Solution Of Optimal Control Problem Research Articles

Numerical solution of fractional variational and optimal control problems via fractional-order Chelyshkov functions

<abstract><p>In this paper, we present a new numerical method based on the fractional-order Chelyshkov functions (FCHFs) for solving fractional variational problems (FVPs) and fractional optimal control problems (FOCPs). The fractional derivatives are considered in the Caputo sense. The operational matrix of fractional integral for FCHFs, together with the Lagrange multiplier method, are used to reduce the fractional optimization problem into a system of algebraic equations. Some results concerning the approximation errors are discussed and the convergence of the presented method is also demonstrated. The performance of the introduced method is tested through several examples. Some comparisons with recent numerical methods are introduced to show the accuracy and effectiveness of the presented method.</p></abstract>

Numerical solutions for fractional optimal control problems by using generalised fractional-order Chebyshev wavelets

This paper proposes a new numerical method for solving fractional optimal control problems (FOCPs). The method is based on generalised fractional-order Chebyshev wavelets (GFOCW). The exact value of the Riemann–Liouville fractional integral operator of the GFOCW is given by applying the incomplete beta function. By using the properties of GFOCW and the collocation method, the FOCP is reduced to a parameter optimisation problem. The last problem is solved by known algorithms. Six numerical examples are given. One of them is an application example in a cancer model. Through these numerical examples, we will show that for some cases of our examples, we will get the exact solutions. These solutions were not obtained previously in the literature. In addition, our method gives more accurate results in comparison with the existing methods.

Time recursive control of stochastic dynamical systems using forward dynamics and applications

The solution of a stochastic optimal control problem may be associated with that of the Hamilton–Jacobi–Bellman (HJB) equation, which is a second order partial differential equation subject to a terminal condition. When this equation is semilinear and satisfies certain other constraints, it can be solved via a nonlinear version of the Feynman–Kac formula. According to this approach, the solution to the HJB equation can be obtained by simulating an associated pair of partly coupled forward–backward stochastic differential equations. Although an elegant way to interpret and solve a partial differential equation, simulating the system of forward–backward equations can be computationally inefficient. In this work, the HJB equation pertaining to the optimal control problem is reformulated such that instead of the given terminal condition, it is now subject to an appropriate initial condition. In the process, while the total cost associated with the control problem remains unchanged, pathwise solutions may not. Associated with the new partial differential equation, we then derive a set of stochastic differential equations whose solutions move only forward in time. This approach has a significant computational advantage over the original formulation. Moreover, since the forward–backward approach generally requires simulating stochastic differential equations over the current to the terminal time at every step, the integration errors may accumulate and carry forward in estimating the control. This error is particularly high initially since the time of integration is the longest there. The proposed method, numerically implemented for the control of stochastically excited oscillators, is free of such errors and hence more robust. Unsurprisingly, it also exhibits lower sampling variance.

Fast iterative solver for the optimal control of time‐dependent PDEs with Crank–Nicolson discretization in time

AbstractIn this article, we derive a new, fast, and robust preconditioned iterative solution strategy for the all‐at‐once solution of optimal control problems with time‐dependent PDEs as constraints, including the heat equation and the non‐steady convection–diffusion equation. After applying an optimize‐then‐discretize approach, one is faced with continuous first‐order optimality conditions consisting of a coupled system of PDEs. As opposed to most work in preconditioning the resulting discretized systems, where a (first‐order accurate) backward Euler method is used for the discretization of the time derivative, we employ a (second‐order accurate) Crank–Nicolson method in time. We apply a carefully tailored invertible transformation for symmetrizing the matrix, and then derive an optimal preconditioner for the saddle‐point system obtained. The key components of this preconditioner are an accurate mass matrix approximation, a good approximation of the Schur complement, and an appropriate multigrid process to apply this latter approximation—these are constructed using our work in transforming the matrix system. We prove the optimality of the approximation of the Schur complement through bounds on the eigenvalues, and test our solver against a widely‐used preconditioner for the linear system arising from a backward Euler discretization. These demonstrate the effectiveness and robustness of our solver with respect to mesh‐sizes, regularization parameter, and diffusion coefficient.

