Fractional Control Problems Research Articles

Cauchy discretization method for solve fractional linear optimal control problems

This paper presents a numerical scheme designed for solving fractional linear optimal control problems (FLOCPs) governed by Caputo fractional derivatives (CFDs) of order 0<α≤1. The method transforms the original fractional control problem into an equivalent linear programming problem (LPP), enabling a more straightforward solution process. To evaluate the performance and accuracy of this approach, several numerical experiments were conducted using MATLAB. These experiments highlight the method’s computational efficiency and its simplicity in terms of implementation. The proposed approach offers accurate results, particularly in scenarios where traditional methods may fall short due to the complexities introduced by fractional derivatives. A detailed numerical example is presented to further illustrate the method’s effectiveness in addressing FLOCPs. The outcomes suggest that this approach is not only practical but also provides valuable insights for researchers dealing with fractional calculus in optimal control theory. This contribution is expected to be beneficial for applications in engineering and the physical sciences, where fractional-order systems play a significant role.

Solving convex uncertain PDE-constrained multi-dimensional fractional control problems via a new approach

Solving convex uncertain PDE-constrained multi-dimensional fractional control problems via a new approach

On the Optimality Condition for Optimal Control of Caputo Fractional Differential Equations with State Constraints

We consider the fractional optimal control problem with state constraints. The fractional calculus of derivatives and integrals can be viewed as generalizations of their classical ones to any arbitrary real order. In our problem setup, the dynamic constraint is captured by the Caputo fractional differential equation with order α ∈ (0, 1), and the objective functional is formulated by the left Riemann-Liouville fractional integral with order β ≥ 1. In addition, there are terminal and running state constraints; while the former is described by initial and final states within a convex set, the latter is given by an explicit instantaneous inequality state constraint. We obtain the maximum principle, the first-order necessary optimality condition, for the problem of this paper. Due to the inherent complex nature of the fractional control problem, the presence of the terminal and running state constraints, and the generalized standing assumptions, the maximum principle of this paper is new in the optimal control problem context, and its proof requires to develop new variational and duality analysis using fractional calculus and functional analysis, together with the Ekeland variational principle and the spike variation.

Analysis of nonlinear fractional optimal control systems described by delay Volterra–Fredholm integral equations via a new spectral collocation method

We aim to introduce a new spectral collocation method for investigating and analyzing nonlinear delay control systems governed by the fractional mixed Volterra–Fredholm integral equations (MVFIEs). The generalized fractional Legendre basis (GFLB) is used as a complete orthogonal basis, and the fractional Legendre–Gaussian nodes are introduced and employed as the fractional collocation points. These nodes correspond to the zeros of the fractional-order Legendre function of degree M. The convergence of the generalized orthogonal basis is discussed in detail based on the Sobolev and L2 norms. Additionally, new fractional integral and delay operators are implemented to reduce the primary control system to a static optimization system. Two benchmark fractional control problems, including delay, are considered to demonstrate the powerfulness and superiority of the new methodology. The introduced collocation approach can be successfully performed for solving even those fractional control problems having any irregularities in the control function, including jump discontinuities and bang–bang behavior.

Optimal control of nonlinear fractional order delay systems governed by Fredholm integral equations based on a new fractional derivative operator

We aim to develop a direct transcription approach for solving a notable category of optimal control problems governed by nonlinear fractional Fredholm integral equations having delays in both input and output signals. The foundation of the new methodology is based on a multi domains decomposition scheme by utilizing the fractional-order Legendre functions. A new fractional derivative operator associated with the fractional basis is introduced by using the Caputo fractional derivative operator. With the use of derivative and delay operators, one can transform the dynamical system related to the fractional control problem into a new system containing algebraic equations. A wide variety of challenging test problems are studied to provide a detailed explanation of the designed approach.

A generalization of Müntz-Legendre polynomials and its implementation in optimal control of nonlinear fractional delay systems

A hybrid of Müntz-Legendre polynomials (MLPs) and block-pulse functions (BPFs) is defined and carried out to analyze nonlinear fractional optimal control problems consisting of multiple delays. Instead of using the Caputo fractional derivative, an alternative fractional derivative operator is utilized. The primary optimization problem reduces to an alternative optimization one involving unknown parameters. For this purpose, the fractional Legendre-Gauss quadrature formula is utilized for approximating the associated cost functional and the fractional Legendre-Gaussian nodes are taken as the collocation points. Some new aspects of the proposed basis are demonstrated to verify the effectiveness of the proposed approach based on the Müntz-Legendre basis (MLB) against the classical orthogonal bases. The simulation results certify the feasibility and reliability of the proposed method. Numerous fractional control problems, e.g., bang-bang controls and control systems with any irregularities in the structure of control input can be handled successfully by employing the new fractional basis.

