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A survey on fuzzy fractional differential and optimal control nonlocal evolution equations
Ravi P. Agarwal, Dumitru Baleanu, Juan J. Nieto, Delfim F. M. Torres, Yong Zhou
TL;DR
The paper surveys fuzzy fractional differential equations and fractional control problems, including controllability, optimal control, feedback control, and relaxed multiple-control systems. It organizes foundational definitions, numerical and evolution-equation methods, and representative existence and controllability results. The survey concludes that fractional dynamical systems remain a rapidly developing field with substantial open work.
Problem
The survey addresses how fractional differential and control equations can represent uncertainty, nonlocality, and infinite-dimensional dynamics across fuzzy and evolution-system settings.
Method
The paper synthesizes definitions, numerical schemes, controllability results, and optimal or relaxed control formulations for several fractional evolution-equation classes.
Results
Under stated hypotheses, the survey reports controllability of system (4.17), existence-related results for multiple control systems, and a complete metric space of fuzzy numbers.
Takeaways & Limitations
Fractional-order models provide a framework for representing nonlocal relations and uncertain phenomena in applications spanning science, technology, and control.
Takeaways & Limitations
The survey concludes that much remains to be done despite the large number of existing results.
Abstract
from arXiv · showhide
We survey some representative results on fuzzy fractional differential equations, controllability, approximate controllability, optimal control, and optimal feedback control for several different kinds of fractional evolution equations. Optimality and relaxation of multiple control problems, described by nonlinear fractional differential equations with nonlocal control conditions in Banach spaces, are considered.
1. Introduction
The introduction surveys fuzzy fractional modeling, numerical solution methods, controllability, and fractional optimal control, emphasizing applications to uncertain and infinite-dimensional systems. It motivates nonlocal control problems and summarizes a contribution on relaxed multiple-control systems.
- Applications: Fractional differential equations model memory and hereditary effects across fields including biology, biomechanics, electrochemistry, control, porous media, and fluid dynamics.The introduction also notes applications to nonlinear oscillations, seepage flow, and traffic models.
- Fuzzy fractional differential equations: Fuzzy fractional operators combine fractional calculus with fuzzy modeling to represent uncertain parameters and initial values.Early work introduced fuzzy fractional calculus for fractional-order systems with uncertainty, followed by existence and uniqueness results.
- Fuzzy fractional differential equations: Approximate and numerical methods are important because most fuzzy fractional differential equations lack exact solutions.Reviewed approaches include Euler, operational-matrix, spectral, Legendre, Laplace-transform, and multistep methods.
- Controllability: Approximate controllability is emphasized because exact controllability is generally too strong for infinite-dimensional fractional diffusion equations.The introduction distinguishes exact from approximate controllability in evolution systems.
- Optimal control: The survey covers fractional optimal control for distributed and infinite-dimensional systems, including delay evolution systems and nonlocal control conditions.It also considers optimality and relaxation for nonlinear fractional multiple-control systems with nonconvex control integrands.
2. Basic definitions and notations
This section establishes the mathematical framework for fuzzy fractional analysis, including fuzzy numbers, metrics, fractional operators, differentiability, and multivalued maps. These definitions provide the spaces and notions used in later equations and control results.
- Framework: The section reviews fractional calculus, fuzzy-number concepts, semigroup theory, and multivalued analysis as foundational tools.The reviewed material includes Bochner integration and Hausdorff-metric constructions.
- Fuzzy numbers: A fuzzy number is defined as an upper-semicontinuous, fuzzy-convex, normal mapping with compactly supported positive-level set.The set of fuzzy numbers is denoted by E.
- Fuzzy numbers: The metric D on fuzzy numbers is based on the Hausdorff distance, and (E, D) is a complete metric space.Completeness supplies the metric-space setting for subsequent fuzzy analysis.
- Differentiability and multivalued analysis: Hukuhara difference, fuzzy Caputo differentiability, and measurable upper- or lower-semicontinuous multifunctions are introduced for fuzzy and control-valued analysis.These notions support the treatment of fuzzy-valued functions and multivalued controls.
