Full paper in PDF:
$%A. Yu. Khapalov, Global controllability properties for the semilinear heat equation with superlinear term, Rev. Mat. Complut. 12 (1999), no. 2, 511535.%$

Global Controllability Properties for the Semilinear Heat Equation with Superlinear Term
Alexander Yu. KHAPALOV
Department of Pure and Applied Mathematics
Washignton State University
Pullman, WA 99164-3113 — USA

Received: October 15, 1997
Revised: March 22, 1999

ABSTRACT

We discuss several global approximate controllability properties for the semilinear heat equation with superlinear reaction-convection term, governed in a bounded domain by locally distributed controls. First, based on the asymptotic analysis in vanishing time, we study the steering of the projections of its solution on any finite dimensional space spanned by the eigenfunctions for the truncated linear part. We show that, if the control-supporting area is properly chosen, then they can approximately be controlled globally at any time in the topology induced by L2(_O_)  . Then, based on the L2(QT)  -estimates as T --> 0  for the control functions solving the first problem, we prove that its global approximate controllability from any u0  (- L2(_O_)  is also possible at any time in certain topology, which is weaker than that of L2(_O_)  . (It is known that this result does not hold in L2(_O_)  .) Finally, based on Altman’s fixed point theorem and some of the above asymptotic-type results, we show that if the nonlinearity is purely of reaction type and is locally Lipschitz, then the global exact controllability in finite dimensions holds as well.

1991 Mathematics Subject Classification: 35, 93.