PhD Position F/M Filippov Solutions for Discontinuous Differential-Algebraic Equations (DAEs): Control and Simulation

The Inria Rennes - Bretagne Atlantique Centre is one of Inria's eight centres and has more than thirty research teams. The Inria Center is a major and recognized player in the field of digital sciences. It is at the heart of a rich R&D and innovation ecosystem: highly innovative PMEs, large industrial groups, competitiveness clusters, research and higher education players, laboratories of excellence, technological research institute, etc.

Collaboration & Contacts

• This PhD is conducted in collaboration with Prof. Stephan Trenn at the University of Groningen (The Netherlands). The selected candidate is expected to visit Groningen regularly.
• There is also the possibility of pursuing a double doctoral degree, which would include a 1-year extension to meet Dutch PhD requirements.
• This position will be funded by ANR JCJC Project: GFdDAE (ANR-25-CE48-4916) 
• Contacts: yahao.chen@inria.fr, benoit.caillaud@inria.fr, s.trenn@rug.nl

Context & Background

Differential-algebraic equations (DAEs) arise naturally when modeling dynamical systems from first principles. In many cases, physical laws are expressed as combinations of differential and algebraic equations. This modeling approach is common in constrained mechanics, chemical and biological processes, power systems, and especially analog circuit design—where idealized components (e.g., resistors, capacitors, inductors) and Kirchhoff’s laws define the system dynamics. When these systems experience abrupt changes—such as switching in electric circuits, mechanical contacts, or discontinuous control inputs—discontinuous DAEs emerge. However, there is currently no comprehensive theoretical foundation for studying such systems. Challenges include:

• Their hybrid behaviors, which differ significantly from ODE counterparts,
• The inconsistent initialization problem caused by switching and algebraic constraints,
• The occurrence of Dirac impulses due to state jumps.

Without a rigorous solution concept, tasks such as simulation, stability analysis, and control design lack solid justification.

Discontinuous DAEs are relevant across many research areas, including systems and control, hybrid systems, and computer-aided simulation. A notable example is switched DAEs. While time-dependent switching has been extensively studied [2-5], progress on state-dependent switching, a subclass of discontinuous DAEs, remains limited.

The Hycomes team at Inria Rennes has contributed to related research through the concept of multi-mode DAEs, in the context of the Modelica language [6-7]. Despite these advancements, challenges persist, including:

• Computing consistent initial values and jumps,
• Managing sliding and chattering behaviors,
• Addressing scalability for large-scale, high-dimensional systems.

These issues emphasize the need for refined mathematical foundations and advanced control methods compatible with Modelica-based simulation platforms.

[1] D. Liberzon. Switching in Systems and Control. Systems and Control: Found. and Appl. Boston: Birkhäuser, 2003.

[2] D. Liberzon and S. Trenn. “On stability of linear switched differential algebraic equations”. In: Proc. IEEE CDC 2009, pp. 2156–2161.

[3] D. Liberzon and S. Trenn. “Switched nonlinear differential algebraic equations: Solution theory, Lyapunov functions, and stability”. In: Automatica 48.5, pp. 954–963.

[4] Y. Chen and S. Trenn. “Impulse-free jump solution of nonlinear differential algebraic equation”. In: Nonlinear Analysis: Hybrid Systems 46 (2022), p. 101238.

[5] Y. Chen and S. Trenn. “On impulse-free solutions and stability of switched nonlinear differential–algebraic equations”. In: Automatica 156 (2023), p. 111208

[6] A. Benveniste, B. Caillaud, and M. Malandain. “The mathematical foundations of physical systems modeling languages”. In: Ann. Rev. in Control 50 (2020), pp. 72–118.

[7] A. Benveniste, B. Caillaud, M. Malandain, and J. Thibault. “Algorithms for the structural analysis of multimode modelica models”. In: Electronics 11.17 (2022), p. 2755.

[8] A.F. Filippov. Differential Equations with Discontinuous Right-hand Sides. English (Transl. from the Russian). Mathematics and Its Applications: Soviet Series, 18. Dordrecht etc.: Kluwer Academic Publishers, 1988.

[9] Y. Shtessel, C. Edwards, L. Fridman, A. Levant, et al. Sliding Mode Control and Observation. Vol. 10. Springer, 2014

For discontinuous ODEs, the Filippov solution framework [8] plays important roles both theoretically (e.g., in switching ODE systems [1] and sliding mode control [9]) and practically (e.g., via Filippov-type solvers in MATLAB). This PhD project aims to:

1. Extend the Filippov solution concept to discontinuous DAEs,
2. Integrate the proposed framework into simulation tools, particularly Modelica.

The PhD student will focus on the following tasks:

1. Conduct a thorough literature review of discontinues DAEs and related systems (e.g., complementarity systems, switching DAEs, hybrid systems).
2. Starting with discontinuous linear DAEs, propose a solution concept and prove well-posedness.
3. Extend the theory to nonlinear systems and compare it with other existing frameworks.
4. Perform stability analysis and develop stabilization or control strategies for discontinuous DAEs.
5. Implement simulation methods in Modelica tools, and test them on benchmark examples.

Deliverables include scientific reports, papers submitted to international conferences and journals, and prototype simulation code.

Technical Skills

• Solid understanding of system modeling, control theory, and differential equations
• Familiarity with any of the following topics would be appreciated:
• DAEs, switched systems, complementarity systems, sliding mode control;
• hybrid systems simulation, Modelica;
• power electronics, contact mechanics, multiphysics modeling.

• • Subsidized meals
  • Partial reimbursement of public transport costs
  • Possibility of teleworking (90 days per year) and flexible organization of working hours
  • Partial payment of insurance costs

monthly gross salary 2200 euros