem (CRP), and Continuous Stirred-Tank Chemical Reactor Problem (CSTCRP). The final results and statistical outcomes suggest that the proposed technique generates a better solution and surpasses the recently represented methods in the literature, which eventually verifies the efficiency and credibility of the recommended approach.

A Neural Network Approach for High-Dimensional Optimal Control Applied to Multiagent Path Finding

We propose a neural network approach that yields approximate solutions for high-dimensional optimal control problems and demonstrate its effectiveness using examples from multi-agent path finding. Our approach yields controls in a feedback form, where the policy function is given by a neural network (NN). Specifically, we fuse the Hamilton-Jacobi-Bellman (HJB) and Pontryagin Maximum Principle (PMP) approaches by parameterizing the value function with an NN. Our approach enables us to obtain approximately optimal controls in real-time without having to solve an optimization problem. Once the policy function is trained, generating a control at a given space-time location takes milliseconds; in contrast, efficient nonlinear programming methods typically perform the same task in seconds. We train the NN offline using the objective function of the control problem and penalty terms that enforce the HJB equations. Therefore, our training algorithm does not involve data generated by another algorithm. By training on a distribution of initial states, we ensure the controls' optimality on a large portion of the state-space. Our grid-free approach scales efficiently to dimensions where grids become impractical or infeasible. We apply our approach to several multi-agent collision-avoidance problems in up to 150 dimensions. Furthermore, we empirically observe that the number of parameters in our approach scales linearly with the dimension of the control problem, thereby mitigating the curse of dimensionality.

Personalized Constrained MPC for glucose regulation

The technological advances reached in the last years overcome the relevant hardware limitations of the past artificial pancreas releasing portable devices with high computational capabilities. In view of this, the use of a Constrained MPC (CMPC) that needs the solution of a finite horizon optimal control problem can be considered. The real advantage of the explicit inclusion of the state constraints in the optimization problem is mainly related to the quality of the model used for the predictions. In a recent work, a CMPC based on an average model has been tested on the 100 in silico patients of the new UVA/Padova simulator, showing satisfactory results in terms of average glucose, time spent in tight range and time above 180 mg/dl. A percentage of time below 70 mg/dl grater than 3% is present for more than 25% of the patients. In this work, a new personalized CMPC is proposed to mitigate the hypoglycemia phenomena for these patients, using personalized impulse-response models for glucose prediction.

Variational Principles for Mirror Descent and Mirror Langevin Dynamics

Mirror descent, introduced by Nemirovski and Yudin in the 1970s, is a primal-dual convex optimization method that can be tailored to the geometry of the optimization problem at hand through the choice of a strongly convex potential function. It arises as a basic primitive in a variety of applications, including large-scale optimization, machine learning, and control. This paper proposes a variational formulation of mirror descent and of its stochastic variant, mirror Langevin dynamics. The main idea, inspired by the classic work of Brezis and Ekeland on variational principles for gradient flows, is to show that mirror descent emerges as a closed-loop solution for a certain optimal control problem, and the Bellman value function is given by the Bregman divergence between the initial condition and the global minimizer of the objective function.

The Numerical Solution for Quadratic Optimal Control Problems by Using Chebyshev and Legendre Polynomials

الغرض من هذا البحث هو حل مشاكل التحكم الأمثل من الدرجة الثانية (QOCP) عدديًا بمساعدة متعددة حدود Chebyshev و Legendre كوظائف أساسية لإيجاد حل للتحكم الأمثل (QOC) تقريبًا. سنشرح خوارزميات الحل بالأمثلة ونستخدم برنامج Mathcad للوصول إلى النتيجة الدقيقة.مقدمةتتمثل مشكلة التحكم المثلى في العثور على عنصر تحكم u ^ * (t) يقلل من مؤشر أداء معين مع تلبية معادلات حالة النظام والقيود. [1]نستخدم طرق التقريب لحل مشكلة التحكم الأمثل اعتمادًا على متعدد حدود Chebyshev في المرة الأولى ومتعدد حدود Legendre ، وبعد ذلك سنقترب من هذه الحلول ذات الوقت الخطي المستمر. للوصول إلى الحلول التقريبية ، نستخدم المعادلات التفاضلية الخطية متعددة المدى لـ u (t) و x (t) لكل من متعدد حدود Chebyshev و Legendre ونجعل شروط هذه المعادلات كمصفوفة مربعة للعثور على هذه القيم بواسطة نظام المصفوفات.عندما نستخدم هذه كثيرات الحدود في حلول تقريبية ، تم تقييم النتائج باستخدام الفهرس مع n = 5.سنشرح هذه الخوارزميات من خلال أخذ بعض الأمثلة لمشاكل التحكم التربيعي.تم تحديد المشكلة التربيعية الخطية على النحو التالي ؛قلل من الوقت المستمر التربيعيدالة التكلفة J = ∫_ (t_0) ^ (t_f) ▒ 〖(x ^ T Qx + u ^ T Ru) □ (24 &amp; dt)〗 ... (1)مع مراعاة معادلات حالة النظام الخطي ؛س ̇ (t) = Dx (t) + Eu (t) ، ... (2)حيث الشرط الأولي x (0) = x_0 والمصفوفات (D و E و Q و R) هي

An Algorithm for Finding Feedback in a Problem with Constraints for One Class of Nonlinear Control Systems

We consider the construction of a feedback according to the Kalman algorithm for a continuous nonlinear control system on a finite time interval with control constraints where the right-hand side of the dynamics equations is linear in control and linearizable in the vicinity of the zero equilibrium position. The solution of an auxiliary optimal control problem with a quadratic functional is used for this task by analogy with the SDRE approach. Because this approach is used in the literature to find suboptimal synthesis in optimal control problems with a quadratic functional with formally linear systems where all coefficient matrices in differential equations and criteria can contain state variables, on a finite time interval it becomes necessary to solve a complicated matrix differential Riccati equations with state-dependent coefficient matrices. Due to the nonlinearity of the system this issue significantly increases the number of calculations for obtaining the coefficients of the gain matrix in the feedback and for obtaining synthesis with a given accuracy in comparison with the Kalman algorithm for linear-quadratic problems. The proposed synthesis construction algorithm is constructed using the extension principle proposed by V.F. Krotov and developed by V.I. Gurman and allows one not only to expand the scope of the SDRE approach to nonlinear control problems with control constraints in the form of closed inequalities, but also to propose a more efficient computational algorithm for finding the matrix of feedback gains in control problems on a finite interval. This article establishes the correctness of the application of the extension principle by introducing analogs of the Lagrange multipliers, which depend on the state and time, and also derives a formula for the suboptimal value of the quality criterion. The presented theoretical results are illustrated by calculating suboptimal feedbacks in the problems of managing three-sector economic systems.

A numerical study on fractional optimal control problems described by Caputo‐Fabrizio fractional integro‐differential equation

AbstractThis paper provides a numerical technique for evaluating the approximate solution of fractional optimal control problems with the Caputo‐Fabrizio (CF) fractional integro‐differential equation. First, due to the Gegenbauer polynomials definition and CF‐fractional derivative, we present the modified operational matrix and complement vector of integration and CF‐fractional derivative. Then, the corresponding discretization of the problem is obtained with the help of the optimization method and the Legendre‐Gauss‐Lobatto (LGL) quadrature rule. The technique of obtaining the proposed matrices and LGL‐quadrature method leads to obtaining the approximate solution with high precision. Moreover, the error of the performance index obtained by the computational method is investigated. Finally, we implement the methodology in several examples. So that, the effect of various parameters defined in the method is illustrated in tables and plots.