Null-controllability Of Parabolic Equations Research Articles

Observability and null-controllability for parabolic equations in $ L_p $-spaces

<p style='text-indent:20px;'>We study (cost-uniform approximate) null-controllability of parabolic equations in <inline-formula><tex-math id="M2">\begin{document}$ L_p( \mathbb{R}^d) $\end{document}</tex-math></inline-formula> and provide explicit bounds on the control cost. In particular, we consider systems of the form <inline-formula><tex-math id="M3">\begin{document}$ \dot{x}(t) = -A_px(t) + {\bf{1}}_ E u(t) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ x(0) = x_0\in L_p ( \mathbb{R}^d) $\end{document}</tex-math></inline-formula>, with interior control on a so-called thick set <inline-formula><tex-math id="M5">\begin{document}$ E \subset \mathbb{R}^d $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M6">\begin{document}$ p\in [1,\infty) $\end{document}</tex-math></inline-formula>, and where <inline-formula><tex-math id="M7">\begin{document}$ A $\end{document}</tex-math></inline-formula> is an elliptic operator of order <inline-formula><tex-math id="M8">\begin{document}$ m \in \mathbb{N} $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M9">\begin{document}$ L_p(\mathbb{R}^d) $\end{document}</tex-math></inline-formula>. We prove null-controllability of this system via duality and a sufficient condition for observability. This condition is given by an uncertainty principle and a dissipation estimate. Our result unifies and generalizes earlier results obtained in the context of Hilbert and Banach spaces. In particular, our result applies to the case <inline-formula><tex-math id="M10">\begin{document}$ p = 1 $\end{document}</tex-math></inline-formula>.</p>

On the null controllability of parabolic equations with nonlinear memory

ABSTRACT In this paper, we study the global null controllability of nonlinear parabolic integro-differential equations arising in reactor dynamics with internal distributed control. Unlike the known results for nonlinear parabolic equations, controllability is analyzed in the frame of strong solutions. The result implies the nonlinear growth of the system with memory which is null controllable can be higher than that of classical parabolic equations. This phenomenon is caused by the memory effects.

Null controllability of degenerate parabolic equations of Grushin and Kolmogorov type

The goal of this note is to present the results of the references [5] and [4]. We study the null controllability of the parabolic equations associated with the Grushin-type operator ∂ x + |x|∂ y (γ > 0) in the rectangle (x, y) ∈ (−1, 1)×(0, 1) or with the Kolmogorov-type operator v∂xf+∂ vf (γ ∈ {1, 2}) in the rectangle (x, v) ∈ T×(−1, 1), under an additive control supported in an open subset ω of the space domain. We prove that the Grushin-type equation is null controllable in any positive time for γ 1. In the transition regime γ = 1 and when ω is a strip ω = (a, b)× (0, 1) , (0 < a, b ≤ 1), a positive minimal time is required for null controllability. For the Kolmogorov-type equation with γ = 1 and periodic-type boundary conditions (in v), we prove that null controllability holds in any positive time, with any control support ω. This improves the previous result [6], in which the control support was a strip ω = T× (a, b). For the Kolmogorov-type equation with Dirichlet boundary conditions and a strip ω = T × (a, b) (0 < a < b < 1) as control support, we prove that null controllability holds in any positive time if γ = 1, and only in large time if γ = 2. Our approach, inspired from [8, 33], is based on 2 key ingredients: the observability of the Fourier components of the solution of the adjoint system (a heat equation with potential), uniformly with respect to the frequency, and the explicit exponential decay rate of these Fourier components.

Null controllability of a parabolic equation involving the Grushin operator in some multi-dimensional domains

We study the null controllability of the linear parabolic equation involving the Grushin operator Gs=Δx+|x|2sΔy(s>0), in the domain Ω=(−1,1)N1×(0,1)N2⊂RN1×RN2,N1,N2≥1, under an additive control supported in an open subset ω of Ω. We prove that the equation is null controllable in any positive time for s<1 and not null controllable for s>1. When s=1, a positive minimal time is required for null controllability.

The L∞-null controllability of parabolic equation with equivalued surface boundary conditions

In this paper, we obtain the L∞−null controllability of the parabolic equation with equivalued surface boundary conditions in Ω× [0, T ]. The control is supported in the product of an open subset of Ω and a subset of [0, T ] with positive measure. The main result is obtained by the method of Lebeau-Robianno-type iteration, based on a new estimate for partial sum of the eigenfunctions of the elliptic operator with equivalued surface boundary conditions.

Single input controllability of a simplified fluid-structure interaction model

In this paper we study a controllability problem for a simplified one dimensional model for the motion of a rigid body in a viscous fluid. The control variable is the velocity of the fluid at one end. One of the novelties brought in with respect to the existing literature consists in the fact that we use a single scalar control. Moreover, we introduce a new methodology, which can be used for other nonlinear parabolic systems, independently of the techniques previously used for the linearized problem. This methodology is based on an abstract argument for the null controllability of parabolic equations in the presence of source terms and it avoids tackling linearized problems with time dependent coefficients.