407 research outputs found
Explicit output-feedback boundary control of reaction-diffusion PDEs on arbitrary-dimensional balls
This paper introduces an explicit output-feedback boundary feedback law that stabilizes
an unstable linear constant-coefficient reaction-diffusion equation on an n-ball (which in 2-D reduces to
a disk and in 3-D reduces to a sphere) using only measurements from the boundary. The backstepping
method is used to design both the control law and a boundary observer. To apply backstepping the
system is reduced to an infinite sequence of 1-D systems using spherical harmonics. Well-posedness and
stability are proved in the L2 and H1 spaces. The resulting control and output injection gain kernels are
the product of the backstepping kernel used in control of one-dimensional reaction-diffusion equations
and a function closely related to the Poisson kernel in the n-ball.Ministerio de Economía y Competitividad MTM2015-65608-
Boundary control of a singular reaction-diffusion equation on a disk
Recently, the problem of boundary stabilization for unstable linear
constant-coefficient reaction-diffusion equation on N-balls has been solved by
means of the backstepping method. However, the extension of this result to
spatially-varying coefficients is far from trivial. This work deals with
radially-varying reaction coefficients under revolution symmetry conditions on
a disk (the 2-D case). Under these conditions, the equations become singular in
the radius. When applying the backstepping method, the same type of singularity
appears in the backstepping kernel equations. Traditionally, well-posedness of
the kernel equations is proved by transforming them into integral equations and
then applying the method of successive approximations. In this case, the
resulting integral equation is singular. A successive approximation series can
still be formulated, however its convergence is challenging to show due to the
singularities. The problem is solved by a rather non-standard proof that uses
the properties of the Catalan numbers, a well-known sequence frequently used in
combinatorial mathematics.Comment: Submitted to the 2nd IFAC Workshop on Control of Systems Governed by
Partial Differential Equations (CPDE 2016
Inertial manifolds and finite-dimensional reduction for dissipative PDEs
These notes are devoted to the problem of finite-dimensional reduction for
parabolic PDEs. We give a detailed exposition of the classical theory of
inertial manifolds as well as various attempts to generalize it based on the
so-called Man\'e projection theorems. The recent counterexamples which show
that the underlying dynamics may be in a sense infinite-dimensional if the
spectral gap condition is violated as well as the discussion on the most
important open problems are also included.Comment: This manuscript is an extended version of the lecture notes taught by
the author as a part of the crash course in the Analysis of Nonlinear PDEs at
Maxwell Center for Analysis and Nonlinear PDEs (Edinburgh, November, 8-9,
2012
Boundary control of a singular reaction-diffusion equation on a disk
Recently, the problem of boundary stabilization for unstable linear constant-coefficient reaction-diffusion equation on n-balls (in particular, disks and spheres) has been solved by means of the backstepping method. However, the extension of this result to spatially-varying coefficients is far from trivial. As a first step, this work deals with radially-varying reaction coefficients under revolution symmetry conditions on a disk (the 2-D case). Under these conditions, the equations become singular in the radius. When applying the backstepping method, the same type of singularity appears in the backstepping kernel equations. Traditionally, well-posedness of the kernel equations is proved by transforming them into integral equations and then applying the method of successive approximations. In this case, the resulting integral equation is singular. A successive approximation series can still be formulated, however its convergence is challenging to show due to the singularities. The problem is solved by a rather non-standard proof that uses the properties of the Catalan numbers, a well-known sequence frequently appearing in combinatorial mathematics.Ministerio de Economía y Competitividad MTM2015-65608-
Machine Learning Accelerated PDE Backstepping Observers
State estimation is important for a variety of tasks, from forecasting to
substituting for unmeasured states in feedback controllers. Performing
real-time state estimation for PDEs using provably and rapidly converging
observers, such as those based on PDE backstepping, is computationally
expensive and in many cases prohibitive. We propose a framework for
accelerating PDE observer computations using learning-based approaches that are
much faster while maintaining accuracy. In particular, we employ the
recently-developed Fourier Neural Operator (FNO) to learn the functional
mapping from the initial observer state and boundary measurements to the state
estimate. By employing backstepping observer gains for previously-designed
observers with particular convergence rate guarantees, we provide numerical
experiments that evaluate the increased computational efficiency gained with
FNO. We consider the state estimation for three benchmark PDE examples
motivated by applications: first, for a reaction-diffusion (parabolic) PDE
whose state is estimated with an exponential rate of convergence; second, for a
parabolic PDE with exact prescribed-time estimation; and, third, for a pair of
coupled first-order hyperbolic PDEs that modeling traffic flow density and
velocity. The ML-accelerated observers trained on simulation data sets for
these PDEs achieves up to three orders of magnitude improvement in
computational speed compared to classical methods. This demonstrates the
attractiveness of the ML-accelerated observers for real-time state estimation
and control.Comment: Accepted to the 61st IEEE Conference on Decision and Control (CDC),
202
On an exponential attractor for a class of PDEs with degenerate diffusion and chemotaxis
In this article we deal with a class of strongly coupled parabolic systems
that encompasses two different effects: degenerate diffusion and chemotaxis.
Such classes of equations arise in the mesoscale level modeling of biomass
spreading mechanisms via chemotaxis. We show the existence of an exponential
attractor and, hence, of a finite-dimensional global attractor under certain
'balance conditions' on the order of the degeneracy and the growth of the
chemotactic function
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