Professor Francisco Jurado | Taming the Dynamics of Nonlinear Partial Differential Equations

Controlling partial differential equations (PDEs) with parametric uncertainties is vital for system stability in science and engineering. Professor Francisco Jurado from Tecnológico Nacional de México (TecNM)/La Laguna addresses this challenging problem with a novel approach using adaptive backstepping boundary control for a one-dimensional modified Burgers’ equation. His innovative work tackles uncertainties in reactive and viscosity terms and considers Robin and Neumann boundary conditions, offering valuable insights into nonlinear PDE control.

How Mathematics Can Explain Our Physical World

Partial differential equations are indispensable for understanding the physical world because they provide mathematical models describing how various quantities change and interact over space and time. From heat transfer, fluid dynamics, and electrostatics to wave phenomena and vibrations, PDEs capture the fundamental laws governing the behaviour of natural systems.

These equations enable researchers to predict and analyse complex processes such as the flow of fluids, the propagation of waves, and the distribution of heat or electric fields, which are crucial for engineering, physics, and environmental science. Moreover, PDEs have expanded their reach into novel areas like traffic flow control, neural networks, machine learning, and quantum systems, offering insights into classical and cutting-edge problems across diverse fields. This versatility makes PDEs indispensable for advancing technology, understanding natural phenomena, and solving real-world challenges across traditional and emerging scientific disciplines.

However, the true power of PDEs is only fully realised when they are paired with appropriate initial and boundary conditions, ensuring that the problems they model are well-posed – meaning their solutions exist, are unique, and stable. While exact solutions to PDEs can provide deep insights, finding these solutions, especially for nonlinear PDEs and complex real-world problems, is often highly challenging. Consequently, various numerical methods have been developed to approximate solutions. These methods enable scientists and engineers to address multifaceted real-world problems that would otherwise be unsolvable.

Professor Francisco Jurado from TecNM/La Laguna now offers us a new approach to tackling the complexities of PDEs.

The Burgers’ Equation

The Burgers’ equation, also known as the Bateman–Burgers equation, is a fundamental partial differential equation with significant applications in applied mathematics, particularly fluid mechanics, nonlinear acoustics, gas dynamics, and traffic flow analysis. Originally introduced by Harry Bateman in 1915 and later extensively studied and popularised by Dutch mathematician and physicist Johannes Martinus Burgers in 1948, this equation has become a cornerstone in the mathematical modelling of various physical phenomena.

Burgers’ equation is especially valuable in modelling wave phenomena in hydrodynamics and acoustics. Initially derived as a prototype to provide analytical insights into turbulence and its complexities, it has since been widely applied to the study of shock waves and wave transmission. Despite its utility, Burgers’ equation is not globally controllable and can exhibit neutral stability under certain conditions, highlighting its intricate and context-dependent behaviour. The challenge of mastering the dynamics of the Burgers’ equation remains a central focus of mathematical research, fuelled by its wide-ranging applications in science and engineering. Gaining a deeper understanding and achieving precise control of this equation is essential, given its critical role in addressing a diverse array of scientific and engineering problems.

Building on this, Professor Jurado’s research stands at the forefront of exploring the complexities of the Burgers’ equation, particularly addressing its lack of global controllability and the potential for neutral stability under certain conditions. In recent research, Professor Jurado investigates the adaptive backstepping boundary control methodology applied to a modified Burgers equation, specifically dealing with unknown but constant parameters in the reactive and nonlinear convective terms. The research delves into scenarios involving Robin and Neumann boundary conditions, with the control signal directly influencing the latter. Professor Jurado’s work pushes the boundaries of our understanding and control of the Burgers equation, advancing the field expressively.

Tackling a Class of Modified Burgers’ Equations

The Burgers’ equation is a key tool for modelling wave behaviour and turbulence and serves as an analogue to the Navier-Stokes equation. While it can be related to the heat equation through transformations, it is not always stable or controllable, making its behaviour unpredictable, especially with certain boundary conditions. Exact solutions for nonlinear PDEs are often difficult to find, particularly in real-world applications, so most researchers focus on numerical methods to manage the equation’s complexity. Professor Jurado’s recent research applies the adaptive backstepping boundary control method to a class of Burgers’ equations, offering a promising solution.

Backstepping boundary control is an advanced technique for stabilising and regulating systems governed by partial differential equations, such as Burgers’ equations. This method involves designing feedback control laws specifically for the system’s boundaries, which is particularly advantageous for managing the nonlinear dynamics inherent in Burgers’ equation – dynamics that can be difficult to control with traditional approaches. By focusing on boundary control, backstepping directly addresses the equation’s nonlinearities and intricate boundary interactions, effectively stabilising the entire system even in the presence of complex phenomena like shock waves and wave propagation.

Professor Jurado shows that an adaptive backstepping boundary control scheme improves the management of Burgers’ equation’s complex dynamics, offering vital insights and solutions for a wide range of technical challenges.

Promising Results

Professor Jurado’s insight significantly advances the management of the complexities associated with unknown but constant parameters in reactive and nonlinear convective terms, particularly under Robin and Neumann boundary conditions where the control signal influences the Neumann boundary condition. By designing a nominal controller based on a linear PDE and adapting it for the modified Burgers’ equation, Professor Jurado offers a robust method for controlling nonlinear PDE systems. The adaptive version of the controller demonstrated in simulations effectively stabilises the system, achieving parameter convergence near ideal values and reducing estimation errors to zero.

These results confirm the efficacy of the proposed method in this class of nonlinear PDE systems and suggest its potential for application to high-dimensional and real-world problems. Thus, Professor Jurado’s approach not only addresses theoretical challenges but also promises practical utility in complex and dynamic scenarios.

Adaptive Control for Burgers’ Equation: Challenges and Future Steps

Professor Jurado’s adaptive control scheme for Burgers’ equation shows promise, with simulations indicating convergence to near-ideal parameter estimates. However, the scheme assumes that a nominal controller for a reaction-advection-diffusion system can stabilise the modified Burgers’ equation, which may not always hold due to differing system dynamics.

Future research offers exciting opportunities to validate this assumption across a wide range of conditions and to explore the development of controllers specifically designed for the unique dynamics of the modified Burgers’ equation. By conducting thorough testing, both in simulations and real-world scenarios, researchers can further enhance the scheme’s effectiveness, ensuring it performs optimally in diverse and complex environments.