Title: Control of coupled hyperbolic-parabolic PDE systems
Laboratory: Laboratoire des signaux et systemes, CNRS, CentraleSupelec, Universite Paris
Saclay, 91190, Gif-sur-Yvette, France
Postdoctoral Advisors: Jean AURIOL, Hugo LHACHEMI 1.
Keywords: partial differential equations, coupled hyperbolic-parabolic PDEs, boundary control.
Description of the Thesis
I. Context
The interconnection of Partial Differential Equations (PDEs) is a well-established topic due to its
relevance in various industrial applications. Examples include traffic networks [1], electric power
transmission systems [2], and thermal heat exchanger tubes [3]. Such systems arise in a broad range
of applications, including fluid-structure interactions [4], thermoacoustic instabilities, and chemotaxis
models [5]. In these configurations, the finite propagation speed characteristic of hyperbolic dynamics
interacts with the smoothing and dissipative effects associated with parabolic dynamics.
Most contributions in the literature typically focus on either hyperbolic or parabolic intercon-
nections. For hyperbolic systems, many constructive control strategies rely on the backstepping ap-
proach [6], which has recently proven effective in designing stabilizing controllers for interconnections
of hyperbolic PDEs with chain structures [7, 8].
For parabolic PDEs, backstepping has been extended to coupled systems with spatially varying
coefficients in [9, 10]. More recently, [11] presented results on the stabilization of interconnected
parabolic systems with a chain structure, employing a recursive backstepping approach. Alternative
methods for parabolic systems have been developed using spectral reduction techniques [12], where
system trajectories are projected onto the modes of the underlying unbounded operator.
Relatively few results are available for mixed hyperbolic-parabolic PDE systems. Although Lyapunov-
based methods [13] and flatness approaches [14] are theoretically applicable, such systems have largely
been overlooked in the literature [15, 16]. However, spectral approaches appear promising for address-
ing coupled parabolic-hyperbolic interconnections, particularly in the case of parabolic PDEs with
input delays [17, 18]. These methods have been shown to be robust against small spatial variations
in the input delay [19]. However, the ability of spectral reduction methods to handle spatially vary-
ing delays undergoing significant variations—especially those induced by in-domain couplings from
transport equations—has yet to be demonstrated.
II. Scientific objectives and methodology
Postdoc goal: The general objective of this postdoctoral project is to analyze the controllability
and design stabilizing control design for a class of hyperbolic-parabolic interconnections.
To establish necessary and sufficient controllability conditions and design stabilizing controllers
for a broad class of coupled hyperbolic-parabolic systems, we propose an original methodology that
integrates concepts from both backstepping and spectral approaches. Specifically, by applying an ap-
propriate integral transformation, we aim to simplify the structure of the PDE system. More precisely,
this transformation should eliminate certain in-domain couplings while introducing integral distributed
terms at the boundaries, thereby making the resulting system more amenable to spectral reduction
techniques. By projecting the system trajectories onto the modes of the underlying unbounded op-
erator governing the reaction-diffusion dynamics, the stabilization problem is reduced to stabilizing a
finite set of ordinary differential equations that incorporate both pointwise and distributed input de-
lays. This is achieved through the use of predictor feedback methods [20]. Implementing this approach
raises several scientific challenges that will be addressed during the postdoctoral research.
• Developing an appropriate backstepping transformation and proving its existence.
Applying the backstepping methodology to coupled hyperbolic-parabolic systems introduces a
new class of kernel equations, whose well-posedness may be challenging to establish.
1The advisors are with Universit´e Paris-Saclay, CNRS, CentraleSup´elec, Laboratoire des signaux et syst`emes, 91190,
Gif-sur-Yvette, France. Emails: {jean.auriol,lucas.brivadis,antoine.chaillet}@centralesupelec.fr.
• Conducting a spectral analysis in the presence of distributed terms. The coupling
with hyperbolic systems leads to distributed delays, whose impact on the parabolic dynamics
must be carefully assessed.
• Numerically implementing the proposed controllers. Practical validation through nu-
merical simulations will be essential to evaluate the effectiveness of the designed control laws.
III. Desired profile
This postdoctoral position requires a strong background in control systems and applied mathematics.
The ideal candidate should hold a Ph.D. in control theory with experience in control design for infinite-
dimensional systems. Expertise in backstepping design and/or spectral reduction methods would be
highly valued. Additionally, the candidate should have a proven track record of publishing high-quality
research articles.
National Security Clearance is needed, and it can require approximately 2 months.
IV. Application
To apply, write an email with your CV and a summary of your research activities to Jean Auriol and
Hugo Lhachemi: firstname.lastname@centralesupelec.fr.
V. References
[1] H. Yu and M. Krsti´c. Traffic Congestion Control by PDE Backstepping. Springer, 2023.
[2] C. Schmuck, F. Woittennek, A. Gensior, and J. Rudolph. Feed-forward control of an HVDC power trans-
mission network. IEEE Transactions on Control Systems Technology, 22(2):597–606, 2014.
