arXiv subjects

Marius Tucsnak

Publications and source records attributed to Marius Tucsnak.

15 recordsLinked to original sources

arXiv2026

Aclass of incrementally scattering-passive nonlinear systems

We investigate a special class of nonlinear infinite dimensional systems. These are obtained by subtracting a nonlinear maximal monotone (possibly multi-valued) operator M from the semigroup generator of a scattering passive linear system. While the linear system may have unbounded linear damping (for instance, boundary damping) which is only densely defined, the nonlinear damping operator M is assumed to be defined on the whole state space. We show that this new class of nonlinear infinite dimensional systems is well-posed and incrementally scattering passive. Our approach uses the theory of maximal monotone operators and the Crandall-Pazy theorem about nonlinear contraction semigroups, which we apply to a Lax-Phillips type nonlinear semigroup that represents the whole system.

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arXiv2026

Bergman-space regularity for the heat equation with white-noise boundary forcing

We introduce a Bergman-space framework for the study of boundary-forced heat equations and show that, in the one-dimensional case, boundary white noise gives rise to a sharp holomorphic regularity phenomenon. More precisely, we consider the heat equation on a bounded interval with Dirichlet or Neumann boundary conditions driven by independent white noises at the endpoints, and we prove that for every positive time the corresponding state extends holomorphically to a rhombus in the complex plane having the original interval as one of its diagonals. Moreover, the resulting process admits a continuous version with values in a scale of weighted Bergman spaces on that rhombus, depending on two parameters $\delta\in(0,1)$ and $\Theta\in\left(0,\frac{\pi}{4}\right)$. To our knowledge, this is the first systematic use of Bergman spaces as state spaces for parabolic equations with stochastic boundary forcing. We also prove that the result is optimal, in the sense that the conclusion fails at the critical values $\delta=0$ and $\Theta=\frac{\pi}{4}$.

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arXiv2025

Approximate Tracking Controllability of Systems with Quadratic Nonlinearities

Given a finite-dimensional time continuous control system and $\varepsilon>0$, we address the question of the existence of controls that maintain the corresponding state trajectories in the $\varepsilon$-neighborhood of any prescribed path in the state space. We investigate this property, called approximate tracking controllability, for linear and quadratic time invariant systems. Concerning linear systems, our answers are negative: by developing a systematic approach, we demonstrate that approximate tracking controllability of the full state is impossible even in a certain weak sense, except for the trivial situation where the control space is isomorphic to the state space. Motivated by these negative findings for linear systems, we focus on nonlinear dynamics. In particular, we prove weak approximate tracking controllability on any time horizon for a general class of systems with arbitrary linear part and quadratic nonlinear terms. The considered weak notion of approximate tracking controllability involves the relaxation metric. We underline the relevance of this weak setting by developing applications to coupled systems (including motion planning problems) and by remarking obstructions that would arise for natural stronger norms. The exposed framework yields global results even if the uncontrolled dynamics might exhibit singularities in finite time.

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arXiv2025

Relaxation enhancement by controlling incompressible fluid flows

We propose a PDE-controllability based approach to the enhancement of diffusive mixing for passive scalar fields. Unlike in the existing literature, our relaxation enhancing fields are not prescribed $\textit{ab initio}$ at every time and at every point of the spatial domain. Instead, we prove that time-dependent relaxation enhancing vector fields can be obtained as $\textit{state trajectories of control systems described by the incompressible Euler equations}$ either driven by finite-dimensional controls or by controls localized in space. The main ingredient of our proof is a new approximate controllability theorem for the incompressible Euler equations on $\mathbb{T}^2$, ensuring the approximate tracking of the full state all over the considered time interval. Combining this with a continuous dependence result yields enhanced relaxation for the passive scalar field. Another essential tool in our analysis is the exact controllability of the incompressible Euler system driven by spatially localized forces.

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arXiv2025

On an Initial Value Problem Describing the Small Oscillations of a Floating Cylinder

We study a coupled PDE-ODE system modeling the small oscillations of a floating cylinder interacting with small water waves. We consider the case when the floating body is an infinite circular cylinder, so that the equations of the free surface of the fluid can be written in one space dimension. The governing equations are formulated as an abstract evolution equation in a suitable Hilbert space, and we establish the well-posedness of the associated initial value problem. A key element of the proof is the analysis of a partial Dirichlet-to-Neumann map on an unbounded domain with a non-smooth boundary.

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arXiv2025

Global Exponential Stabilization for a Simplified Fluid-Particle Interaction System

This work considers a system coupling a viscous Burgers equation (aimed to describe a simplified model of $1D$ fluid flow) with the ODE describing the motion of a point mass moving inside the fluid. The point mass is possibly under the action of a feedback control. Our main contributions are that we prove two global exponential stability results. More precisely, we first show that the velocity field corresponding to the free dynamics case is globally exponentially stable. We next show that, in the presence of the feedback control both the velocity field and the distance from the mass point to a prescribed target position decay exponentially. The proofs of these results heavily rely on the use of a special test function allowing both to prove that the mass point stays away from the boundary and to construct a perturbed Lyapunov function.

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arXiv2024

LQR control for a system describing the interaction between a floating solid and the surrounding fluid

This paper studies an infinite time horizon LQR optimal control problem for a system describing, within a linear approximation, the vertical oscillations of a floating solid, coupled to the motion of the free boundary fluid on which it floats. The fluid flow is described by a viscous version of the linearized Saint-Venant equations (shallow water regime). The major difficulty we are facing is that the domain occupied by the fluid is unbounded so that the system is not exponentially stable. This fact firstly raises challenges in proving the wellposedness, requiring the combined use of analytic semigroup theory and of an interpolation technique. The main contribution of the paper is that we show that, in spite of the lack of exponential stabilizability, we can define a wellposed LQR problem for which a Riccati based approach to design feedback controls can be implemented.