A numerical approach based on fractional-order hybrid functions of block-pulse and Bernoulli polynomials for numerical solutions of fractional optimal control problems

We present, an accurate and efficient computational method based on the fractional-order hybrid of block-pulse functions and Bernoulli polynomials for solving fractional optimal control problems. The Riemann–Liouville fractional integral operator for the fractional-order hybrid of block-pulse functions and Bernoulli polynomials is constructed. The original problem is transformed to a system of algebraic equations which can be solved easily. The method is very accurate and is computationally very attractive. Examples are included to provide the capacity of the proposal method.

Multilevel Techniques for the Solution of HJB Minimum-Time Control Problems

The solution of minimum-time feedback optimal control problems is generally achieved using the dynamic programming approach, in which the value function must be computed on numerical grids with a very large number of points. Classical numerical strategies, such as value iteration (VI) or policy iteration (PI) methods, become very inefficient if the number of grid points is large. This is a strong limitation to their use in real-world applications. To address this problem, the authors present a novel multilevel framework, where classical VI and PI are embedded in a full-approximation storage (FAS) scheme. In fact, the authors will show that VI and PI have excellent smoothing properties, a fact that makes them very suitable for use in multilevel frameworks. Moreover, a new smoother is developed by accelerating VI using Anderson’s extrapolation technique. The effectiveness of our new scheme is demonstrated by several numerical experiments.

Research on a collocation approach and three metaheuristic techniques based on MVO, MFO, and WOA for optimal control of fractional differential equation

Exploiting a comprehensive mathematical model for a class of systems governed by fractional optimal control problems is the significant focal point of the current paper. The efficiency index is a function of both control and state variables and the dynamic control system relies on Caputo fractional derivatives. The attributes of Bernoulli polynomials and their operational matrices of fractional Riemann–Liouville integrations are applied to convert the optimization problem to the nonlinear programing problem. Executing multi-verse optimizer, moth-flame optimization, and whale optimization algorithm terminate to the most excellent solution of fractional optimal control problems. A study on the advantage and performance between these approaches is analyzed by some examples. Comprehensive analysis ascertains that moth-flame optimization significantly solves the example. Furthermore, the privilege and advantage of preference with its accuracy are numerically indicated. Finally, results demonstrate that the objective function value gained by moth-flame optimization in comparison with other algorithms effectively decreased.

A new high accurate approximate approach to solve optimal control problems of fractional order via efficient basis functions

Investigation of spectral (collocation or Galerkin) methods for the solution approximation of different classes of optimal control problems have had been increased in recent years. In this research work, we proposed and analyzed an applicable Legendre spectral collocation approach, based on an efficient space of fractional basis functions, to compute numerically the solution of fractional optimal control problems. We first introduce a fractional Lagrange interpolation and then approximate the unknown control and state by this new set of fractional interpolation functions. The problem is discretized in terms of shifted Legendre–Gauss collocation points and two approaches are suggested to calculate the exact fractional differentiation matrix. Also, we prove that the approximate solutions of the obtained discretized problem converge to the optimal solution of the main problem by assuming some mild conditions. The method is implemented for three numerical test problems to show the claimed efficiency and capability.

An adaptive wavelet collocation method for the optimal heat source problem

PurposeThis paper aims to propose a new adaptive numerical method to find more accurate numerical solution for the heat source optimal control problem (OCP).Design/methodology/approachThe main aim of this paper is to present an adaptive collocation approach based on the interpolating wavelets to solve an OCP for finding optimal heat source, in a two-dimensional domain. This problem arises when the domain is heated by microwaves or by electromagnetic induction.FindingsThis paper shows that combination of interpolating wavelet basis and finite difference method makes an accurate structure to design adaptive algorithm for such problems which usually have non-smooth solution.Originality/valueThe proposed numerical technique is flexible for different OCP governed by a partial differential equation with box constraint over the control or the state function.

