Bilinear control for global controllability of the semilinear parabolic equations with superlinear terms. (English) Zbl 0983.93023

Khapalov, Alexander

Chen, Goong (ed.) et al., Control of nonlinear distributed parameter systems. Partly proceedings of the conference advances in control of nonlinear distributed parameter systems, Texas A & M Univ., College Station, TX, USA. Dedicated to Prof. David L. Russell on the occasion of his 60th birthday. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 218, 139-155 (2001).

This is an interesting and well-written paper on controllability for the semilinear heat equation. The control system is \[ \frac{\partial u}{\partial t}=\Delta u+k(t)u-f(x,t,u,\nabla u)+v(x,t)\chi_{\omega} (x) \text{ in} \quad Q_T=\Omega\times (0,T), \]
\[ u=0 \text{ in} \quad\Sigma_T =\partial \Omega \times (0,T), u(0)=u_0\in L^2 (\Omega). \] The controls are \(k\in L^{\infty}(0,T)\) and \(v\in L^2 (Q_T)\). The main specific feature of this setting is the fact that the nonlinearity \(f\) is superlinear in both the third and the fourth variables. In this case there is a well-known lack of controllability when the control \(v\) acts only. This is why the control \(k\) appears here. Due to the possible nonuniqueness of solutions, two approximate controllability concepts are defined and two results are stated and proved. For the one dimensional problem, a controllability result is proved in case the control \(v\) depends on \(t\) only.
For the entire collection see [Zbl 0959.00047].

Reviewer: Ovidiu Cârjá (Iaşi)

MSC:

93C20	Control/observation systems governed by partial differential equations
93B05	Controllability
93C10	Nonlinear systems in control theory

Keywords:

semilinear heat equation; approximate controllability; bilinear control; lumped control; asymptotic analysis