On necessary optimality conditions and exact penalization for a constrained fractional optimal control problem

AbstractThe purpose of this article is twofold. We first develop the first‐order necessary optimality conditions for a general constrained fractional optimal control problem using calculus of variation. These conditions are of the form of fractional ordinary differential equations which reduce to the conventional Euler–Lagrange equations when the dynamical system becomes a first‐order one. Then, we prove, via the optimality conditions established, that a popular penalty method used for conventional constrained optimal control and optimization is an “exact” penalty method for the fractional control problem by showing that an optimal solution to the penalized problem satisfies the optimality conditions of the original one when the penalty parameter is sufficiently large. An example is also provided to verify the necessary optimality conditions.

Harvest Management Problem with a Fractional Logistic Equation

In this article, we study a fractional control problem that models the maximization of the profit obtained by exploiting a certain resource whose dynamics are governed by the fractional logistic equation. Due to the singularity of this problem, we develop different resolution techniques, both for the classical case and for the fractional case. We perform several numerical simulations to make a comparison between both cases.

Approximate controllability results for Sobolev‐type delay differential system of fractional order without uniqueness

AbstractThe main aim of this article is to focus on approximate controllability for Sobolev‐type fractional control problems in Hilbert spaces without uniqueness. By using the fixed point theorem for multivalued maps with nonconvex values, the main results of our article are proved. Moreover, we obtain some results on the continuity of the solution map and the topological structure of the solution set of the considered Sobolev‐type fractional differential system. Finally, we provide a theoretical application to assist in the effectiveness of our discussion.

A Novel Proof on the Existence of the Solution of Fractional Control Problem Governed by Burgers Equations

A Novel Proof on the Existence of the Solution of Fractional Control Problem Governed by Burgers Equations

Duality with (ρ, b) – quasiinvexity for multidimensional vector fractional control problems

In this paper, by using the recently introduced concept of invexity associated with multiple integral vector functionals, we introduce several results of duality for a class of multiobjective fractional control problems involving multiple integrals. Under (ρ, b) – quasiinvexity assumptions, we formulate and prove weak, strong and converse duality results.

Diffusive representation of a fractional control using adaptive partitioning algorithm

This article presents optimal fractional control. This control is based on the property of the invariance of a fractional order differential equation. The problem formulation of the used control is expressed by diffusivere presentation. The fractional control problem is described in a minimization form, where the global optimum represents the diffusive realization of the controller. To determine the optimal fractional diffusive control, an adaptive partitioning algorithm is used. As an application, we have chosen the control of a DC motor with uncertain parameters

A Class of Nonlocal Fractional Evolution Equations and Optimal Controls

In this paper we study the existence of solutions for a class of semilinear fractional differential equations with nonlocal conditions and involving abstract Volterra operators. The existence of an optimal solution for a class of fractional control problem involving Caputo fractional derivatives is obtained. An example is presented to illustrate our main result.

Modified Adomian decomposition method for solving fractional optimal control problems

In this study, we use the modified Adomian decomposition method to solve a class of fractional optimal control problems. The performance index of a fractional optimal control problem is considered as a function of both the state and the control variables, and the dynamical system is expressed in terms of a Caputo type fractional derivative. Some properties of fractional derivatives and integrals are used to obtain Euler–Lagrange equations for a linear tracking fractional control problem and then, the modified Adomian decomposition method is used to solve the resulting fractional differential equations. This technique rapidly provides convergent successive approximations of the exact solution to a linear tracking fractional optimal control problem. We compare the proposed technique with some numerical methods to demonstrate the accuracy and efficiency of the modified Adomian decomposition method by examining several illustrative test problems.

A Fast Gradient Projection Method for a Constrained Fractional Optimal Control

Optimal control problems governed by a fractional diffusion equation tends to provide a better description than one by a classical second-order Fickian diffusion equation in the context of transport or conduction processes in heterogeneous media. However, the fractional control problem introduces significantly increased computational complexity and storage requirement than the corresponding classical control problem, due to the nonlocal nature of fractional differential operators. We develop a fast gradient projection method for a pointwise constrained optimal control problem governed by a time-dependent space-fractional diffusion equation, which requires the computational cost from $$O(M N^3)$$O(MN3) of a conventional solver to $$O(M N\log N)$$O(MNlogN) and memory requirement from $$O(N^2)$$O(N2) to O(N) for a problem of size N and of M time steps. Numerical experiments show the utility of the method.

