92 research outputs found
Entropy-Preserving Coupling Conditions for One-dimensional Euler Systems at Junctions
This paper is concerned with a set of novel coupling conditions for the
one-dimensional Euler system with source terms at a junction of
pipes with possibly different cross-sectional areas. Beside conservation of
mass, we require the equality of the total enthalpy at the junction and that
the specific entropy for pipes with outgoing flow equals the convex combination
of all entropies that belong to pipes with incoming flow. Previously used
coupling conditions include equality of pressure or dynamic pressure. They are
restricted to the special case of a junction having only one pipe with outgoing
flow direction. Recently, Reigstad [SIAM J. Appl. Math., 75:679--702, 2015]
showed that such pressure-based coupling conditions can produce non-physical
solutions for isothermal flows through the production of mechanical energy. Our
new coupling conditions ensure energy as well as entropy conservation and also
apply to junctions connecting an arbitrary number of pipes with flexible flow
directions. We prove the existence and uniqueness of solutions to the
generalised Riemann problem at a junction in the neighbourhood of constant
stationary states which belong to the subsonic region. This provides the basis
for the well-posedness of the homogeneous and inhomogeneous Cauchy problems for
initial data with sufficiently small total variation.Comment: 17 pages, 2 figure
On the controllability of entropy solutions of scalar conservation laws at a junction via lyapunov methods
In this note, we prove a controllability result for entropy solutions of scalar conservation laws on a star-shaped graph. Using a Lyapunov-type approach, we show that, under a monotonicity assumption on the flux, if u and v are two entropy solutions corresponding to different initial data and same in-flux boundary data (at the exterior nodes of the star-shaped graph), then u ≡ v for a sufficiently large time. In order words, we can drive u to the target profile v in a sufficiently large control time by inputting the trace of v at the exterior nodes as in-flux boundary data for u. This result can also be shown to hold on tree-shaped networks by an inductive argument. We illustrate the result with some numerical simulationsThis work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement NO: 694126-DyCon), the Air Force Office of Scientific Research (AFOSR) under Award NO: FA9550-18-1-0242, the Grant MTM2017-92996-C2-1-R COSNET of MINECO (Spain), the Alexander von Humboldt-Professorship program, the European Unions Horizon 2020 research and innovation programme under the Marie SklodowskaCurie grant agreement No.765579-ConFlex, and the Transregio 154 Project “Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks” of the Deutsche Forschungsgemeinschaft. N. De Nitti is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilita e le loro ` Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). We gratefully acknowledge M. Musch for implementing the numerical simulations of Section 4. We also thank B. Andreianov, J.-A. Barcena-Petisco, G. M. Coclite, C. Donadello, and V. Perrollaz for helpful ´ conversations related to the topic of this work. Finally, we express our gratitude to the anonymous referees for their careful reports, which greatly improved the quality of the manuscrip
Switching rules for stabilization of linear systems of conservation laws
International audienceIn this paper, the exponential convergence in L 2-norm is analyzed for a class of switched linear systems of conservation laws. The boundary conditions are subject to switches. We investigate the problem of synthesizing stabilizing switching controllers. By means of Lyapunov techniques, three control strategies are developed based on steepest descent selection, possibly combined with a hysteresis and a low-pass filter. For the first strategy we show the global exponential stabilizability, but no result for the existence and uniqueness of trajectories can be stated. For the other ones, the problem is shown to be well posed and global exponential convergence can be obtained. Moreover, we consider the robustness issues for these switching rules in presence of measurement noise. Some numerical examples illustrate our approach and show the merits of the proposed strategies. Particularly, we have developped a model for a network of open channels, with switching controllers in the gate operations
Lyapunov techniques for stabilization of switched linear systems of conservation laws
http://cdc2013.units.it/International audienceIn this paper, the exponential stability in L2 - norm is investigated for a class of switched linear systems of conservation laws. The state equations and the boundary conditions are both subject to switching. We consider the problem of synthesizing stabilizing switching controllers. By means of Lyapunov techniques, three control strategies are developed based on steepest descent selection, possibly combined with a hysteresis and a low-pass filter. Some numerical examples are considered to illustrate our approach and to show the merits of the proposed strategies
Event-based control of linear hyperbolic systems of conservation laws
International audienceIn this article, we introduce event-based boundary controls for 1-dimensional linear hyperbolic systems of conservation laws. Inspired by event-triggered controls developed for finite-dimensional systems, an extension to the infinite dimensional case by means of Lyapunov techniques, is studied. The main contribution of the paper lies in the definition of two event-triggering conditions, by which global exponential stability and well-posedness of the system under investigation is achieved. Some numerical simulations are performed for the control of a system describing traffic flow on a roundabout
Event-triggered boundary control of constant-parameter reaction-diffusion PDEs: a small-gain approach
This paper deals with an event-triggered boundary control of
constant-parameters reaction-diffusion PDE systems. The approach relies on the
emulation of backstepping control along with a suitable triggering condition
which establishes the time instants at which the control value needs to be
sampled/updated. In this paper, it is shown that under the proposed
event-triggered boundary control, there exists a minimal dwell-time
(independent of the initial condition) between two triggering times and
furthermore the well-posedness and global exponential stability are guaranteed.
The analysis follows small-gain arguments and builds on recent papers on
sampled-data control for this kind of PDE. A simulation example is presented to
validate the theoretical results.Comment: 10 pages, to be submitted to Automatic
Entropy-Preserving Coupling of Hierarchical Gas Models
This paper is concerned with coupling conditions at junctions for transport
models which differ in their fidelity to describe transient flow in gas
pipelines. It also includes the integration of compressors between two pipes
with possibly different models. A hierarchy of three one-dimensional gas
transport models is built through the 3x3 polytropic Euler equations, the 2x2
isentropic Euler equations and a simplified version of it for small velocities.
To ensure entropy preservation, we make use of the novel entropy-preserving
coupling conditions recently proposed by Lang and Mindt [Netw. Heterog. Media,
13:177-190, 2018] and require the continuity of the total enthalpy at the
junction and that the specific entropy for pipes with outgoing flow equals the
convex combination of all entropies that belong to pipes with incoming flow. We
prove the existence and uniqueness of solutions to generalised Riemann problems
at a junction in the neighbourhood of constant coupling functions and
stationary states which belong to the subsonic region. This provides the basis
for the well-posedness of certain Cauchy problems for initial data with
sufficiently small total variation.Comment: 28 pages, 3 figures. arXiv admin note: text overlap with
arXiv:1704.0403
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