Optimal installation of renewable electricity sources: the case of Italy

Starting from the model in Koch-Vargiolu (2019), we test the real impact of current renewable installed power in the electricity price in Italy, and assess how much the renewable installation strategy which was put in place in Italy deviated from the optimal one obtained from the model in the period 2012--2018. To do so, we consider the Ornstein-Uhlenbeck (O-U) process, including an exogenous increasing process influencing the mean reverting term, which is interpreted as the current renewable installed power. Using real data of electricity price, photovoltaic and wind energy production from the six main Italian price zones, we estimate the parameters of the model and obtain quantitative results, such as the production of photovoltaic energy impacts the North zone, while wind is significant for Sardinia and the Central North zone does not present electricity price impact. Then we implement the solution of the singular optimal control problem of installing renewable power production devices, in order to maximize the profit of selling the produced energy in the market net of installation costs. We extend the results of \cite{KV} to the case when no impact on power price is presented, and to the case when $N$ players can produce electricity by installing renewable power plants. We are thus able to describe the optimal strategy and compare it with the real installation strategy that was put in place in Italy.

Boundary optimal control of time-space SIR model with nonlinear Robin boundary condition.

A boundary optimal control problem arising in time-space SIR epidemic models is treated. In this work we aim with the control of the flux of infected individuals crossing part of boundary. On the other side of the domain, we suppose a nonlinear boundary condition of third kind: nonlinear Robin boundary condition, this condition models immersing individual crossing this part of the boundary of the domain of study. We give the existence and uniqueness of the solution of both state and optimal control problem ending some numerical tests throughout a simple example.

Numerical solution of time-delay optimal control problems by the operational matrix based on Hartley series

In this paper, a numerical technique based on the Hartley series for solving a class of time-delayed optimal control problems (TDOCPs) is introduced. The main idea is converting such TDOCPs into a system of algebraic equations. Thus, we first expand the state and control variables in terms of the Hartley series with undetermined coefficients. The delay terms in the problem under consideration are expanded in terms of the Hartley series. Applying the operational matrices of the Hartley series including integration, differentiation, dual, product, delay, and substituting the estimated functions into the cost function, the given TDOCP is reduced to a system of algebraic equations to be solved. The convergence of the proposed method is extensively investigated. At last, the precision and applicability of the proposed method is studied through different types of numerical examples.

SENTIMENT ANALYSIS OF KASPI PRODUCT REVIEWS

This article discusses the study of problems of optimal control for electric power systems. The numerical solution of optimal control problems for complex electric power systems using an iterative algorithm is shown. Also considered are issues of solving the optimal control of a nonlinear system of ordinary differential equations in two different cases. The proposed solution methods follow the principle of continuation of extremal problems based on sufficient conditions for optimality of V. F. Krotov. A special case of optimal control problems is considered. Numerical experiments showed sufficient efficiency of the implemented algorithms. The problem of optimal motion control of a two-system electric power system is graphically illustrated in the proposed numerical example.

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

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, we consider the construction of a feedback according to the Kalman algorithm. For this, the solution of an auxiliary optimal control problem with a quadratic functional is used by analogy with the SDRE approach.Since 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, then on a finite time interval it becomes necessary to solve a complicated matrix differential Riccati equations, with state-dependent coefficient matrices. This circumstance, due to the nonlinearity of the system, in comparison with the Kalman algorithm for linear-quadratic problems, 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. The proposed synthesis construction algorithm is constructed using the extension principle proposed by V. F. Krotov and developed by V. I. Gurman and allows 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. The article establishes the correctness of the application of the extension principle by introducing analogs of the Lagrange multipliers, depending 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.

Discussion on Optimality Conditions of Fuzzy Optimal Control Problems

In this study, a novel methodology for detecting fuzzy optimal solutions of fuzzy optimal control problems governed by fuzzy differential equations is presented. For this purpose, first the concept of optimal solution is proposed by considering a partial ordering on fuzzy numbers space which is based on a new parametric representation for α-level set of fuzzy numbers. Using this partial ordering and parametric representation, the discussed fuzzy optimal control problems are reduced to general optimal control problems in parametric form. Then by Pontryagin’s principle, a candidate for the solution of the original problem is derived. The procedure of extracting candidate solution is illustrated for two experimental problems, which improve the recent results of Refs. 10 and 17.

