Solution Of Optimal Control Problem Research Articles

Optimal finite element error estimates for an optimal control problem governed by the wave equation with controls of bounded variation

Abstract This work discusses the finite element discretization of an optimal control problem for the linear wave equation with time-dependent controls of bounded variation. The main focus lies on the convergence analysis of the discretization method. The state equation is discretized by a space-time finite element method. The controls are not discretized. Under suitable assumptions optimal convergence rates for the error in the state and control variable are proven. Based on a conditional gradient method the solution of the semi-discretized optimal control problem is computed. The theoretical convergence rates are confirmed in a numerical example.

Solving optimal control problems using the Picard’s iteration method

In this paper, the Picard’s iteration method is proposed to obtain an approximate analytical solution for linear and nonlinear optimal control problems with quadratic objective functional. It consists in deriving the necessary optimality conditions using the minimum principle of Pontryagin, which result in a two-point-boundary-value-problem (TPBVP). By applying the Picard’s iteration method to the resulting TPBVP, the optimal control law and the optimal trajectory are obtained in the form of a truncated series. The efficiency of the proposed technique for handling optimal control problems is illustrated by four numerical examples, and comparison with other methods is made.

Determination of an optimal control strategy for vaccine administration in COVID-19 pandemic treatment

Background and objectiveFor decades, mathematical models have been used to predict the behavior of physical and biological systems, as well as to define strategies aiming at the minimization of the effects regarding different types of diseases. In the present days, the development of mathematical models to simulate the dynamic behavior of the novel coronavirus disease (COVID-19) is considered an important theme due to the quantity of infected people worldwide. In this work, the objective is to determine an optimal control strategy for vaccine administration in COVID-19 pandemic treatment considering real data from China. Two optimal control problems (mono- and multi-objective) to determine a strategy for vaccine administration in COVID-19 pandemic treatment are proposed. The first consists of minimizing the quantity of infected individuals during the treatment. The second considers minimizing together the quantity of infected individuals and the prescribed vaccine concentration during the treatment. MethodsAn inverse problem is formulated and solved in order to determine the parameters of the compartmental Susceptible-Infectious-Removed model. The solutions for both optimal control problems proposed are obtained by using Differential Evolution and Multi-objective Optimization Differential Evolution algorithms. ResultsA comparative analysis on the influence related to the inclusion of a control strategy in the population subject to the epidemic is carried out, in terms of the compartmental model and its control parameters. The results regarding the proposed optimal control problems provide information from which an optimal strategy for vaccine administration can be defined. ConclusionsThe solution of the optimal control problem can provide information about the effect of vaccination of a population in the face of an epidemic, as well as essential elements for decision making in the economic and governmental spheres.

Strong-order conditions of Runge-Kutta method for stochastic optimal control problems

In this work, we obtain strong-order conditions of the Runge-Kutta method for the optimal control of the stochastic differential equations. We follow the discretize-then-optimize approach in order to get the optimality system. We compare Stratonovich-Taylor expansions of the exact solution and approximation method of the stochastic optimal control problem defined by the Runge-Kutta scheme successively to get the strong-order conditions. We derive the strong order-1.5 conditions and verify by a numerical example.

Numerical solution for fractional optimal control problems by Hermite polynomials

In this study, a numerical method based on Hermite polynomial approximation for solving a class of fractional optimal control problems is presented. The order of the fractional derivative is taken as less than one and described in the Caputo sense. Operational matrices of integration by using such known formulas as Caputo and Riemann–Liouville operators for computing fractional derivatives and integration of polynomials is introduced and used to reduce the problem of a system of algebraic equations. The convergence of the proposed method is analyzed, and the error upper bound for the operational matrix of the fractional integration is obtained. To confirm the validity and accuracy of the proposed numerical method, three numerical examples are presented along with a comparison between our numerical results and those obtained using Legendre polynomials. Illustrative examples are included to demonstrate the validity and applicability of the new technique.

