Solution Of Control Problem Research Articles

A minimum principle for stochastic control of hepatitis C epidemic model

In this paper, we discussed a stochastic optimal control of hepatitis C that minimizes the side effect and reduces the viral load. The control variables represent the drug therapy used for blocking a new infection and virus production. The solution of control problem is solved using the stochastic minimum principle and a four-step scheme. The numerical simulation is carried out to justify the theoretical analysis. The result shows that using both types of drugs for therapy is much more effective.

Numerical solution of the control problem on the depletion of gas reservoirs with weakly permeable top

The current stage in the development of mathematical and software support for the processes of designing the development of hydrocarbon fields is characterized not only by the improvement of the means of geological and hydrodynamic modeling of reservoir fluid filtration but also by the use of algorithms for optimizing the development of gas deposits. The paper considers the problem of optimal control of the depletion of a gas reservoir with a low-permeability top. Using the so-called Myatiev-Girinsky hydraulic scheme, a two-dimensional equation describing the unsteady gas flow in a reservoir with a jumper is averaged over the capacity of the productive reservoir. This comes down to a one-dimensional equation with an additional term, taking into account gas-dynamic relationships between the reservoir and the jumper. For the numerical solution of process control problems, a formula for the gradient of the functional characterizing the reservoir depletion is found, and the method of successive approximations based on Pontryagin’s maximum principle is applied. In this case, the direct and conjugate boundary value problems are solved by the method of straight lines, and the required flow rate, without taking it beyond the maximum and minimum possible, is found by the gradient projection method with a special choice of step. A brief block diagram of the algorithm for solving the problem is shown; on its basis, a computer program was compiled. The results of calculations are presented to identify the influence of the values of the complex communication parameter not only on the state of the object but also on the operating mode of the well. The expediency of using the presented optimization tool is dictated by an increase in the share of deposits

Neural Networks Simulation of Distributed SEIR System

In this paper, a neural network-based optimal control synthesis is presented for distributed optimal control problems. We deal with solutions of systems controlled by parabolic differential equations with control and state constraints and discrete time delays. The given optimal control problem is transformed into a discrete nonlinear problem and then implemented into a feed-forward adaptive critic neural network. We propose a new algorithm to reach optimal control and an optimal trajectory using a feed-forward neural network. We present a concrete application of this simulation method on the SEIR (Susceptible—Exposed—Infectious—Recovered) optimal control problem of a distributed system for disease control. The results show that the adaptive-critic-based neural network approach is suitable for the solution of optimal distributed control problems with delay in state and control variables subject to control-state constraints and simulates the spread of disease in the SEIR system.

Statics and Dynamics of Continuum Robots Based on Cosserat Rods and Optimal Control Theories

This article explores the relationship between optimal control and Cosserat beam theory from the perspective of solving the forward and inverse dynamics (and statics as a subcase) of continuous manipulators and snake-like bioinspired locomotors. By invoking the principle of minimum potential energy and the Gauss principle of least constraint, it is shown that the quasi-static and dynamic evolutions of these robots are the solutions of optimal control problems in the space variable, which can be solved at each step (of loading or time) of a simulation with the shooting method. In addition to offering an alternative viewpoint on several simulation approaches proposed in the recent past, the optimal control viewpoint allows us to improve some of them while providing a better understanding of their numerical properties. The approach and its properties are illustrated through a set of numerical examples validated against a reference simulator.

Minimizing sequences for a linear-quadratic control problem with three-tempo variables under weak nonlinear perturbations

The paper deals with the construction of minimizing sequences for the problem of minimizing a weakly nonlinearly perturbed quadratic performance index on trajectories of a weakly nonlinear system with threetempo state variables. For this purpose, the so-called direct scheme for constructing an asymptotic solution is used, which consists in immediate substituting the postulated asymptotic expansion of the solution into the problem conditions and constructing a series of optimal control problems (in the case under consideration, linear-quadratic ones), the solutions of which are terms of the asymptotic expansion of the solution of the original nonlinear control problem. An estimate is obtained for the proximity of the optimal trajectory to the trajectory of the equation of state when some asymptotic approximation to the optimal control is used as a control. An example is given that illustrates in detail the proposed scheme for constructing minimizing sequences.

