insight - Partial Differential Equations Control - # Inverse Optimal Control Design for PDEs with Convection

Core Concepts

The authors develop inverse optimal feedback controllers for partial differential equations (PDEs) with quadratic convection, counter-convection, and diffusion, where the derivative of the control Lyapunov function (CLF) has a depressed cubic or quadratic dependence on the control input. The inverse optimal controllers minimize meaningful cost functionals that penalize the state and control.

Abstract

The paper considers the problem of inverse optimal control design for systems that are not affine in the control, focusing on classes of partial differential equations (PDEs) with quadratic convection, counter-convection, and diffusion.
Key highlights:
For PDEs with quadratic convection, the L2 norm is a CLF whose derivative has a depressed cubic dependence on the control input. The authors construct an inverse optimal controller, called the Cardano-Lyapunov controller, that minimizes a meaningful cost functional.
For PDEs with linear counter-convection, a weighted L2 norm is a CLF whose derivative has a quadratic dependence on the control input. The authors construct two distinct inverse optimal controllers that minimize a different cost functional.
The authors show that the inverse optimal controllers achieve asymptotic stabilization of the origin and provide explicit formulas for the cost functionals that are minimized.
For the quadratic case, the authors also propose a switching strategy between the two inverse optimal controllers to reduce the control effort.
The paper provides a comprehensive framework for inverse optimal control design of PDEs with convection and diffusion, extending the existing results on inverse optimality, which were mostly limited to control-affine systems.

Stats

The paper does not contain any explicit numerical data or statistics. The focus is on the theoretical development of inverse optimal control design for PDEs with convection and diffusion.

Quotes

"Inverse optimal control design for systems that are not affine in the control is a challenging problem that has not been fully addressed in the literature."
"The authors construct a (family of) feedback law(s) and show that some cost functional is minimized in closed-loop. We say in this case that the (family of) control law(s) is inverse optimal."
"The common point in the aforementioned references is that the underlying control system is affine in the control. In this case, the inverse optimal controllers can be designed using Sontag's universal formula [17]. This approach is specific to control-affine systems, and cannot be applied if the derivative of the CLF is given, e.g., by (1)."

Key Insights Distilled From

by Mohamed Cami... at **arxiv.org** 04-02-2024

Deeper Inquiries

To extend the inverse optimal control design approach to handle more general classes of PDEs, such as those with nonlinear convection or diffusion terms, several modifications and adaptations can be made. One approach could involve incorporating additional terms in the control Lyapunov function (CLF) derivative to account for the nonlinearity introduced by the convection or diffusion terms. By appropriately structuring the CLF derivative to capture the nonlinear dynamics of the system, inverse optimal controllers can be designed to stabilize these more complex PDEs. Additionally, the feedback laws may need to be adjusted to accommodate the new terms and ensure asymptotic stability in the presence of nonlinearity. Techniques from nonlinear control theory and Lyapunov-based control design can be leveraged to address the challenges posed by nonlinear convection or diffusion terms in PDEs.

The inverse optimal controllers presented in this work have potential applications in various fields where PDE-constrained systems are prevalent. One key application could be in the field of fluid dynamics, where PDEs with convection play a significant role in modeling fluid flow phenomena. By implementing the inverse optimal controllers in fluid flow systems, it may be possible to achieve robust stabilization and control of flow patterns, leading to improved efficiency and performance. Other potential applications include chemical reactors, manufacturing processes, and traffic flow control, where PDEs with convection are commonly used to model system behavior. The controllers could be implemented in real-world systems using numerical simulations and experimental validation to demonstrate their effectiveness in practical settings.

The switching strategy between the two inverse optimal controllers in the quadratic case can potentially be generalized to other PDE systems with higher-order polynomial dependence of the CLF derivative on the control input. By analyzing the structure of the CLF derivative and the corresponding feedback laws, similar switching mechanisms can be devised for PDEs with different polynomial dependencies. The key lies in identifying the critical points where switching between controllers is beneficial and ensuring that the overall control strategy remains effective in stabilizing the system. This generalization may involve adapting the switching conditions and feedback laws based on the specific characteristics of the CLF derivative in each case, allowing for a flexible and adaptive control approach across a broader range of PDE systems.

0