Linear Parabolic Systems Research Articles

Boundary null controllability of some multi-dimensional linear parabolic systems by the moment method

Boundary null controllability of some multi-dimensional linear parabolic systems by the moment method

Elastic flow of curves with partial free boundary

We consider a curve with boundary points free to move on a line in R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{\\mathbb {R}}}}^2$$\\end{document}, which evolves by the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-gradient flow of the elastic energy, that is, a linear combination of the Willmore and the length functional. For this planar evolution problem, we study the short and long-time existence. Once we establish under which boundary conditions the PDE’s system is well-posed (in our case the Navier boundary conditions), employing the Solonnikov theory for linear parabolic systems in Hölder space, we show that there exists a unique flow in a maximal time interval [0, T). Then, using energy methods we prove that the maximal time is T=+∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T= + \\infty $$\\end{document}.

Cooperative Control and Performance Evaluation of a Linear MIMO Parabolic Spatiotemporal Dynamic System on Directed and Switching Topologies.

The current work concerns on cooperative control of exponential stabilization and control performance improvement in the spatial domain for a linear spatiotemporal dynamic system associated with multiple control actuators and multiple collaborative measurement sensors. By assuming that the system dynamics is modeled by a MIMO parabolic partial differential equation (PPDE) and each sensor can share measurement information with its topological neighbors in directed and switching topological networks, a cooperative control protocol is proposed to achieve the control aim of this article. With the help of a combination of multiagent consensus theory, Lyapunov's method, and integral inequality technique, sufficient conditions are presented for the closed-loop exponential stability of the PPDE in the norm |·|2 . Moreover, some performance indexes are defined to evaluate the closed-loop control performance improvement in the spatial domain. Extensive simulation results are finally presented for a simple numerical example and a practical heat treatment process to verify the effectiveness and performance improvement capability of the proposed cooperative control protocol.

Well posedness of linear parabolic partial differential equations posed on a star-shaped network with local time Kirchhoff's boundary condition at the vertex

The main purpose of this work is to provide an existence and uniqueness result for the solution of a linear parabolic system posed on a star-shaped network, which presents a new type of Kirchhoff's boundary transmission condition at the junction. This new type of Kirchhoff's condition - that we decide to call here local-time Kirchhoff's condition - induces a dynamical behavior with respect to an external variable that may be interpreted as a local time parameter, designed to drive the system only at the singular point of the network. The seeds of this study point towards a theoretical inquiry of a particular generalization of Walsh's random spider motions, whose spinning measures would select the available directions according to the local time of the motion at the junction of the network.

Stackelberg exact controllability of a class of nonlocal parabolic equations

This paper deals with a multi-objective control problem for a class of nonlocal parabolic equations, where the non-locality is expressed through an integral kernel. We present the Stackelberg strategy that combines the concepts of controllability to trajectories with optimal control. The strategy involves two controls: a main control (the leader) and a secondary control (the follower). The leader solves a controllability to trajectories problem which consists to drive the state of the system to a prescribed target at a final time while the follower solves an optimal control problem which consists to minimize a given cost functional. The paper considers two cases: in the first case, both the leader and the follower act in the interior of the domain, and in the second case, the leader acts in the interior of the domain and the follower acts on a small part of the boundary. These results are applied to both linear and nonlinear nonlocal parabolic systems.

Finite-dimensional adaptive observer design for linear parabolic systems

A new finite-dimensional adaptive observer is proposed for some linear parabolic systems. The observer is based on the modal decomposition approach and uses a classical persistent excitation condition to ensure practical exponential convergence of both states and parameters estimation.

Lyapunov-based stabilization mobile control design of linear parabolic PDE systems

This article considers the exponential mobile stabilization control of linear parabolic partial differential equation (PDE) systems. Firstly, a stabilization scheme based on static output feedback (SOF) control and mobile actuator/sensor guidance is proposed. Next, the operator theory is used to discuss the well-posedness of closed-loop PDE systems. Thus, the SOF control and mobile guidance design approach is simultaneously proposed to guarantee that the closed-loop PDE system is exponentially stable. In final, the effectiveness of the design method is verified by numerical simulation.

