Solution Of Optimal Control Problem Research Articles

A two-level variational algorithm in the Sobolev–Orlicz space to predict daily surface reflectance at LANDSAT high spatial resolution and MODIS temporal frequency

We propose a new two-level variational model in Sobolev–Orlicz spaces with nonstandard growth conditions of the objective functional and discuss its applications to the spatiotemporal interpolation of multispectral satellite images. At the first level, we deal with the temporal interpolation problem that can be cast as a state constrained optimal control problem for anisotropic convection–diffusion equation, whereas at the second level we solve a constrained minimization problem with a nonstandard growth energy functional that lives in variable Sobolev–Orlicz spaces. The characteristic feature of the proposed model is the fact that the variable exponent, which is associated with nonstandard growth in spatial interpolation problem, is unknown a priori and it depends on the solution of the first-level optimal control problem. It makes this spatiotemporal interpolation problem rather challenging. In view of this, we discuss the consistency of the proposed model, study the existence of optimal solutions, and derive the corresponding optimality systems. In particular, we apply this approach to the well-known prediction problem of the Daily MODIS Surface Reflectance at the Landsat-Like Resolution.

Ergodic control of McKean–Vlasov SDEs and associated Bellman equation

We consider the ergodic control problem for McKean-Vlasov stochastic differential equations and prove the existence and uniqueness of the viscosity solution to the associated fully nonlinear HJB equation in a lifted sense. Furthermore, as the time horizon goes to infinity, we show that the solutions of finite-horizon time-averaging optimal control problems converge to that of the ergodic control problem. Our results require dissipativity conditions and dissipativity-like conditions on distribution variables of both drift and diffusion coefficients.

On Weak Variational Control Inequalities via Interval Analysis

This paper deals with the connections between the interval-valued optimal control problem and the associated weak variational control inequality. More precisely, by considering the (strictly) LU-convexity and path independence properties of the involved curvilinear integral functionals, we establish a result on the existence of LU-optimal solutions for the interval-valued optimal control problem under study, and a result on the existence of solutions for the associated weak variational control inequality.

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.

Nonlinear optimal control for a gas compressor driven by an induction motor

The article proposes a nonlinear optimal control approach for the dynamic model of a gas centrifugal compressor being driven by an induction motor (IM). The dynamic model of the integrated compressor-induction motor system, being initially expressed in a nonlinear and multivariable state-space form, undergoes approximate linearization around a temporary operating point that is recomputed at each time-step of the control method. The linearization relies on first-order Taylor series expansion and on the computation of the associated Jacobian matrices. For the linearized state-space model of the compressor-IM a stabilizing optimal (H-infinity) feedback controller is designed. This controller stands for the solution to the nonlinear optimal control problem of the compressor-IM under model uncertainty and external perturbations. To compute the controller’s feedback gains an algebraic Riccati equation is repetitively solved at each iteration of the control algorithm. The global stability properties of the control method are proven through Lyapunov analysis. Finally, to implement state estimation-based control of the compressor-IM, without the need to measure its entire state vector, the H-infinity Kalman Filter is used as a robust state estimator. The differential flatness properties of the compressor-IM system are proven. The article’s method provides one of the few algorithmically simple and computationally efficient solutions for the nonlinear optimal control problem of the compressor-IM 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.

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.

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.