Calculus of Variations, Mathematical Aspects of Systems Theory and Control Theory

Calculus of variations provides a unifying language for problems in which the state or trajectory of a system is determined by extremising an energy or action functional. Rooted in classical mechanics, it yields the Euler–Lagrange equations that govern geodesics, minimal surfaces and the optimal‐path problems of physics and geometry. In parallel, systems theory and control theory examine the regulation and estimation of dynamical processes under uncertainty or disturbance, using tools such as Lie brackets, distributions and Hamiltonian systems. The Pontryagin maximum principle formulates optimal control problems as boundary‐value problems in an extended phase space, while robust H∞ methods design feedback laws that minimise worst‐case amplification of disturbances. Modern applications span from generative modelling in machine learning—where variational formulations and gradient flows underlie continuous normalising flows—to observer design for nonlinear electrical drives, and from fluid‐flow transport problems in biomedical imaging to nonlocal variational inequalities in continuum mechanics. Interconnections between these fields emerge through common reliance on convex duality, Γ-convergence, geometric measure theory and Hamiltonian dynamics, enabling the systematic transfer of ideas between pure analysis, numerical schemes and real‐world engineering challenges.

Research from Nature Portfolio

Hyperparameter tuning in continuous normalising flows has been addressed by recasting the flow‐based density estimation problem into a Wasserstein gradient‐flow framework. By integrating the Jordan–Kinderlehrer–Otto (JKO) scheme directly into the training of neural ordinary differential equations, this approach removes the need for manual selection of soft‐penalty weights, instead performing iterative transport updates that guarantee stability and accurate density estimation. In brain‐fluid imaging, an unbalanced regularized optimal mass transport model has been developed to handle nonconserved mass distributions under advection–diffusion dynamics. By introducing entropy penalties alongside kinetic‐energy minimisation, the method accommodates local sources and sinks, yielding stable reconstructions of fluid flow on complex anatomical geometries and revealing detailed transport pathways in neurological applications.

Calculus of Variations, Mathematical Aspects of Systems Theory and Control Theory publication trend

The graph below shows the total number of articles in calculus of variations, mathematical aspects of systems theory and control theory across all publications each year (not limited to Nature Index journals).

Technical terms

Variational principle: A statement that the true evolution or configuration of a system extremises an action or energy functional.

Lagrangian: A function of configuration and rate variables whose integral over time or space is made stationary in variational formulations.

Euler–Lagrange equation: The necessary differential condition arising from setting the first variation of a functional to zero.

Pontryagin maximum principle: A set of necessary conditions for optimal control expressed via a Hamiltonian boundary‐value problem.

H∞ control: A robust design framework aiming to minimise the worst‐case gain from disturbances to designated outputs.

Γ-convergence: A mode of convergence for sequences of functionals that ensures convergence of minimisers and minimal values.

Continuous normalising flow: A continuous mapping of probability densities parameterised by neural ODEs, trained to match target distributions via variational objectives.

Regularized optimal mass transport: An extension of classical transport that incorporates entropy or diffusion penalties to allow for nonconserved mass transfer under dynamic constraints.

References

1. Taming hyperparameter tuning in continuous normalizing flows using the JKO scheme. Scientific Reports (2023).
2. Unbalanced regularized optimal mass transport with applications to fluid flows in the brain. Scientific Reports (2024).
3. DNN-Based H∞ Control Scheme of Nonlinear Time-Varying Dynamic Systems With External Disturbance and its Application to UAV Tracking Design. IEEE Access (2021).
4. SOS-Based Nonlinear Observer Design for Simultaneous State and Disturbance Estimation Designed for a PMSM Model. Sustainability (2022).
5. Optimal incompatible Korn–Maxwell–Sobolev inequalities in all dimensions. Calculus of Variations and Partial Differential Equations (2023).