Controllability of a fractional output linear system with constraints

AbstractThe primary objective of this research is to generalize the concept of controllability with constraints to cases where the output function is a Riemann–Liouville fractional derivative of order More precisely, if , we obtain the enlarged controllability of the system state and enlarged controllability of the gradient state of the system is obtained with Moreover, we aim to characterize the optimal control that enables us to guide a linear parabolic system to a fractional final state within the evolution system's domain. We adopt two approaches to solve the aforementioned problem: The first is based on the Lagrangian technique, and the second one employs the subdifferential theory. Subsequently, we develop an algorithm to compute the minimum energy control. Finally, we provide numerical simulations to illustrate the validity of the obtained theoretical results.

Convergence to equilibrium for linear parabolic systems coupled by matrix‐valued potentials

AbstractWe consider systems of parabolic linear equations, subject to Neumann boundary conditions on bounded domains in , that are coupled by a matrix‐valued potential V, and investigate under which conditions each solution to such a system converges to an equilibrium as . While this is clearly a fundamental question about systems of parabolic equations, it has been studied, up to now, only under certain positivity assumptions on the potential V. Without positivity, Perron–Frobenius theory cannot be applied and the problem is seemingly wide open. In this paper, we address this problem for all potentials that are ‐dissipative for some . While the case can be treated by classical Hilbert space methods, the matter becomes more delicate for . We solve this problem by employing recent spectral theoretic results that are closely tied to the geometric structure of ‐spaces.

The principal eigenvalue for partially degenerate and periodic reaction-diffusion systems with time delay

In this paper, we first establish the theory of principal eigenvalues for a large class of partially degenerate, linear and periodic parabolic cooperative systems with time delay. Then we apply these theoretical results to study the global dynamics of a blacklegged tick Ixodes scapularis population model. To present a threshold-type result in terms of basic reproduction ratio R0 for such a model, we also extend the earlier theory of R0 to abstract functional differential equations with time-delayed internal transition.

Robust Boundary Controller Design with Proportional Integral Observer for a Linear 1D Heat Equation*

The problem of state estimation with boundary disturbance reconstruction and output regulation for linear 1D parabolic systems with a boundary measurement and unknown constant disturbance input on the opposite boundary is addressed. As the disturbance is unmatched it cannot be directly compensated by the measurement. The problem is addressed with an approach that consists of a backstepping observer for the nominal part with an asymptotic compensation term for the disturbance reconstruction and control offset compensation. The main difference to similar studies relies in the fact that the state space does not need to be extended, leading to a reduced-order observer setup. The effectiveness of the approach is illustrated using numerical simulations.

Approximate and Exact Controllability of Switched Infinite-Dimensional Linear Systems

This article addresses the approximate and exact controllability issues of switched infinite-dimensional linear systems. First, a controllability map that is related to switching sequence and a controllability Gramian operator are proposed, based on which two necessary and sufficient conditions are established for approximate and exact controllability, respectively. Second, it shows that the obtained conditions can degenerate into the existing controllability criteria of infinite-dimensional linear systems, switched finite-dimensional linear systems, and a single finite-dimensional linear system. Finally, the new results are applied to analyze the approximate controllability of switched linear parabolic systems and the exact controllability of switched linear wave systems. Two examples are given to illustrate the effectiveness of the proposed approaches.

Results on Boundary Control for Parabolic Systems Using Backstepping Method

This paper focuses on the stabilization problem for the linear parabolic system using the backstepping method. The exponentially stability results for considered parabolic system are derived in two cases with Dirichlet and Neumann local terms. Also, the boundary conditions for the problem is assumed to be mixed or Robin-type boundary conditions. The main aim is to achieve the stability of the considered system using the backstepping method with help of Volterra integral transformation. The explicit solutions of kernel functions in integral transformation is obtained by using Laplace transform and designed a boundary control law to the closed-loop system. Finally, the effectiveness and applicability of the derived results are validated through a single-species pattern generation model.

Internal stabilization of an underactuated linear parabolic system via modal decomposition

This work concerns the internal stabilization of underactuated linear systems of m heat equations in cascade, where the control is placed internally in the first equation only and the diffusion coefficients are distinct. Combining the modal decomposition method with a recently introduced state-transformation approach for observation problems, a proportional-type stabilizing control is given explicitly. It is based on a transformation for the ODE system corresponding to the comparatively unstable modes into a target one, where calculation of the stabilization law is independent of the arbitrarily large number of them and it is achieved by solving generalized Sylvester equations recursively. This provides a finite-dimensional counterpart of a recently introduced infinite-dimensional one, which led to Lyapunov stabilization. The present approach answers to the problem of stabilization with actuators not appearing in all the states and when boundary control results do not apply.

