DAE-based Contracts

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In the field of systems engineering, contract-based design [1] is a modular methodology that enables independent component development while ensuring correct system-wide integration. A specific instance is the assume--guarantee contract: Contract = (A, G). Here, Assumptions A describe what the component expects from its environment, while Guarantees G specify what the component promises to deliver, provided that the assumptions hold. Formally, a contract can be represented as an implication:

                                                                E ≼ A ⇒ (Σ ∧ E)≼ G

meaning that if the environment satisfies the assumptions A, the system under the environment must ensure the guarantees G. This contract-based perspective supports modular and compositional system design.

In recent years, the design and analysis of large-scale control systems have become increasingly challenging. To address this, contract-based design has been introduced into the control systems domain. Two notable studies [2,3] develop contract frameworks for linear time-invariant (LTI) control systems:

                                                  Σ: \dot{x} = A x + B u, \\
                                                               y = C x + D u.

In [2], the classical behavioral theory introduced by Jan Willems is used to formalize key contract-theoretic notions such as assumptions, guarantees, refinement, and composition---for Σ. In [3], geometric control theory is employed to define simulation relations between two control systems, providing a foundation for implementing assume--guarantee contracts.A contract-based control design algorithm is then proposed based on these results.

[1] A. Benveniste, B. Caillaud, et all, Contracts for System Design. Foundations and Trends in Electronic Design Automation, Now Publishers, 2018.

[2] B. M. Shali, A. van der Schaft, and B. Besselink, “Composition of behavioural assume-guarantee contracts,” IEEE Transactions on Automatic Control, pp. 1–16, 2022.

[3] B. M. Shali, A. van der Schaft, and B. Besselink, “Design and control for implementation of
simulation-based assume-guarantee contracts,” IEEE Transactions on Automatic Control, pp. 1–15, 2025.

[4] Y. Chen, and W. Respondek. "Geometric analysis of differential-algebraic equations via linear control theory." SIAM Journal on Control and Optimization 59.1 (2021): 103-130.

[5] F. L. Lewis "A survey of linear singular systems." Circuits, systems and signal processing 5.1 (1986): 3-36.

The goal of this internship is to extend contract theory to linear differential--algebraic equations (DAEs):

                            E\dot{z} = A z.

As a modular modeling approach derived from first-principle physics, DAEs frequently appear in constrained mechanical systems, power networks, and analog circuit design. Mathematically, DAEs offer several potential advantages for contract-based analysis:

• System interconnections can be naturally expressed as algebraic equations, supporting a compositional framework.
• DAEs treat all variables uniformly---states, inputs, and outputs---aligning well with the behavioral approach.
   
• The geometric analysis of DAEs is well established [4] [5], providing effective tools for describing relations between systems and specifications.
  
  

• Conduct literature reviews on both contract theory and linear DAEs.

• Define notions from contract theory for DAE systems and develop theories on their verifications.

• Apply the proposed DAE-based contracts to simple examples.

• Participate actively in team meetings and brainstorming sessions.

• Strong mathematical reasoning and problem-solving skills.
  
  
• Familiarity with one or more of the following topics is an advantage: DAEs, contract theory, geometric control

• Scientific curiosity, autonomy, and the ability to work independently

• Subsidized meals
• Partial reimbursement of public transport costs
• Leave: 7 weeks of annual leave + 10 extra days off due to RTT (statutory reduction in working hours) + possibility of exceptional leave (sick children, moving home, etc.)

Gratification