2026 Sino-French Autumn School
on PDE Control-Chengdu
Under the framework of the China-France Applied Mathematics Research Network, Sichuan University will host the Sino-French Fall School on Control of Partial Differential Equations from October 12 to 24, 2026, with the first day (October 12) designated for registration. The four invited lecturers are internationally renowned scholars in the field of Control Theory for Partial Differential Equations, who will deliver the latest cutting-edge knowledge in this discipline. Graduate students and young scholars in related fields are welcome to attend, and the lectures will be conducted in English.
1) Scientific Committee
Professor Jean-Michel Coron (Sorbonne University, France)
Professor Tatsien Li (Fudan University, China)
Professor Katharina Schratz (Sorbonne Université, France)
Professor Emmanuel Trélat (Sorbonne University, France)
Professor Ping Zhang (Chinese Academy of Sciences Institute of Mathematics and Systems Science, China)
Professor Xu Zhang (Sichuan University, China)
2) Organizing Committee
Professor Xiaoyu Fu (Sichuan University)
Professor Qi Lü (Sichuan University)
Professor Tianxiao Wang (Sichuan University)
Professor Xu Zhang (Sichuan University)
3) Courses
Course 1: Nonlinear controllability and Lie brackets
Lecturer: Professor Karine Beauchard (École normale supérieure de Rennes, France)
Summary：We study the controllability of a nonlinear control system locally around an equilibrium. A natural strategy (the "linear test") consists of proving the controllability of the linearized system and then concluding by applying the inverse mapping theorem. When the linearized system is not controllable, it is necessary to study higher-order terms. In Chapter 1, we present the power series expansion method to address such situations. We also examine the effects of nonlinearities on the local controllability result: its sensitivity to time and to the norm used to measure the controls. We also study Hölder cost estimates. In Chapter 2, we present a fundamental result by Krener: local controllability is encoded in iterated Lie brackets of the vector fields, evaluated at the equilibrium. In Chapter 3, we state the Lie algebra rank condition equivalent to the accessibility property (which is necessary for local controllability). We identify several frameworks in which this condition is also sufficient for local controllability and give applications to quantum control.
References:
[1] F. Albertini and D. D’Alessandro. Notions of Controllability for Quantum Mechanical Systems. Proceedings of the 40th IEEE Conference on Decision and Control, vol. 2. Orlando, FL, USA, 2001, 1589–1594.
[2] J.-M. Coron. Control and nonlinearity, vol. 136. Providence, RI: American Mathematical Society (AMS), 2007.
[3] V. Jurdjevic and H. Sussmann. Controllability of nonlinear systems. J. Differential Equations. 12 (1972), 95–116.
[4] V. Jurdjevic and H. Sussmann. Control Systems on Lie Groups. J. Differential Equations. 12 (1972), 313–329.
[5] A. Krener. On the equivalence of control systems and the linearization of nonlinear systems. SIAM J. Control. 11 (1973), 670–676.
Course 2: A Concise Introduction to Control Theory for Stochastic Partial Differential Equation
Lecturer: Professor Qi Lü(Sichuan University, China)
Summary：In this series of talks, I will give a concise introduction to control theory for systems governed by stochastic partial differential equations. We will be mainly concerned with controllability and optimal control problems for these systems. We shall focus on stochastic parabolic equations and stochastic hyperbolic equations in one spatial dimension, which allow us to avoid many technical difficulties. In particular, we shall see that both the formulation of the corresponding stochastic control problems and the tools for solving them may differ considerably from their deterministic or finite-dimensional counterparts. More importantly, one must develop new tools—such as the stochastic transposition method introduced in our previous work—to solve some problems in this field.
References:
[1] Q. Lü and X. Zhang. Mathematical Control Theory for Stochastic Partial Differential Equations. Probab. Theory Stoch. Model., vol. 101. Springer, Cham, 2021.
[2] Q. Lü and X. Zhang. Control theory for stochastic distributed parameter systems, an engineering perspective. Annu. Rev. Control. 51 (2021), 268–330.
[3] Q. Lü and X. Zhang. A concise introduction to control theory for stochastic partial differential equations. Math. Control Relat. Fields. 12 (2022), 847–954.
