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Well-posedness and passivity for a class of bilinear control systems

This paper establishes the well-posedness and continuous dependence of mild and classical solutions for a class of bilinear infinite-dimensional systems in Banach spaces, demonstrates passivity in Hilbert spaces for co-located outputs, and applies these theoretical results to the bilinear Schrödinger equation, the Fokker–Planck equation, and district heating systems.

Original authors: Abdelhakim Dahmani, Hannes Gernandt, René Hosfeld, Timo Reis, Tolgahan Tasci

Published 2026-08-13

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CC BY 4.0

Original authors: Abdelhakim Dahmani, Hannes Gernandt, René Hosfeld, Timo Reis, Tolgahan Tasci

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to steer a very complex machine, like a giant robot or a fleet of drones. In the world of engineering, there are two main ways to describe how these machines move. The first is "linear," where if you push the button twice as hard, the machine moves twice as fast. It's predictable, like pushing a shopping cart. The second is "nonlinear," where things get messy and chaotic; a tiny push might send the machine spinning wildly, or a huge push might do almost nothing. These are the hard ones to control.

But there is a fascinating middle ground called "bilinear." Think of it like a dance between the machine's current position and your control inputs. The machine's movement depends on where it is and what you tell it to do, but in a specific, structured way. It's not as simple as a shopping cart, but it's not as wild as a storm. This middle ground shows up everywhere: in the quantum world where we try to control atoms, in the pipes that heat our cities, and in the diffusion of particles. The big challenge for scientists is that while we know how to control the simple linear machines and have some tools for the chaotic nonlinear ones, the "bilinear" dance has been tricky to master, especially when the machines are huge and described by complex math. We need to know two things: first, does the machine actually follow our instructions without breaking down (mathematically called "well-posedness"), and second, does it obey the laws of energy conservation so we can keep it stable (called "passivity")?

This paper steps into that middle ground to tidy up the rules for these bilinear systems. The authors, a team of mathematicians, tackle a class of abstract systems that are infinite in size (meaning they describe things like heat spreading through a pipe or a wave function in space, rather than just a few moving parts). Their main job is to prove that if you set up these systems correctly, they will behave nicely: a unique solution will exist, and if you tweak the starting conditions or the controls slightly, the result won't suddenly explode or change wildly. They call this "well-posedness."

Once they proved the systems behave, they moved on to the energy question. They showed that for these bilinear systems, if you define the "output" (what the system tells you back) in a specific way—matching it up with the "input" (what you put in)—the system acts like a passive energy bucket. It can't create energy out of nowhere; it can only store what you give it or lose it to the environment. This is a huge deal because it means engineers can use this energy balance to design controllers that keep these complex systems stable, like keeping a quantum computer from falling apart or ensuring a district heating pipe doesn't overheat.

The paper doesn't just stay in the realm of abstract math; it proves these rules work for real-world examples. They applied their new theory to the Schrödinger equation (which describes how quantum particles move), the Fokker–Planck equation (which tracks how particles drift and spread), and a model for a district heating pipe. In all these cases, they showed that the systems are well-behaved and passive, provided the controls are chosen carefully. They didn't just guess; they provided rigorous mathematical proofs that these systems have unique solutions and that their energy balances hold true. This gives a solid foundation for future engineers to build better, safer, and more efficient control systems for everything from quantum tech to city infrastructure.

Technical Summary: Well-Posedness and Passivity for a Class of Bilinear Control Systems

Problem Statement
The paper addresses the analysis of abstract infinite-dimensional bilinear control systems, which occupy an intermediate position between linear and genuinely nonlinear systems. These systems arise in diverse applications such as district heating, quantum control, and diffusion processes. The specific mathematical challenge lies in establishing the well-posedness (existence, uniqueness, and continuous dependence on data) and passivity properties for systems where the control operators may be unbounded. While passivity is well-understood for linear infinite-dimensional systems and finite-dimensional bilinear systems, the theory for infinite-dimensional bilinear systems with unbounded control and observation operators remains underdeveloped. A primary difficulty is defining a suitable co-located output and handling the non-autonomous nature of the system, where the generator depends on time through the control input.