Efficient Solution of Discrete Subproblems Arising in Integer Optimal Control with Total Variation Regularization

We consider a class of integer linear programs (IPs) that arise as discretizations of trust-region subproblems of a trust-region algorithm for the solution of control problems, where the control input is an integer-valued function on a one-dimensional domain and is regularized with a total variation term in the objective, which may be interpreted as a penalization of switching costs between different control modes. We prove that solving an instance of the considered problem class is equivalent to solving a resource-constrained shortest-path problem (RCSPP) on a layered directed acyclic graph. This structural finding yields an algorithmic solution approach based on topological sorting and corresponding run-time complexities that are quadratic in the number of discretization intervals of the underlying control problem, the main quantifier for the size of a problem instance. We also consider the solution of the RCSPP with an [Formula: see text] algorithm. Specifically, the analysis of a Lagrangian relaxation yields a consistent heuristic function for the [Formula: see text] algorithm and a preprocessing procedure, which can be employed to accelerate the [Formula: see text] algorithm for the RCSPP without losing optimality of the computed solution. We generate IP instances by executing the trust-region algorithm on several integer optimal control problems. The numerical results show that the accelerated [Formula: see text] algorithm and topological sorting outperform a general-purpose IP solver significantly. Moreover, the accelerated [Formula: see text] algorithm is able to outperform topological sorting for larger problem instances. We also give computational evidence that the performance of the superordinate trust-region algorithm may be improved if it is initialized with a solution obtained with the combinatorial integral approximation. History: Accepted by Andrea Lodi, Area Editor for Design and Analysis of Algorithms–Discrete. Supplemental Material: The online supplement is available at https://doi.org/10.1287/ijoc.2023.1294 .

On the solution stability of parabolic optimal control problems

The paper investigates stability properties of solutions of optimal control problems constrained by semilinear parabolic partial differential equations. Hölder or Lipschitz dependence of the optimal solution on perturbations are obtained for problems in which the equation and the objective functional are affine with respect to the control. The perturbations may appear in both the equation and in the objective functional and may nonlinearly depend on the state and control variables. The main results are based on an extension of recently introduced assumptions on the joint growth of the first and second variation of the objective functional. The stability of the optimal solution is obtained as a consequence of a more general result obtained in the paper–the metric subregularity of the mapping associated with the system of first order necessary optimality conditions. This property also enables error estimates for approximation methods. A Lipschitz estimate for the dependence of the optimal control on the Tikhonov regularization parameter is obtained as a by-product.

A deep learning-based predictive controller for the optimal charging of a lithium-ion cell with non-measurable states

Battery charging is a complex task, which needs to be addressed by a proper control methodology to find the highest charging current while guaranteeing safety. Among the different approaches, model predictive control appears particularly suitable due to its ability in dealing with nonlinear systems and constraints. However, its use in a realistic scenario is limited due to the high computational burden required by the online solution of an optimal control problem. A neural network-based algorithm is here proposed to significantly reduce the real-time computational effort by approximating the predictive control law. In addition, for the first time to the authors’ knowledge, an adaptation of the proposed deep learning-based algorithm is presented for the case in which the battery’s internal states are not measurable. The superiority of proposed methodology is highlighted in simulation by comparing it with a predictive controller coupled with a properly designed state observer.