Robust Multi-Objective Singular Optimal Control Ofpenicillin Fermentation Process

The determination of optimal feeding profile of fed-batch fermentation requires the solution of a singular optimal control problem. The complexity in obtaining the solution to this singular problem is due to the nonlinear dynamics of the system model, the presence of control variables in linear form and the existence of constraints in both the state and control variables. Traditionally, during the optimization process, uncertainties associated with design variables, control parameters and mathematical model are not considered. In this contribution, a systematic methodology to evaluate uncertainties during the resolution of a singular optimal control problem is proposed. This approach consists of the Multi-objective Optimization Differential Evolution algorithm associated with Effective Mean Concept. The proposed methodology is applied to determine the feed substrate concentration in fed-batch penicillin fermentation process. The robust multi- objective singular optimal control problem consists of maximizing the productivity and minimizing the operation total time.

Trajectory Planning of Quadrotor Systems for Various Objective Functions

SUMMARYQuadrotors are unmanned aerial vehicles with many potential applications ranging from mapping to supporting rescue operations. A key feature required for the use of these vehicles under complex conditions is a technique to analytically solve the problem of trajectory planning. Hence, this paper presents a heuristic approach for optimal path planning that the optimization strategy is based on the indirect solution of the open-loop optimal control problem. Firstly, an adequate dynamic system modeling is considered with respect to a configuration of a commercial quadrotor helicopter. The model predicts the effect of the thrust and torques induced by the four propellers on the quadrotor motion. Quadcopter dynamics is described by differential equations that have been derived by using the Newton–Euler method. Then, a path planning algorithm is developed to find the optimal trajectories that meet various objective functions, such as fuel efficiency, and guarantee the flight stability and high-speed operation. Typically, the necessary condition of optimality for a constrained optimal control problem is formulated as a standard form of a two-point boundary-value problem using Pontryagin’s minimum principle. One advantage of the proposed method can solve a wide range of optimal maneuvers for arbitrary initial and final states relevant to every considered cost function. In order to verify the effectiveness of the presented algorithm, several simulation and experiment studies are carried out for finding the optimal path between two points with different objective functions by using MATLAB software. The results clearly show the effect of the proposed approach on the quadrotor systems.

Numerical Solutions for the Optimal Control Governing by Variable Coefficients Nonlinear Hyperbolic Boundary Value Problem Using the Gradient Projection, Gradient and Frank Wolfe Methods

This paper is concerned with studying the numerical solution for the discrete classical optimal control problem (NSDCOCP) governed by a variable coefficients nonlinear hyperbolic boundary value problem (VCNLHBVP). The DSCOCP is solved by using the Galerkin finite element method (GFEM) for the space variable and implicit finite difference scheme (GFEM-IFDS) for the time variable to get the NS for the discrete weak form (DWF) and for the discrete adjoint weak form (DSAWF) While, the gradient projection method (GRPM), also called the gradient method (GRM), or the Frank Wolfe method (FRM) are used to minimize the discrete cost function (DCF) to find the DSCOC. Within these three methods, the Armijo step option (ARMSO) or the optimal step option (OPSO) are used to improve the discrete classical control (DSCC). Finally, some illustrative examples for the problem are given to show the accuracy and efficiency of the methods.

Two‐grid methods for –P1 mixed finite element approximation of general elliptic optimal control problems with low regularity

AbstractIn this paper, we present a two‐grid mixed finite element scheme for distributed optimal control governed by general elliptic equations. –P1 mixed finite elements are used for the discretization of the state and co‐state variables, whereas piecewise constant function is used to approximate the control variable. We first use a new approach to obtain the superclose property between the centroid interpolation and the numerical solution of the optimal control u with order h2 under the low regularity. Based on the superclose property, we derive the optimal a priori error estimates. Then, using a postprocessing projection operator, we get a second‐order superconvergent result for the control u. Next, we construct a two‐grid mixed finite element scheme and analyze a priori error estimates. In the two‐grid scheme, the solution of the elliptic optimal control problem on a fine grid is reduced to the solution of the elliptic optimal control problem on a much coarser grid and the solution of a linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy. Finally, a numerical example is presented to verify the theoretical results.