Methodology
The authors employ a functional-analytic framework within Banach and Hilbert spaces.

System Formulation: The study focuses on the abstract system $(\Sigma)$ :
$\dot{x}(t) = Ax(t) + B_1 F(u_1(t), x(t)) + B_2 u_2(t), \quad x(0) = x_0$
where $A$ generates a $C_0$ -semigroup on a Banach space $X$ , $B_1$ and $B_2$ are control operators mapping into the extrapolation space $X_{-1}$ , and $F$ is a bounded bilinear mapping.
Admissibility: The analysis relies heavily on the concept of $L^p$ -admissible control operators. This allows the treatment of unbounded operators $B_1$ and $B_2$ (typical in boundary control) by ensuring that the input-to-state map remains well-defined in the state space $X$ .
Solution Theory:
- Mild Solutions: Existence and uniqueness of global mild solutions are established using fixed-point arguments in the space of continuous functions, leveraging the admissibility of the control operators.
- Classical Solutions: The authors extend the existence results to classical solutions ( $C^1$ in time) by imposing higher regularity on the inputs ( $W^{1,p}_{loc}$ ) and a compatibility condition on the initial data. This involves a detailed analysis of the derivative of the solution and the use of Gronwall's inequality to ensure global existence.
- Continuous Dependence: A priori estimates are derived to prove that the mild solution depends continuously on the initial state and the input functions in the $C([0, T]; X)$ topology.
Passivity Analysis:
- The framework is specialized to Hilbert spaces to define a natural energy storage function and co-located output.
- For bounded operators, passivity is derived directly from the dissipativity of the generator $A$ .
- For unbounded operators, the authors utilize fractional interpolation and extrapolation spaces ( $X_{1/2}$ and $X_{-1/2}$ ) associated with a self-adjoint, negative operator $A$ . This allows the formulation of an energy balance identity using duality pairings, even when individual terms in the equation do not reside in $X$ .

Key Contributions and Results

Well-Posedness in Banach Spaces: The paper provides a rigorous existence and uniqueness theorem for global mild solutions of abstract bilinear systems with admissible control operators. Furthermore, it establishes conditions under which these solutions are classical, requiring inputs in Sobolev spaces and a specific compatibility condition.
Continuous Dependence: It is proven that the solution map is continuous with respect to perturbations in the initial data and the control inputs, a crucial property for control design and stability analysis.
Passivity Theorem: The main contribution regarding passivity is a theorem stating that for systems in Hilbert spaces with admissible control operators, the system is passive with respect to a suitably defined co-located output. The storage function is identified as the squared norm of the state (or a weighted norm in the case of unbounded operators).
- For bounded operators, the result holds for classical solutions and extends to mild solutions via density arguments.
- For unbounded operators (specifically where $A$ is self-adjoint and negative), the result is established using fractional spaces, showing that the energy balance holds in the sense of duality pairings between $X_{-1/2}$ and $X_{1/2}$ .
Applications: The theoretical results are applied to verify the passivity of three specific physical models:
- Bilinear Schrödinger Equation: Demonstrated to be passive with bounded multiplication operators.
- Fokker–Planck Equation: Passivity is shown by transforming the system via an isomorphism to a form where the generator is self-adjoint and negative, allowing the application of the unbounded operator passivity theorem.
- District Heating Pipe: The bilinear model describing temperature evolution in a pipe is shown to be passive, with the flow velocity acting as the bilinear control term.

Significance and Claims
The paper claims to contribute to the theory of infinite-dimensional bilinear systems by bridging the gap between well-posedness and passivity in the presence of unbounded control operators. The authors state that their results provide a foundation for passivity-based feedback design and stabilization of such systems. By incorporating the bilinear structure into the passivity framework without sacrificing well-posedness, the work complements existing theories on finite-dimensional bilinear systems, time-varying passive systems, and port-Hamiltonian systems with boundary control. The authors modestly note that while the theory is developed for Hilbert spaces, the extension to Banach spaces for passivity remains a delicate open problem, with only partial results currently available in the literature. The work does not propose new experimental setups but rather provides the mathematical justification for the energy-based analysis of existing physical models.

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