3. Fuzzy fractional differential equations
The section surveys fuzzy fractional differential-equation models and numerical methods under uncertain initial data, parameters, and forcing. It highlights spectral tau techniques, application models, and related analytical and numerical developments.
- Related methods: The section also covers predictor–corrector, multistep, interval-uncertainty, and contraction-principle approaches for fuzzy fractional equations.These works address approximation, stability, uniqueness, and solution theory across different fuzzy fractional settings.
- Numerical methods: A shifted-Chebyshev spectral tau method reduces fuzzy fractional differential equations to a fuzzy algebraic linear system.The method constructs fuzzy residual equations, determines unknown coefficient vectors, and solves the resulting matrix system.
- Numerical methods: Generalized fractional Legendre polynomials support fuzzy approximate functions and an effective spectral tau method for uncertain fractional models.The reviewed work derives fuzzy Caputo derivatives of the generalized polynomials and applies them to physical models.
- Applications under uncertainty: Fuzzy fractional models replace deterministic initial conditions or parameters with fuzzy quantities to represent uncertainty in physical and chemical systems.Examples include fuzzy stress–strain and kinetic hydrolysis models with fuzzy initial data and concentration.
- Applications under uncertainty: The reviewed applications include non-Newtonian viscosity, viscoelastic behavior, acid hydrolysis, and drug- or chemical-process modeling under uncertainty.The models use fuzzy coefficients, set-valued functions, or fuzzy concentrations depending on the application.
4. Controllability
The section formulates a Sobolev-type fractional evolution system and establishes controllability using exponentially bounded propagation families, under stated operator, control, and nonlinear-term assumptions.
- System formulation: The controlled system uses a Caputo derivative, two closed linear operators, a bounded control operator, and a nonlinear term in a separable Banach space.The state evolves in X, while controls belong to a Banach space of admissible control functions.
- System formulation: A propagation family generated by (A, E) provides the analytical framework for defining mild solutions of the fractional system.The family is strongly continuous and exponentially bounded, with norm continuity for positive times.
- Mild solutions: Mild solutions are continuous functions satisfying the system’s integral representation for each admissible control and initial state.The representation includes the initial state, nonlinear contribution, and control contribution through the fractional solution operator.
- Controllability results: Under hypotheses (H1)–(H5) and an additional condition involving ℓ = aqM1 Γ(q+1), system (4.17) is controllable on J.Controllability means every initial state and target state in D(E) can be connected by an admissible control.
- Controllability results: Under the same theorem assumptions, the set of mild solutions is nonempty and compact in C(J, X) for the specified control.The control is given by the control formula introduced before the theorem.
5. Approximate controllability
This section develops approximate controllability for Sobolev-type fractional evolution systems by combining operator-semigroup representations, control formulas, and fixed-point arguments. Under stated structural and boundedness assumptions, approximate controllability follows from linear-system criteria and nonlinear fixed-point results.
- System formulation: The Sobolev-type system uses Caputo fractional dynamics with operators E and A, bounded controls through B, and a mild solution represented using a Green function.The operator assumptions ensure that -AE^-1 generates a uniformly continuous semigroup, which supports the mild-solution formulation.
- Linear systems: For the linear system, approximate controllability is equivalent to the resolvent condition εR(ε; Γ_a^0) → 0 under the stated continuity and boundedness assumptions.The associated Gramian criterion also characterizes approximate controllability through positivity and the triviality of the adjoint kernel.
- Approximate controllability: The nonlinear analysis constructs fixed points of P_ε in bounded closed convex subsets of C(J, X) for every ε > 0.This fixed-point construction is used after imposing approximate controllability of the corresponding linear system.
- Controllability concept: Approximate controllability means that the closure of the terminal-time reachability set equals the state space X.The reachability set consists of terminal states generated by admissible controls under the nonlocal initial condition.
- Approximate controllability: If the nonlinear term is bounded by N(t) with N ∈ L^r(J, R+) and rq > 1, the Sobolev-type fractional system is approximately controllable on J.The result assumes the operator, continuity, and linear approximate-controllability hypotheses stated earlier.