Biorthogonal multiwavelets on the interval for solving multidimensional fractional optimal control problems with inequality constraint

SummaryThis article proposes a new numerical approach for solving fractional optimal control problems including state and control inequality constraints using new biorthogonal multiwavelets. The properties of biorthogonal multiwavelets are first given. The Riemann‐Liouville fractional integral operator for biorthogonal multiwavelets is utilized to reduce the solution of optimal control problems to a nonlinear programming one, to which existing, well‐developed algorithms may be applied. In order to save the memory requirement and computation time, a threshold procedure is applied to obtain algebraic equations. The method is computationally very attractive and gives very accurate results.

Solving fractional optimal control problems by new Bernoulli wavelets operational matrices

SummaryIn this article, a new numerical method based on Bernoulli wavelet basis has been applied to give the approximate solution of the fractional optimal control problems. The new operational matrices of multiplication and fractional integration are constructed. The proposed method is applied to reduce the problem to the solution of a system of algebraic equations. The fractional derivative is considered in the Caputo sense. Convergence of the algorithm is proved and some results concerning the error analysis are obtained. Approximate solutions are given and in the cases when we have an exact solution, a comparison with the exact solution is presented to demonstrate the validity and applicability of the proposed method. In addition, we compare the obtained results with the results of other methods. Comparison shows the more accuracy of presented technique in comparison to other published methods.

Numerical Solution of Fractional Optimal Control Problems with Inequality Constraint Using the Fractional-Order Bernoulli Wavelet Functions

This paper studies the fractional optimal control problems (FOCPs) with inequality constraints. Using the Caputo definition, an optimization method based on a set of basis functions, namely the fractional-order Bernoulli wavelet functions (F-BWFs), is proposed. The solution is expanded in terms of the F-BWFs with unknown coefficients. In the first step, we convert the inequality conditions to equality conditions. In the second step, we use the operational matrix (OM) of fractional integration and the product OM of F-BWFs, with the help of the Lagrange multipliers technique for converting the FOCPs into an easier one, described by a system of nonlinear algebraic equations. Finally, for illustrating the efficiency and accuracy of the proposed technique, several numerical examples are analysed and the results compared with the analytical or the approximate solutions obtained by other techniques.

Convergence analysis of numerical solutions for optimal control of variational–hemivariational inequalities

Variational–hemivariational inequalities and hemivariational inequalities form a powerful mathematical tool in modeling and studying problems in science and engineering where non-smooth, non-monotone and multi-valued relations among different physical quantities are present. In this paper, we consider the numerical solution of optimal control problems for variational–hemivariational inequalities or hemivariational inequalities, and prove the convergence of numerical solutions under rather general assumptions.

Optimal control of discrete-time switched linear systems

This paper considers the optimal control problems of switched autonomous linear systems and switched non-autonomous linear systems in discrete-time, where the objective is that of minimizing a linear quadratic performance index function or multiple linear quadratic performance index function. The subsystems switching sequence and the number of switchings are free in this paper. By employing an embedding technique, we embed the origin switched autonomous or non-autonomous linear systems into a larger family of systems. Owing to the embedding technique, the optimal control problem of the origin switched linear systems can be relaxed into a traditional optimal control problem for the larger family systems. We thus derive the optimal switching control law and the optimal control input for the bang-bang type optimal solution of the relaxed optimal control problem simultaneously. The optimal solution of the origin switched optimal control problem can be obtained by the bang-bang type optimal solution of the relaxed optimal control problem. Finally, results of numerical simulation examples are given to demonstrate the remarkable effectiveness and superiority of the developed method.

Numerical solution of bilateral obstacle optimal control problem, where the controls and the obstacles coincide

This work is deals with the numerical solution of a bilateral obstacle optimal control problem which is similar to the one given in Bergounioux et al [9] with some modifications. It can be regarded as an extension of our previous work [18], where the main feature of the present work is that the controls and the two obstacles are the same. For the numerical resolution we follow the idea of our previous work [18]. We begin by discretizing the optimality system of the underlying problem by using finite differences schemes, then we propose an iterative algorithm. Finally, numerical examples are provides to show the efficiency of the proposed algorithm and the used scheme.

