Journal Article

A broad class of evolution equations are approximately controllable, but never exactly controllable

Diomedes Bárcenas, Hugo Leiva and Zoraida Sívoli

in IMA Journal of Mathematical Control and Information

Published on behalf of Institute of Mathematics and its Applications

Volume 22, issue 3, pages 310-320
Published in print September 2005 | ISSN: 0265-0754
Published online September 2005 | e-ISSN: 1471-6887 | DOI: http://dx.doi.org/10.1093/imamci/dni029
A broad class of evolution equations are approximately controllable, but never exactly controllable

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As we have announced in the title of this work, we show that a broad class of evolution equations are approximately controllable but never exactly controllable. This class is represented by the following infinite-dimensional time-varying control system:

z = A(t)z + B(t)u(t), t > 0, zZ

where Z, U are Banach spaces, the control function u belong to Lp(0, t1; U), t1 > 0, 1 < p < ∞, B ∈, L (0, t1; L(U, Z)) and A(t) generates a strongly continuous evolution operator U(t, s) according to Pazy (1983; Semigroups of Linear Operators with Applications to Partial Differential Equations). Specifically, we prove the following statement: If U(t, s) is compact for 0 ⩽ s < tt1, then the system can never be exactly controllable on [0, t1]. This class is so large that includes diffusion equations, damped flexible beam equation, some thermoelastic equations, strongly damped wave equations, etc.

Keywords: evolution equations; approximate and exact controllability; compact operators

Journal Article.  0 words. 

Subjects: Mathematics

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