Advances in Differential Equations

Uniform stabilization of spherical shells by boundary dissipation

I. Lasiecka, R. Triggiani, and V. Valente

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Abstract

Under homogeneous boundary conditions, the "energy" remains constant in time (conservative problem). We then introduce suitable dissipative feedback controls on the boundary (forces, shears, moments) and show that

(i) the resulting closed loop feedback problem generates a s.c. semigroup of contractions on a natural function space;

the corresponding "energy" (norm of the semigroup solutions) decays exponentially in the uniform topology.

As a consequence of the above uniform stabilization result, we obtain---via a result of [15]---a corresponding exact controllability result by explicit boundary controls, which improves upon the recent result of [4, 5]. Energy (multipliers) methods are used, along with semigroup methods. In the process of absorbing lower-order terms to obtain the final energy inequality, we establish a unique continuation result for the system of strongly coupled equations describing the dynamics of the shell, which is of interest in its own right. To this end, Carleman estimates are used after a suitable change of variable.

Article information

Source
Adv. Differential Equations Volume 1, Number 4 (1996), 635-674.

Dates
First available in Project Euclid: 25 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.ade/1366896031

Mathematical Reviews number (MathSciNet)
MR1401407

Zentralblatt MATH identifier
0857.35017

Subjects
Primary: 93D15: Stabilization of systems by feedback
Secondary: 35Q72 73K15 73K50 93C20: Systems governed by partial differential equations

Citation

Lasiecka, I.; Triggiani, R.; Valente, V. Uniform stabilization of spherical shells by boundary dissipation. Adv. Differential Equations 1 (1996), no. 4, 635--674. https://projecteuclid.org/euclid.ade/1366896031.


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