Application of periodic Bernoulli polynomials in approximating the solution of linear time-delay optimal control problems

In this paper, a numerical method based on application of periodic Bernoulli polynomials for solving optimal control of linear time-delay systems with quadratic performance index is presented. Some important properties of Bernoulli polynomials are reviewed and a new operational matrix of delay based on these functions is introduced. Also an error bound for approximating a function with periodic Bernoulli polynomials is presented. Using operational matrices of differentiation, delay and the integration of the cross product of two Bernoulli polynomials, the linear time delay optimal control problem with quadratic objective function is converted to a nonlinear optimization problem. Finally, numerous illustrative examples are studied to show the efficiency and accuracy of the proposed method compared with some other methods.

New class of hybrid explicit methods for numerical solution of optimal control problems

Forward-backward sweep method (FBSM) is an indirect numerical method used for solving optimal control problems, in which the differential equation arising from this method is solved by the Pontryagin’s maximum principle. In this paper, a set of hybrid methods based on explicit 6th-order RungeKutta method is presented for the FBSM solution of optimal control problems. Order of truncation error, stability region, and numerical results of the new hybrid methods were compared with those of the 6th-order Runge Kutta method. Numerical results show that new hybrid methods are more accurate than the 6th-order Runge–Kutta method and that their stability regions are also wider than that of the 6th-order Runge–Kutta method.

Solving quantum optimal control problems by wavelets method

We present the quantum equation and synthesize an optimal control proce dure for this equation. We develop a theoretical method for the analysis of quantum optimal control system given by the time depending Schrodinger equation. The Legendre wavelet method is proposed for solving this problem. This can be used as an efficient and accurate computational method for obtaining numerical solutions of different quantum optimal control problems. The distinguishing feature of this paper is that it makes the method, previously used to solve non-quantum control equations based on Legendre wavelets, usable by using a change of variables for quantum control equations.

Optimization problem of attitude control of a spacecraft with bounded rotary energy using quaternions

The original control problem of optimal reorientation from a state of rest to a state of rest is considered and solved. The control function is torque vector. Problem of optimal control is investigated in detail for statement when control is restricted and the used functional of optimality includes kinematical rotary energy and time of maneuver. For solving and synthesis of the optimal control program, the quaternion method and the Pontryagin’s maximum principle are applied. Analytic solution of the proposed problem is presented basing on the differential equation connecting the angular velocity vector and quaternion of spacecraft attitude. It is shown that a chosen criterion of optimality provides a turn of a spacecraft with rotation energy which does not exceed the required value. This property of proposed control increases safety of flight. The time-optimal problem was solved also. The control law is formulated in the form of an explicit dependence of the control variables on the phase coordinates. All key expressions and equations are written in quaternion form which is convenient for onboard realization and implementation. The analysis of the special control regime of the spacecraft was made. Analytical formulas were written for duration of acceleration and braking. For specific cases of spacecraft configurations (dynamically symmetric spacecraft and spheric-symmetrical spacecraft as particular cases), complete solution of optimal control problem in closed form is given. Numerical example and the results of mathematical simulation for spacecraft motion under optimal control are demonstrated. This data supplements the made theoretical descriptions, and illustrates the practical feasibility of the designed algorithm for controlling the spatial orientation of the spacecraft showing reorientation process in visual form.

On the estimation of optimal state-of-charge trajectory for plug-in hybrid electric buses using trip information

Plug-in hybrid electric buses are a viable solution to increase the fuel economy. In this framework, precise estimation of optimal state-of-charge trajectory along the upcoming driving cycle appears to play a pivotal role in the way to approach the globally optimal fuel economy. This paper aims to conduct a parametric study on the key factors affecting the estimation of optimal state-of-charge trajectory, including trip information availability and trip segment distance, and to provide a guideline for the design and implementation of predictive energy management systems. To accomplish this, the dynamic programming algorithm is employed to obtain the solution of optimal control problem for the sampled driving cycles in a particular bus route. A large database comprising of driving features of the cycles and the optimal solution is developed which then is used to construct a neural network based estimator for obtaining the optimal state-of-charge trajectory. The main results show promising performance of the proposed method with about 76% reduction in the root mean square error of the estimated trajectory comparing to the linear state-of-charge trajectory assumption. Moreover, the robustness of the estimator is verified through simulation and it is observed that appropriate choice of trip segment distance is vital to improve the estimation accuracy, especially in case of uncertain prediction of trip information.