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arXiv2024

Input-to-state stability for bilinear feedback systems

Input-to-state stability estimates with respect to small initial conditions and input functions for infinite-dimensional systems with bilinear feedback are shown. We apply the obtained results to controlled versions of a viscous Burger equation with Dirichlet boundary conditions, a Schrödinger equation, a Navier--Stokes system and a semilinear wave equation.

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arXiv2021

Shallow water waves generated by a floating object: a control theoretical perspective

We consider a control system describing the interaction of water waves with a partially immersed rigid body constraint to move only in the vertical direction. The fluid is modeled by the shallow water equations. The control signal is a vertical force acting on the floating body. We first derive the full governing equations of the fluid-body system in a water tank and reformulate them as an initial boundary value problem of a first-order evolution system. We then linearize the equations around the equilibrium state and we study its well-posedness. Finally we focus on the reachability and stabilizability of the linear system. Our main result asserts that, provided that the floating body is situated in the middle of the tank, any symmetric waves with appropriate regularity can be obtained from the equilibrium state by an appropriate control force. This implies, in particular, that we can project this system on the subspace of states with appropriate symmetry properties to obtain a reduced system which is approximately controllable and strongly stabilizable. Note that, in general, this system is not controllable (even approximately).

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arXiv2020

Stabilizability properties of a linearized water waves system

We consider the strong stabilization of small amplitude gravity water waves in a two dimensional rectangular domain. The control acts on one lateral boundary, by imposing the horizontal acceleration of the water along that boundary, as a multiple of a scalar input function $u$, times a given function $h$ of the height along the active boundary. The state $z$ of the system consists of two functions: the water level $ζ$ along the top boundary, and its time derivative $\dotζ$. We prove that for suitable functions $h$, there exists a bounded feedback functional $F$ such that the feedback $u=Fz$ renders the closed-loop system strongly stable. Moreover, for initial states in the domain of the semigroup generator, the norm of the solution decays like $(1+t)^{-\frac{1}{6}}$. Our approach uses a detailed analysis of the partial Dirichlet to Neumann and Neumann to Neumann operators associated to certain edges of the rectangular domain, as well as recent abstract non-uniform stabilization results by Chill, Paunonen, Seifert, Stahn and Tomilov (2019).

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arXiv2017

$L^p$-$L^q$ Maximal Regularity for some Operators Associated with Linearized Incompressible Fluid-Rigid Body Problems

We study an unbounded operator arising naturally after linearizing the system modelling the motion of a rigid body in a viscous incompressible fluid. We show that this operator is $\mathcal{R}$ sectorial in $L^q$ for every $q\in (1,\infty)$, thus it has the maximal $L^p$-$L^q$ regularity property. Moreover, we show that the generated semigroup is exponentially stable with respect to the $L^q$ norm. Finally, we use the results to prove the global existence for small initial data, in an $L^p$-$L^q$ setting, for the original nonlinear problem.

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arXiv2017

Mathematical Analysis of the Motion of a Rigid Body in a Compressible Navier-Stokes-Fourier Fluid

We study an initial and boundary value problem modelling the motion of a rigid body in a heat conducting gas. The solid is supposed to be a perfect thermal insulator. The gas is described by the compressible Navier-Stokes-Fourier equations, whereas the motion of the solid is governed by Newton's laws. The main results assert the existence of strong solutions, in an L p-L q setting, both locally in time and globally in time for small data. The proof is essentially using the maximal regularity property of associated linear systems. This property is checked by proving the R-sectoriality of the corresponding operators, which in turn is obtained by a perturbation method.

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arXiv2015

Perturbations of time optimal control problems for a class of abstract parabolic systems

In this work we study the asymptotic behavior of the solutions of a class of abstract parabolic time optimal control problems when the generators converge, in an appropriate sense, to a given strictly negative operator. Our main application to PDEs systems concerns the behavior of optimal time and of the associated optimal controls for parabolic equations with highly oscillating coefficients, as we encounter in homogenization theory. Our main results assert that, provided that the target is a closed ball centered at the origin and of positive radius, the solutions of the time optimal control problems for the systems with oscillating coefficients converge, in the usual norms, to the solution of the corresponding problem for the homogenized system. In order to prove our main theorem, we provide several new results, which could be of a broader interest, on time and norm optimal control problems.

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arXiv2011

On conditions for asymptotic stability of dissipative infinite-dimensional systems with intermittent damping

We study the asymptotic stability of a dissipative evolution in a Hilbert space subject to intermittent damping. We observe that, even if the intermittence satisfies a persistent excitation condition, if the Hilbert space is infinite-dimensional then the system needs not being asymptotically stable (not even in the weak sense). Exponential stability is recovered under a generalized observability inequality, allowing for time-domains that are not intervals. Weak asymptotic stability is obtained under a similarly generalized unique continuation principle. Finally, strong asymptotic stability is proved for intermittences that do not necessarily satisfy some persistent excitation condition, evaluating their total contribution to the decay of the trajectories of the damped system. Our results are discussed using the example of the wave equation, Schr\"odinger's equation and, for strong stability, also the special case of finite-dimensional systems.

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arXiv2010

Boundary feedback stabilization by piecewise constant time delay for the wave equation

For vibrating systems, a delay in the application of a feedback control can destroy the stabilizing effect of the control. In this paper we consider a vibrating string that is fixed at one end and stabilized with a boundary feedback with delay at the other end. We show that for certain feedback parameters the system is exponentially stable with constant delays of the form 4L/c, 8L/c, 12L/c ... Moreover, we show that the system is exponentially stable with piecewise constant delays that attain the values 4L/c and 8L/c.

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