[3] C.-Z. Xu and G. Sallet. Exponential stability and transfer functions of processes governed by symmetric
hyperbolic systems. ESAIM: Control, Optimisation and Calculus of Variations, 7:421–442, 2002.
[4] X. Zhang and E. Zuazua. Asymptotic behavior of a hyperbolic-parabolic coupled system arising in fluid-
structure interaction. In Free Boundary Problems: Theory and Applications, pages 445–455. Springer,
2007.
[5] T. Li, R. Pan, and K. Zhao. Global dynamics of a hyperbolic-parabolic model arising from chemotaxis.
SIAM Journal on Applied Mathematics, 72(1):417–443, 2012.
[6] M. Krstic and A. Smyshlyaev. Boundary control of PDEs: A course on backstepping designs, volume 16.
Siam, 2008.
[7] J. Deutscher and J. Gabriel. A backstepping approach to output regulation for coupled linear wave–ode
systems. Automatica, 123:109338, 2021.
[8] J. Redaud, J. Auriol, and S.-I. Niculescu. Output-feedback control of an underactuated network of inter-
connected hyperbolic PDE-ODE systems. Systems and Control letters, 2021.
[9] R. Vazquez and M. Krstic. Boundary control of coupled reaction-advection-diffusion systems with spatially-
varying coefficients. IEEE Transactions on Automatic Control, 62:2026–2033, 2017.
[10] L. Camacho-Solorio, R. Vazquez, and M. Krstic. Boundary observers for coupled diffusion–reaction systems
with prescribed convergence rate. Systems & Control Letters, 135:104586, 2020.
[11] X. Xu, L. Liu, M. Krstic, and G. Feng. Stabilization of chains of linear parabolic PDE–ODE cascades.
Automatica, 148:110763, 2023.
[12] H. Lhachemi and R. Shorten. Output feedback stabilization of an ode-reaction–diffusion pde cascade with
a long interconnection delay. Automatica, 147:110704, 2023.
[13] A. Mironchenko and H. Ito. Construction of Lyapunov functions for interconnected parabolic systems: an
iISS approach. SIAM Journal on Control and Optimization, 53(6):3364–3382, 2015.
[14] T. Meurer. Flatness-based trajectory planning for diffusion–reaction systems in a parallelepipedon—a
spectral approach. Automatica, 47(5):935–949, 2011.
[15] I. Karafyllis and M. Krstic. Small-gain stability analysis of certain hyperbolic–parabolic PDE loops. Systems
& Control Letters, 118:52–61, 2018.
[16] R. Vazquez, J. Auriol, F. Bribiesca-Argomedo, and M. Krstic. Backstepping for partial differential equa-
tions. arXiv preprint arXiv:2410.15146, 2024.
[17] H. Lhachemi and C. Prieur. Predictor-based output feedback stabilization of an input delayed parabolic
pde with boundary measurement. Automatica, 137:110115, 2022.
[18] H. Lhachemi, C. Prieur, and R. Shorten. An LMI condition for the robustness of constant-delay linear
predictor feedback with respect to uncertain time-varying input delays. Automatica, 109:108551, 2019.
[19] H. Lhachemi, C. Prieur, and R. Shorten. Robustness of constant-delay predictor feedback for in-domain sta-
bilization of reaction–diffusion pdes with time-and spatially-varying input delays. Automatica, 123:109347,
2021.
[20] Iasson Karafyllis and Miroslav Krstic. Predictor feedback for delay systems: Implementations and approx-
imations. Springer, 2017.
Advisors’ Biographies
Jean AURIOL (supervision load: 50%). He received his Master’s degree in civil engineering
in 2015 (major: applied maths) from MINES ParisTech, part of PSL Research University, and in
2018 his PhD degree in control theory and applied mathematics from the same university (Centre
Automatique et Syst`emes). From 2018 to 2019, he was postdoctoral researcher at the Department
of Petroleum Engineering, University of Calgary, Canada, where he was working on implementing
backstepping control laws to attenuate mechanical vibrations in drilling systems. Since December
2019, he has been an associate researcher (Charg´e de Recherche) at CNRS, Universit´e Paris-Saclay,
CentraleSup´elec, L2S. His research interests include robust control of distributed parameter systems
and interconnected systems.
Hugo LHACHEMI (supervision load: 50%). He received a four-year university degree in math-
ematics from Universit´e Claude Bernard Lyon I, France, in 2011, an engineering degree from Ecole
Centrale de Lyon, France, in 2013, and a M.Sc. degree in Aerospace Engineering from Polytechnique
Montreal, Canada, in 2013. He received his Ph.D. degree in Electrical Engineering from Polytech-
nique Montreal in 2017. He was postdoctoral fellow in Automation and Control at University College
Dublin, Ireland, from 2018 to 2020. Since 2020, he is an Associate Professor of System Control at
CentraleSup´elec, France. His main research interests include nonlinear control, infinite dimensional
systems, and their applications to aerospace and cyber–physical systems.