Comparison of Control Techniques for Harmonic Isolation in Series VSC-Based Power Flow Controller in Distribution Grids

The application of power electronics equipment in medium voltage (MV) distribution grids can provide new management solutions for power flow control, load balancing and voltage problems. A series MV VSC-based power flow controller has recently been presented to interconnect two radial distribution feeders performing active and reactive power transfers to improve the flexibility and utilization of these circuits in a controlled and secure way. Although not previously explored, this power flow controller can integrate the functionality of a series power filter, accomplishing independent control of the fundamental power flow while isolating the harmonic content between the two interconnected feeders. This prevents harmonic pollution from one feeder from propagating to the other, improving the voltage quality. To implement the harmonic isolation, several control strategies can be used. Therefore, this paper provides a comparative analysis between two of the main harmonic control techniques found in the literature: the Synchronous Reference Frame (SRF) controller and the Proportional Resonant (PR) controller. Assessments are conducted both through simulations and experimental results in a meshed network at 13.8 kV with different types of non-linear loads. In the simulation cases, both algorithms showed similar results; however, in the experimental cases, the PR-based solution exhibited better performance in isolating the harmonics from one feeder to the other.

On the Stability of Solutions to Control Problems for a Nonlinear Reaction–Diffusion–Convection Model

The problems of multiplicative control for the reaction–diffusion–convection model with coefficients that are nonlinearly dependent on the solution and also dependent on spatial variables are studied. In the case of a power-law dependence of the model coefficients on the solution, optimality systems are derived for extremum problems. With their help, estimates of the local stability of solutions of specific control problems with respect to small perturbations of both the cost functionals and one of the given functions of the boundary value problem are obtained.

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

Обобщенная задача динамического уравновешивания и перспективные направления ее применения

The paper considers the generalized problem of the machines and mechanisms dynamic balance in terms of ensuring the given laws of altering reactions in the selected links. Representation of equations of the mechanical systems dynamics in the form of differential algebraic equations was used making it possible to obtain mathematical models of the nonlinear mechanical systems dynamics with the arbitrary structure of kinematic and force connections. With this approach, the constraint reactions are determined by algebraic equations from the system coordinates. The problem solution is based on changing the bonds selected reactions due to the impact on reactions in the other selected bonds by adding non-stationary terms to the selected bonds equation. Conditions for rigorous solution of the optimal control problem for a mechanical system are shown for the integral quality criterion not explicitly containing the control functions. The method is aimed at numerical models of the mechanical systems widely used in the programs for dynamic analysis of the coupled systems of bodies. Test examples are provided for manipulator, anthropomorphic robot and controlled suspension of a transport vehicle. The method was realized in the software package for simulating the controlled motion dynamics of the coupled systems of bodies.

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 ).

Online adaptive optimal tracking control for model-free nonlinear systems via a dynamic neural network

This paper presents an online adaptive approximate solution for the optimal tracking control problem of model-free nonlinear systems. Firstly, a dynamic neural network identifier with properly designed weights updating laws is developed to identify the unknown dynamics. Then an adaptive optimal tracking control policy consisting of two terms is proposed, i.e. a steady-state control term is established to ensure the desired tracking performance at the steady state, and an optimal control term is proposed to ensure the optimal tracking error dynamics optimally. The composite Lyapunov method is used to analyse the stability of the closed-loop system. Two simulation examples are presented to demonstrate the effectiveness of the proposed method.

Model Predictive Interaction Control for Robotic Manipulation Tasks

This article presents the concept of model predictive interaction control (MPIC) as a generic, flexible, and comprehensive approach for robotic manipulation tasks. MPIC is based on the repetitive solution of an optimal control problem that includes a robot model for motion prediction as well as an interaction model for force prediction. In order to handle both elastic and rigid contact situations, a cascaded approach with low-level PD control is adopted, which allows to combine the linear-elastic environment model and the limited controller stiffness. Due to its flexibility, MPIC can be favorably used for realizing the elementary manipulation primitives (MP) within a hierarchical task planning framework, where each MP corresponds to a particular parameterization of the cost function and the constraints. The control methodology and the manipulation approach are evaluated in simulations and experiments using a 7-degree-of-freedom industrial robot.

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;