A Formulation and Numerical Scheme for Fractional Optimal Control Problems

This article presents a general formulation and general numerical scheme for a class of fractional optimal control problems (FOCPs). The fractional derivative is described in the Caputo sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of fractional differential equations. The calculus of variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler-Lagrange equations for the FOCP. The formulation presented, and the resulting equations, are very similar to those that appear in classical optimal control theory. Thus, the present formulation essentially extends classical control theory to fractional dynamic systems. The formulation is used to derive the control equations for a quadratic linear fractional control problem. An iterative numerical scheme for finding the approximate numerical solution of the resulting equations is presented. For a linear system, this method results in a set of linear simultaneous equations, which can be solved directly. Numerical results for a FOCP are presented to demonstrate the feasibility of the method. It is shown that the solutions converge as the number of grid points increases, and the solutions approach the classical solutions as the order of the fractional derivatives approaches 1. The formulation presented is simple, and can be extended to other FOCPs.

A Quadratic Numerical Scheme for Fractional Optimal Control Problems

This paper presents a quadratic numerical scheme for a class of fractional optimal control problems (FOCPs). The fractional derivative is described in the Caputo sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of fractional differential equations. The calculus of variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler–Lagrange equations for the FOCP. The formulation presented and the resulting equations are very similar to those that appear in the classical optimal control theory. Thus, the present formulation essentially extends the classical control theory to fractional dynamic systems. The formulation is used to derive the control equations for a quadratic linear fractional control problem. For a linear system, this method results into a set of linear simultaneous equations, which can be solved using a direct or an iterative scheme. Numerical results for a FOCP are presented to demonstrate the feasibility of the method. It is shown that the solutions converge as the number of grid points increases, and the solutions approach to classical solutions as the order of the fractional derivatives approach to 1. The formulation presented is simple and can be extended to other FOCPs.

A FORMULATION AND A NUMERICAL SCHEME FOR FRACTIONAL OPTIMAL CONTROL PROBLEMS

This paper presents a general formulation and a general numerical scheme for a class of Fractional Optimal Control Problems (FOCPs). The fractional derivative is described in the Caputo sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of Fractional Differential Equations (FDEs). The Calculus of Variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler-Lagrange equations for the FOCP. The formulation presented and the resulting equations are very similar to those that appear in the classical optimal control theory. Thus, the present formulation essentially extends the classical control theory to fractional dynamic systems. The formulation is used to derive the control equations for a quadratic linear fractional control problem. An iterative numerical scheme is presented to find the approximate numerical solution of the resulting equations. For a linear system, this method results into a set of linear simultaneous equations, which can be solved directly. Numerical results for a FOCP are presented to demonstrate the feasibility of the method. It is shown that the solutions converge as the number of grid points increases, and the solutions approach to classical solutions as the order of the fractional derivatives approach to 1. The formulation presented is simple and can be extended to other FOCPs.

A General Formulation and Solution Scheme for Fractional Optimal Control Problems

Accurate modeling of many dynamic systems leads to a set of Fractional Differential Equations (FDEs). This paper presents a general formulation and a solution scheme for a class of Fractional Optimal Control Problems (FOCPs) for those systems. The fractional derivative is described in the Riemann–Liouville sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of FDEs. The Calculus of Variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler–Lagrange equations for the FOCP. The formulation presented and the resulting equations are very similar to those that appear in the classical optimal control theory. Thus, the present formulation essentially extends the classical control theory to fractional dynamic system. The formulation is used to derive the control equations for a quadratic linear fractional control problem. An approach similar to a variational virtual work coupled with the Lagrange multiplier technique is presented to find the approximate numerical solution of the resulting equations. Numerical solutions for two fractional systems, a time-invariant and a time-varying, are presented to demonstrate the feasibility of the method. It is shown that (1) the solutions converge as the number of approximating terms increase, and (2) the solutions approach to classical solutions as the order of the fractional derivatives approach to 1. The formulation presented is simple and can be extended to other FOCPs. It is hoped that the simplicity of this formulation will initiate a new interest in the area of optimal control of fractional systems.

Duality for multiobjective fractional generalized control problems with (F,ρ)(F,ρ)-convexity

The concept of efficiency is used to formulate duality for multiobjective fractional control problems. Wolfe and Mond–Weier-type duals are formulated. Using the generalized (F,ρ)-convexity on the functions involved, we establish the weak and strong duality theorems for multiobjective fractional generalized control problems.