14 research outputs found
Backstepping PDE Design: A Convex Optimization Approach
Abstract\u2014Backstepping design for boundary linear PDE is
formulated as a convex optimization problem. Some classes of
parabolic PDEs and a first-order hyperbolic PDE are studied,
with particular attention to non-strict feedback structures. Based
on the compactness of the Volterra and Fredholm-type operators
involved, their Kernels are approximated via polynomial
functions. The resulting Kernel-PDEs are optimized using Sumof-
Squares (SOS) decomposition and solved via semidefinite
programming, with sufficient precision to guarantee the stability
of the system in the L2-norm. This formulation allows optimizing
extra degrees of freedom where the Kernel-PDEs are included
as constraints. Uniqueness and invertibility of the Fredholm-type
transformation are proved for polynomial Kernels in the space
of continuous functions. The effectiveness and limitations of the
approach proposed are illustrated by numerical solutions of some
Kernel-PDEs
Two sided boundary stabilization of two linear hyperbolic PDEs in minimum time
International audienceâ We solve the problem of stabilizing two coupled linear hyperbolic PDEs using actuation at both boundary of the spatial domain in minimum time. We design a novel Fredholm transformation similarly to backstepping approaches. This yields an explicit full-state feedback law that achieves the theoretical lower bound for convergence time to zero
Delay-Adaptive Control of First-order Hyperbolic PIDEs
We develop a delay-adaptive controller for a class of first-order hyperbolic
partial integro-differential equations (PIDEs) with an unknown input delay. By
employing a transport PDE to represent delayed actuator states, the system is
transformed into a transport partial differential equation (PDE) with unknown
propagation speed cascaded with a PIDE. A parameter update law is designed
using a Lyapunov argument and the infinite-dimensional backstepping technique
to establish global stability results. Furthermore, the well-posedness of the
closed-loop system is analyzed. Finally, the effectiveness of the proposed
method was validated through numerical simulation
Stabilization and controllability of first-order integro-differential hyperbolic equations
In the present article we study the stabilization of first-order linear
integro-differential hyperbolic equations. For such equations we prove that the
stabilization in finite time is equivalent to the exact controllability
property. The proof relies on a Fredholm transformation that maps the original
system into a finite-time stable target system. The controllability assumption
is used to prove the invertibility of such a transformation. Finally, using the
method of moments, we show in a particular case that the controllability is
reduced to the criterion of Fattorini
Feedback control on the velocity field and source term of a normal flow equation
open4openA. Alessandri; P. Bagnerini; M. Gaggero; A. RossiAlessandri, A.; Bagnerini, P.; Gaggero, M.; Rossi, A
A Fredholm transformation for the rapid stabilization of a degenerate parabolic equation
This paper deals with the rapid stabilization of a degenerate parabolic equation with a right Dirich-let control. Our strategy consists in applying a backstepping strategy, which seeks to find an invertible transformation mapping the degenerate parabolic equation to stabilize into an exponentially stable system whose decay rate is known and as large as we desire. The transformation under consideration in this paper is Fredholm. It involves a kernel solving itself another PDE, at least formally. The main goal of the paper is to prove that the Fredholm transformation is well-defined, continuous and invertible in the natural energy space. It allows us to deduce the rapid stabilization
Boundary stabilization in finite time of one-dimensional linear hyperbolic balance laws with coefficients depending on time and space
In this article we are interested in the boundary stabilization in finite
time of one-dimensional linear hyperbolic balance laws with coefficients
depending on time and space. We extend the so called "backstepping method" by
introducing appropriate time-dependent integral transformations in order to map
our initial system to a new one which has desired stability properties. The
kernels of the integral transformations involved are solutions to non standard
multi-dimensional hyperbolic PDEs, where the time dependence introduces several
new difficulties in the treatment of their well-posedness. This work
generalizes previous results of the literature, where only time-independent
systems were considered