Evolutionary Differential Equations and Control Theory

Evolutionary differential equations model the time evolution of complex systems in fields ranging from ecology and epidemiology to materials science and economics. When coupled with control theory, these equations enable the design of intervention strategies that steer system trajectories towards desired outcomes, such as stabilising population dynamics or optimising energy consumption. Central to this interplay are fractional derivatives, which capture memory effects and anomalous transport, and semigroup methods, which frame the dynamics in abstract functional spaces. Control objectives typically involve exact or approximate controllability, seeking to guide the state of an infinite-dimensional system using bounded inputs. The development of mild solution frameworks has facilitated the analysis of nonlocal and stochastic perturbations, while advances in fixed-point theorems and resolvent operator techniques have expanded the class of admissible nonlinearities. Collectively, these innovations have sharpened our understanding of stability, robustness and optimisation in systems exhibiting hereditary behaviour or spatial heterogeneity, with applications in climate modelling, robotic swarms and financial networks.

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Research from all publishers

Recent work has extended controllability results for fractional delay integrodifferential systems by establishing existence of mild solutions under nonlocal and Sobolev-type conditions. These studies employ multivalued analysis and cosine families to derive approximate controllability criteria for systems of order between one and two, illustrating applications to viscoelastic materials. Stochastic generalisations have introduced Atangana-Baleanu fractional derivatives into neutral delay models, leveraging compactness of integral operators and martingale techniques to ensure boundedness and equicontinuity of solution maps. Meanwhile, time-varying fractional dynamical systems with a single control delay have been shown to satisfy necessary and sufficient controllability conditions via Grammian matrices and successive approximation, with numerical examples confirming stability of the control schemes. Together, these contributions demonstrate the growing maturity of fractional-order control methodologies and their applicability to engineering, physics and biological modelling.

Evolutionary Differential Equations and Control Theory publication trend

The graph below shows the total number of articles in evolutionary differential equations and control theory across all publications each year (not limited to Nature Index journals).

Technical terms

Evolutionary differential equation: An equation describing the time evolution of a system’s state in an infinite-dimensional space, often governed by an operator semigroup.

Fractional derivative: A generalisation of the classical derivative to non‐integer orders, capturing memory and hereditary properties in dynamical systems.

Controllability: The property that a system’s state can be driven to a desired target within finite time using suitable control inputs.

Mild solution: A solution concept for evolution equations obtained via integral formulations and semigroup theory, accommodating unbounded operators.

Semigroup: A family of linear operators describing the evolution of the homogeneous part of an abstract differential equation over time.

References

1. New results on approximate controllability of fractional delay integrodifferential systems of order 1 . Alexandria Engineering Journal (2023).
2. Approximate controllability of Atangana-Baleanu fractional neutral delay integrodifferential stochastic systems with nonlocal conditions☆. Ain Shams Engineering Journal (2023).
3. Controllability of the time-varying fractional dynamical systems with a single delay in control. Nonlinear Dynamics (2024).