6. Existence and optimal control
This section establishes mild solvability and optimal-control existence for nonlinear fractional delay evolution systems with nonlocal control conditions. Under measurability, growth, operator, and cost-functional assumptions, unique mild solutions and optimal controls exist.
- System formulation: The system is a nonlinear Caputo fractional delay evolution equation on Banach spaces, with semigroup generator A, nonlinear term f, and control operator B.The history segment is defined by z_t(θ) = z(t + θ) for θ ∈ [-r, 0].
- Admissible controls: The admissible control set is bounded, closed, and convex in L^p(J, Y), and Bu belongs to L^p(J, X) for every admissible control.These properties support the existence analysis for the controlled system.
- Existence and uniqueness: For each admissible control u and suitable p satisfying pq > 1, the system has a unique mild solution on [-r, T].The admissible controls are measurable selections from closed, convex, bounded control-value sets.
- Optimal control: Under lower semicontinuity, convexity, measurability, growth, and strong continuity assumptions, the Lagrange optimal-control problem admits at least one optimal pair.The optimal pair consists of an admissible control and its corresponding mild solution.
- Optimal control: The existence theorem also permits replacing the original control-set condition with weak compactness of U and measurable closed convex values U(t).This provides an alternative control-set hypothesis for optimal-control existence.
7. Optimal feedback control
This section formulates semilinear fractional feedback control through mild state trajectories and feasible state-control pairs. Under compact-semigroup, regularity, pseudo-continuity, and cost assumptions, feasible pairs and optimal feedback control pairs exist.
- Feedback system: The feedback system is a semilinear Caputo fractional evolution equation on a reflexive Banach space, with controls selected from a Polish control space.The generator A produces a compact C0-semigroup, and the nonlinear dynamics are represented by f(t, x, u).
- Mild solutions: A mild solution is a continuous state trajectory satisfying the fractional integral equation involving the initial state, fractional resolvent operators, and the controlled nonlinearity.The Wright function appears in the fractional solution representation.
- Mild solutions: Under the stated hypotheses, every initial state and admissible control determine a unique mild solution in C(J, X).The assumptions include reflexivity, compact semigroup generation, measurability, continuity, and local Lipschitz continuity.
- Optimal feedback control: Under the cost, Cesari-property, and system assumptions, the Lagrange feedback-control problem admits at least one optimal control pair.The result applies to feasible state-control pairs and uses the stated lower-semicontinuity and multifunction conditions.
8. Optimal solutions to relaxation in multiple control problems of Sobolev type
The survey studies relaxation and optimality for multiple-control Sobolev-type nonlinear fractional systems with nonlocal conditions and mixed nonconvex control constraints. Under stated operator and data assumptions, the relaxation problem has optimal solutions approximated by minimizing sequences of the original problem.
- Problem formulation: The problem considers multi-integral functionals with nonconvex control integrands and mixed nonconvex constraints in nonlinear fractional systems with nonlocal control conditions.The systems are posed in Banach spaces and use multiple controls constrained through a set-valued map.
- Relaxation: The original controls satisfy u1(t), . . . , ur(t) ∈ U(t, x(t)), while the relaxed system replaces these sets by cl conv U(t, x(t)).The solution and trajectory sets are defined for both the original and convexified control systems.
- Problem formulation: The Sobolev-type formulation uses linear operators L, M, and E, with bounded compact inverses that yield a uniformly bounded compact analytic semigroup.The operator assumptions support rewriting the system in an evolution-equation framework.
- Existence and compactness: The original solution set is nonempty, and the relaxed solution set is compact in C(J, X) × ω-L1/β(I, T).This compactness result provides the solution-space setting used for the relaxation analysis.
- Optimality and approximation: The relaxation problem has a solution, and every relaxed solution is obtained as the limit of a minimizing sequence for the original problem.The convergence occurs for trajectories and controls in the stated function spaces, with the associated functional relation also holding.
- Scope: The survey concludes that fractional differential equations and fractional optimal control remain strongly developing fields despite numerous existing results.It identifies substantial remaining work within the broad area of fractional dynamical systems.