Influence of numerical discretizations on hitting probabilities for linear stochastic parabolic systems

This paper investigates the influence of standard numerical discretizations on hitting probabilities for a system of linear stochastic parabolic equations driven by space-time white noises. We establish lower and upper bounds for hitting probabilities of the associated numerical solutions of both temporal and spatial semi-discretizations in terms of Bessel–Riesz capacity and Hausdorff measure, respectively. Moreover, the critical dimensions of both temporal and spatial semi-discretizations turn out to be halves of those of the exact solution. This reveals that there exist some Borel sets A such that the probability of the event that the paths of the numerical solution hit A cannot converge to that of the exact solution when the stepsizes vanish.

Modeling and Analysis of Energy/Exergy for Absorber Pipes of Linear Parabolic Concentrating Systems

In this paper, the physical parameters of the absorber pipe of a linear parabolic collector have been investigated. The types of solar collectors, specifically the linear parabolic collector, have been comprehensively studied. Then, the mathematical model of heat transfer in the absorber pipe of the collector has been presented based on valid references. Numerical solutions of the equations related to the absorber pipe were performed by MATLAB software, and the effects of the physical parameters of the absorber pipe on its efficiency were investigated. The results show that increasing the length of the absorber pipe causes a nonlinear decrease in the efficiency of the absorber pipe. One of the important results is the increase in fluid temperature due to the increase in the diameter of the adsorbent tube, which increases the diameter of the fluid temperature by 60 K, in which the parameter increases the efficiency by 0.38%.

Regularity of a parabolic system involving curl

This note presents a regularity result with proof for an initial-boundary value problem of a linear parabolic system involving curl of the unknown vector field, subjected to the boundary condition of prescribing the tangential component of the solution.

$$L^{p}$$ Theory for the Interaction Between the Incompressible Navier–Stokes System and a Damped Plate

We consider a viscous incompressible fluid governed by the Navier–Stokes system written in a domain where a part of the boundary can deform. We assume that the corresponding displacement follows a damped beam equation. Our main results are the existence and uniqueness of strong solutions for the corresponding fluid-structure interaction system in an \(L^p\)-\(L^q\) setting for small times or for small data. An important ingredient of the proof consists in the study of a linear parabolic system coupling the non stationary Stokes system and a damped plate equation. We show that this linear system possesses the maximal regularity property by proving the \({\mathcal {R}}\)-sectoriality of the corresponding operator. The proof of the main results is then obtained by an appropriate change of variables to handle the free boundary and a fixed point argument to treat the nonlinearities of this system.

Numerical Approximation of Port-Hamiltonian Systems for Hyperbolic or Parabolic PDEs with Boundary Control

We consider the design of structure-preserving discretization methods for the solution of systems of boundary controlled Partial Differential Equations (PDEs) thanks to the port-Hamiltonian formalism. We first provide a novel general structure of infinite-dimensional port-Hamiltonian systems (pHs) for which the Partitioned Finite Element Method (PFEM) straightforwardly applies. The proposed strategy is applied to abstract multidimensional linear hyperbolic and parabolic systems of PDEs. Then we show that instructional model problems based on the wave equation, Mindlin equation and heat equation fit within this unified framework. Secondly, we introduce the ongoing project SCRIMP (Simulation and Control of Interactions in Multi-Physics) developed for the numerical simulation of infinite-dimensional pHs. SCRIMP notably relies on the FEniCS open-source computing platform for the finite element spatial discretization. Finally, we illustrate how to solve the considered model problems within this framework by carefully explaining the methodology. As additional support, companion interactive Jupyter notebooks are available.

Elastic flow of networks: short-time existence result

In this paper we study the $$L^2$$ -gradient flow of the penalized elastic energy on networks of q-curves in $$\mathbb {R}^{n}$$ for $$q \ge 3$$ . Each curve is fixed at one end-point and at the other is joint to the other curves at a movable q-junction. For this geometric evolution problem with natural boundary condition we show the existence of smooth solutions for a (possibly) short interval of time. Since the geometric problem is not well-posed, due to the freedom in reparametrization of curves, we consider a fourth-order non-degenerate parabolic quasilinear system, called the analytic problem, and show first a short-time existence result for this parabolic system. The proof relies on applying Solonnikov’s theory on linear parabolic systems and Banach fixed point theorem in proper Holder spaces. Then the original geometric problem is solved by establishing the relation between the analytical solutions and the solutions to the geometrical problem.