Semiglobal exponential stabilization of nonautonomous semilinear parabolic-like systems

Sérgio S. Rodrigues

doi:10.3934/eect.2020027

Article Contents

2020, Volume 9, Issue 3: 635-672. Doi: 10.3934/eect.2020027

This issue Previous Article Local null controllability of coupled degenerate systems with nonlocal terms and one control force Next Article Boundary controllability of the Korteweg-de Vries equation on a tree-shaped network

Semiglobal exponential stabilization of nonautonomous semilinear parabolic-like systems

Sérgio S. Rodrigues

Johann Radon Institute for Computational and Applied Mathematics, ÖAW, Altenbergerstraße 69, A-4040 Linz, Austria

Received: March 31, 2019

Revised: May 31, 2019

Early access: December 2019

Published: September 2020

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Abstract

It is shown that an explicit oblique projection nonlinear feedback controller is able to stabilize semilinear parabolic equations, with time-dependent dynamics and with a polynomial nonlinearity. The actuators are typically modeled by a finite number of indicator functions of small subdomains. No constraint is imposed on the sign of the polynomial nonlinearity. The norm of the initial condition can be arbitrarily large, and the total volume covered by the actuators can be arbitrarily small. The number of actuators depends on the operator norm of the oblique projection, on the polynomial degree of the nonlinearity, on the norm of the initial condition, and on the total volume covered by the actuators. The range of the feedback controller coincides with the range of the oblique projection, which is the linear span of the actuators. The oblique projection is performed along the orthogonal complement of a subspace spanned by a suitable finite number of eigenfunctions of the diffusion operator. For rectangular domains, it is possible to explicitly construct/place the actuators so that the stability of the closed-loop system is guaranteed. Simulations are presented, which show the semiglobal stabilizing performance of the nonlinear feedback.

Keywords:

Semiglobal exponential stabilization,
nonlinear feedback,
nonlinear nonautonomous parabolic systems,
finite-dimensional controller,
oblique projections.

Mathematics Subject Classification: 93D15, 93C10, 93B52.

Citation:

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Figure 1. Uncontrolled solutions. Linear and nonlinear systems

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Figure 2. Linear systems and linear feedback

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Figure 3. Nonlinear systems and linear feedback

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Figure 4. Nonlinear systems and nonlinear feedback

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Figure 5. Nonlinear systems and nonlinear feedback. Bigger initial condition

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Figure 6. Nonlinear systems and nonlinear feedback. Increasing the number of actuators

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Figure 7. Nonlinear systems and linear feedback. Increasing the number of actuators

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Figure 8. Nonlinear systems and nonlinear feedback. $ y(0) = c_{\rm ic}\sin(8\pi x)\in E_{ {\mathbb M}_\sigma}^\perp $

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