Optimal Control and Boundary Value Problems for Fractional Differential Equations
Summary
Fractional differential equations generalise classical differential equations by introducing derivatives of non-integer order, thereby capturing memory effects and spatial nonlocality inherent in many complex systems. Their integration into optimal control theory and boundary value problems has yielded powerful tools for modelling anomalous diffusion, viscoelastic materials and finance. In optimal control, fractional dynamics require new variational principles and maximum‐principle formulations that account for history-dependent state equations. Establishing existence, uniqueness and regularity of optimal controls hinges on the interplay between the fractional operator’s nonlocal kernel and admissible control sets. Concurrently, boundary value problems for fractional operators—Dirichlet, Neumann and Robin—demand precise definitions of nonlocal boundary conditions and analysis in fractional Sobolev spaces. Novel numerical schemes, including spectral collocation, finite‐element and integral-equation approaches, have been devised to approximate both control and boundary-value solutions. These developments provide a cohesive framework for practical applications such as thermal regulation in materials with memory, coordinated control of anomalous transport processes and the design of surfaces with prescribed response to external stimuli.
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Optimal Control and Boundary Value Problems for Fractional Differential Equations publication trend
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Technical terms
Fractional differential equation: A differential equation involving derivatives of non-integer order, used to model systems with memory and hereditary characteristics.
Fractional Laplacian: A nonlocal operator extending the classical Laplace operator to fractional order, defined either via integral kernels or spectral decomposition.
Optimal control: A mathematical discipline concerned with determining control functions that steer a dynamical system to optimise a given performance criterion.
Boundary value problem: A problem of finding a solution to a differential equation that satisfies specified conditions on the domain boundary.
Controllability: A property of a dynamical system indicating whether it can be driven from an initial state to a desired final state using admissible controls.
References
- Optimal control of the coefficient for the regional fractional $p$-Laplace equation: Approximation and convergence. Mathematical Control and Related Fields (2019).
- Controllability of the one-dimensional fractional heat equation under positivity constraints. Communications on Pure and Applied Analysis (2020).
- Globally Existing Solutions to the Problem of Dirichlet for the Fractional 3D Poisson Equation. Fractal and Fractional (2023).
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