4,356 research outputs found
Exponential Stability of Switched Linear Hyperbolic Initial-Boundary Value Problems
We consider the initial-boundary value problem governed by systems of linear hyperbolic partial differential equations in the canonical diagonal form and study conditions for exponential stability when the system discontinuously switches between a finite set of modes. The switching system is fairly general in that the system matrix functions as well as the boundary conditions may switch in time. We show how the stability mechanism developed for classical solutions of hyperbolic initial boundary value problems can be generalized to the case in which weaker solutions become necessary due to arbitrary switching. We also provide an explicit dwell-time bound for guaranteeing exponential stability of the switching system when, for each mode, the system is exponentially stable. Our stability conditions only depend on the system parameters and boundary data. These conditions easily generalize to switching systems in the nondiagonal form under a simple commutativity assumption. We present tutorial examples to illustrate the instabilities that can result from switching
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
Control of Homodirectional and General Heterodirectional Linear Coupled Hyperbolic PDEs
Research on stabilization of coupled hyperbolic PDEs has been dominated by
the focus on pairs of counter-convecting ("heterodirectional") transport PDEs
with distributed local coupling and with controls at one or both boundaries. A
recent extension allows stabilization using only one control for a system
containing an arbitrary number of coupled transport PDEs that convect at
different speeds against the direction of the PDE whose boundary is actuated.
In this paper we present a solution to the fully general case, in which the
number of PDEs in either direction is arbitrary, and where actuation is applied
on only one boundary (to all the PDEs that convect downstream from that
boundary). To solve this general problem, we solve, as a special case, the
problem of control of coupled "homodirectional" hyperbolic linear PDEs, where
multiple transport PDEs convect in the same direction with arbitrary local
coupling. Our approach is based on PDE backstepping and yields solutions to
stabilization, by both full-state and observer-based output feedback,
trajectory planning, and trajectory tracking problems
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
An excision scheme for black holes in constrained evolution formulations: spherically symmetric case
Excision techniques are used in order to deal with black holes in numerical
simulations of Einstein equations and consist in removing a topological sphere
containing the physical singularity from the numerical domain, applying instead
appropriate boundary conditions at the excised surface. In this work we present
recent developments of this technique in the case of constrained formulations
of Einstein equations and for spherically symmetric spacetimes. We present a
new set of boundary conditions to apply to the elliptic system in the
fully-constrained formalism of Bonazzola et al. (2004), at an arbitrary
coordinate sphere inside the apparent horizon. Analytical properties of this
system of boundary conditions are studied and, under some assumptions, an
exponential convergence toward the stationary solution is exhibited for the
vacuum spacetime. This is verified in numerical examples, together with the
applicability in the case of the accretion of a scalar field onto a
Schwarzschild black hole. We also present the successful use of the excision
technique in the collapse of a neutron star to a black hole, when excision is
switched on during the simulation, after the formation of the apparent horizon.
This allows the accretion of matter remaining outside the excision surface and
for the stable long-term evolution of the newly formed black hole.Comment: 14 pages, 9 figures. New section added and changes included according
to published articl
Minimum time control of heterodirectional linear coupled hyperbolic PDEs
We solve the problem of stabilization of a class of linear first-order
hyperbolic systems featuring n rightward convecting transport PDEs and m
leftward convecting transport PDEs. Using the backstepping approach yields
solutions to stabilization in minimal time and observer based output feedback
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