Bang-Bang and Multiple Valued Optimal Solutions of Control Problems Related to Quasi-linear Elliptic Equations

This paper investigates two optimal control problems in which the state equation is a quasi-linear elliptic partial differential equation. In the first problem the admissible set comprises functions with range in the interval [0,1] with prescribed integral. We show the optimal solution is unique and it is of bang-bang type. In the second problem the admissible set is a particular class of simple functions. We will show again the optimal solution is unique and derive the corresponding optimality condition.

Encrypted MPC based on ADMM real-time iterations

Encrypted control enables confidential controller evaluations in cloud-based or networked control systems. Technically, an encrypted controller is a modified control algorithm that is capable of computing encrypted control actions based on encrypted system states without intermediate decryption. The realization of such controllers, e.g., using homomorphic encryption, is non-trivial. Nevertheless, even optimization-based model predictive control (MPC) has already been implemented in an encrypted fashion. However, the existing schemes either require an explicit solution of the parametric optimal control problem (OCP) or they can only consider input constraints. In this paper, we present a novel encrypted MPC that allows to include state and input constraints without the requirement of an explicit solution of the OCP. The approach builds on the encrypted implementation of a single iteration of the alternating direction method of multipliers (ADMM) per time step, i.e., ADMM real-time iterations.

Approximate Solution of Quadratic Time Varying Optimal Control Problems via Differential Transform Method

Because there is no general analytical method for solving the quadratic time varying optimal control problems (QTVOCPs), many authors adopt the numerical methods for this purpose. This motivates some authors to use the differential transform method (DTM) for finding an approximate solution for the linear quadratic optimal control problems (LQOCPs) and a class of nonlinear quadratic optimal control. In this paper, we propose an algorithm depend on applying DTM about the two end points of the time horizon to find an approximate solution for the linear or nonlinear QTVOCPs. To show the reliability, accuracy and efficiency of suggested method, some illustrated examples have been provided.

Theoretical and numerical results for some bi-objective optimal control problems

This article deals with the solution of some multi-objective optimal control problems for several PDEs: linear and semilinear elliptic equations and stationary Navier-Stokes systems. More precisely, we look for Pareto equilibria associated to standard cost functionals. First, we study the linear and semilinear cases. We prove the existence of equilibria, we deduce appropriate optimality systems, we present some iterative algorithms and we establish convergence results. Then, we analyze the existence and characterization of Pareto equilibria for the Navier-Stokes equations. Here, we use the formalism of Dubovitskii and Milyutin. In this framework, we also present a finite element approximation of the bi-objective problem and we illustrate the techniques with several numerical experiments.

Autonomous Intersection Management over Continuous Space: A Microscopic and Precise Solution via Computational Optimal Control

Autonomous intersection management (AIM) refers to planning cooperative trajectories for multiple connected and automated vehicles (CAVs) when they pass through an unsignalized intersection. In modeling a generic AIM scheme, the predominant network-level or lane-level methods limit the cooperation potentiality of a multi-CAV team because 1) lane changes are forbidden or only allowed at discrete spots in the intersection, 2) each CAVs travel path is fixed or selected among a few topological choices, and 3) each CAVs travel velocity is fixed or set to a specified pattern. To overcome these limitations, this work models the intersection as a continuous free space and describes an AIM scheme as a multi-CAV trajectory optimization problem. Concretely, a centralized optimal control problem (OCP) is formulated and then numerically solved. To derive a satisfactory initial guess for the numerical optimization, a priority-based decentralized framework is proposed, wherein an x-y-time A* algorithm is adopted to generate a coarse trajectory for each CAV. To facilitate the OCP solution process, 1) the collision-avoidance constraints in the OCP are convexified, and 2) a stepwise computation strategy is adopted. Simulation results show the efficacy of the proposed offline AIM method.