ENACT Méthode du Backstepping en dimension supérieure et IA pour équations non linéaires

Keywords

Stabilization, Control, ODE, PDE, Robotic, AI

Subject details

In control theory, stabilization refers to the ability to return a system to a stable or unstable equilibrium. To do so, an action on the state of the system is done according to its current state. This process is called a feedback loop. Stabilization may be achieved for a variety of physical system, from everyone's childhood experience to hold a pencil straight up on the tip of a finger to the landing of the SpaceX rocket. 

 A wide array of methods exist in control theory to achieve stabilization, and specifically to design a feedback law so that the feedback loop is stable. One that has gained significant attention over the past decade is the backstepping method for partial differential equations (PDE). The idea is simple : assume you want to stabilize a system to an equilibrium, and that you know a similar system that is already stable. The goal is to «learn» from the stability of the latter system to deduce the feedback law to achieve the stability of the former system. From the mathematical point of view, «learning» means to prove the existence of an invertible transformation mapping the state to stabilize to the stable state. 

 This method has been shown [Cor15] to work for linear ordinary differential equations (ODE), and extensive work has been done on the backstepping method for linear PDEs, leading to over 1000 papers in the literature over the past two decades. Despite advances to give necessary and sufficient conditions on the backstepping method to work ([GHXZ22], [GHXZ24]), this method is restricted for the moment to 1-dimensional PDEs in space. 

The first objective of the PhD thesis is to extend this method to multi-dimensional PDEs for self-adjoint operators. The goal here is to modify the target stable system usually used in the literature, designed to shift all the eigenvalues by the same factor, by introducing a spectral projection to only shift the eigenmodes not decaying sufficiently fast. This idea has the advantage of lifting the current limitations of the backstepping method, due to the unbounded growth of the multiplicity of the eigenvalues in spatial dimension greater than 1. The second objective is to explore the extension of the philosophy of the backstepping method to nonlinear ODEs or PDEs using AI. Only local results are known in the literature for the stabilization of nonlinear ODEs and PDEs using the backstepping method, local meaning here that the state of the system has to be already close to the equilibrium. This restriction mainly comes from our inability to use a global approach to prove the existence and invertibility of the transformation. We want to circumvent this restriction by designing Neural Networks able to learn from the stable system, and design a feedback law to stabilize the initial system. 

This present many significant advantages. First, the use of AI for this learning procedure avoids completely the need to formulate the nonlinear equivalence equation on the transformation for it to maps the system to stabilize to the stable target system, a very complicated task for nonlinear equations. Moreover, it may provide significant numerical evidences of the existence of such transformation, helping to determine whether it is a lead to follow theoretically. Finally, this approach could be much more efficient to implement a feedback law in practice, as the equation giving the feedback law is rather complex, and the numerical convergence of the feedback is not ensured by classical numerical approximation methods. Different examples of nonlinear systems will be investigated. First, a two-dimensional nonlinear parabolic PDE will be considered, for which unsupervised Neural Networks were already designed [AGHZ25]. A more challenging application, drawn from robotic applications, will be to train robots to learn from imitating humans movements.