Course 3: Controllability of Coupled PDE Cascades. From Observability Inequalities to Wave-Heat and Heat-Heat Systems
Lecturer: Professor Emmanuel Trélat (Sorbonne Université, France)
Summary：This minicourse will present spectral and functional-analytic methods for controllability of coupled one-dimensional PDE cascades. After recalling the abstract semigroup framework for linear control systems, admissibility of boundary control operators, the duality between controllability and observability, and the Hilbert Uniqueness Method, the course will focus on moment methods and observability inequalities adapted to coupled PDEs. A central role will be played by Ingham--Müntz type inequalities, which combine the hyperbolic non-harmonic Fourier mechanism with the parabolic Müntz–Szász mechanism. The main case study will be the controllability of wave–heat and heat–wave cascade systems. The course will explain how Riesz basis decompositions, adjoint spectral analysis, and coupling-dependent modal coefficients lead to exact controllability results in weighted Hilbert spaces, with a sharp minimal time imposed by the hyperbolic component. A final part will discuss parabolic–parabolic cascades, in particular heat–heat systems, the block moment method, and open problems. The emphasis will be on controllability properties and reachable spaces.
References
[1] A. Benabdallah, F. Boyer and M. Morancey. A block moment method to handle spectral condensation phenomenon in parabolic control problems. Ann. H. Lebesgue. 3 (2020), 717–793.
[2] F. Boyer and M. Morancey. Analysis of non scalar control problems for parabolic systems by the block moment method. C. R. Math. Acad. Sci. Paris. 361 (2023), 1191–1248.
[3] V. Komornik and G. Tenenbaum. An Ingham-Müntz type theorem and simultaneous observation problems. Evol. Equ. Control Theory. 4 (2015), 297–314.
[4] H. Lhachemi, C. Prieur and E. Trélat. Boundary control of heat-heat cascades. arXiv:2506.10497, 2025.
[5] H. Lhachemi, C. Prieur and E. Trélat. Controllability of wave-heat and heat-wave cascades. arXiv:2601.18212, 2026.
[6] J.-L. Lions. Contrȏlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Masson, Paris, 1988.
[7] E. Trélat. Control in Finite and Infinite Dimension. SpringerBriefs on PDEs and Data Science, Springer, Singapore, 2024.
[8] M. Tucsnak and G. Weiss. Observation and Control for Operator Semi-Groups. Birkhäuser, Basel, 2009.
[9] X. Zhang and E. Zuazua. Polynomial decay and control of a 1-d hyperbolic-parabolic coupled system. J. Differential Equations. 204 (2004), 380–438.
Course 4: Time optimal control theory
Lecturer: Professor Gengsheng Wang (Hetao Institute of Mathematics and Interdisciplinary Sciences)
Summary：In this course, we will systematically introduce the theory of time optimal control for evolution equations. The curriculum will progress gradually from ordinary differential equations to partial differential equations, following a logical progression from fundamental to advanced topics. Specific course content includes: the existence analysis of time optimal controls; the core content concerning three types of maximum principles; the demonstration of equivalence among several typical optimal control problems; and several key properties of time optimal controls, including the bang-bang property and dynamic properties.
References
[1] H.O. Fattorini. Infinite Dimensional Linear Control Systems: The Time Optimal and Norm Optmal Problems. North-Holland Mathematics Studies, Vol. 201, North-Holland/Elserier, Amsterdam, 2005.
[2] G. Wang, L. Wang, Y. Xu and Y. Zhang. Time Optimal Control of Evolution Equations. Progress in Nonlinear Differnetial Equations and Their Aopplications. Vol. 92, Birkhäuser /Springer, Cham, 2018.
4) Cost
Participants will be exempted from registration fees, provided with free accommodation and a certain amount of catering subsidies. Participants are generally responsible for their own round-trip transportation expenses.
5) Registration & Training Schedule
Training: October 12–24, 2026
Registration: June 20 – July 30, 2026 (Late applications will not be accepted.)
To Register: Please click the link below to complete your online registration. All interested participants are welcome.
https://www.wjx.top/vm/Qgld1zw.aspx
6) Contact person
Miss Guo Tian, Phone number: 15515888183; Email: gt15515888183@163.com