On a Constrained Approximate Controllability Problem for the Heat Equation

2001 ◽  
Vol 108 (1) ◽  
pp. 29-64 ◽  
Author(s):  
J. H. Ortega ◽  
E. Zuazua
Keyword(s):  
Heat Equation ◽  
Approximate Controllability ◽  
Controllability Problem

scholarly journals Controllability of a one-dimensional fractional heat equation: theoretical and numerical aspects

2018 ◽  
Vol 36 (4) ◽  
pp. 1199-1235 ◽  
Author(s):  
Umberto Biccari ◽  
Víctor Hernández-Santamaría
Keyword(s):  
Heat Equation ◽  
Approximate Controllability ◽  
Fractional Laplacian ◽  
Controllability Problem ◽  
Finite Element Scheme ◽  
Element Scheme ◽  
One Dimensional ◽  
Fractional Heat Equation ◽  
Hilbert Uniqueness Method ◽  
Uniqueness Method

Abstract We analyse the controllability problem for a one-dimensional heat equation involving the fractional Laplacian $(-d_x^{\,2})^{s}$ on the interval $(-1,1)$. Using classical results and techniques, we show that, acting from an open subset $\omega \subset (-1,1)$, the problem is null-controllable for $s>1/2$ and that for $s\leqslant 1/2$ we only have approximate controllability. Moreover, we deal with the numerical computation of the control employing the penalized Hilbert Uniqueness Method and a finite element scheme for the approximation of the solution to the corresponding elliptic equation. We present several experiments confirming the expected controllability properties.

scholarly journals Approximate controllability of the semilinear heat equation

1995 ◽  
Vol 125 (1) ◽  
pp. 31-61 ◽  
Author(s):  
Caroline Fabre ◽  
Jean-Pierre Puel ◽  
Enrike Zuazua
Keyword(s):  
Fixed Point ◽  
Heat Equation ◽  
Bounded Domain ◽  
Variational Approach ◽  
Approximate Controllability ◽  
Nonempty Subset ◽  
Linear Equations ◽  
Fixed Point Method ◽  
Controllability Problem ◽  
Semilinear Heat Equation

This article is concerned with the study of approximate controllability for the semilinear heat equation in a bounded domain Ω when the control acts on any open and nonempty subset of Ω or on a part of the boundary. In the case of both an internal and a boundary control, the approximate controllability in LP(Ω) for 1 ≦ p < + ∞ is proved when the nonlinearity is globally Lipschitz with a control in L∞. In the case of the interior control, we also prove approximate controllability in C0(Ω). The proof combines a variational approach to the controllability problem for linear equations and a fixed point method. We also prove that the control can be taken to be of “quasi bang-bang” form.

scholarly journals Controllability of the Semilinear Heat Equation with Impulses and Delay on the State

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Hugo Leiva
Keyword(s):  
Continuous Function ◽  
Characteristic Function ◽  
Heat Equation ◽  
Bounded Domain ◽  
Distributed Control ◽  
Approximate Controllability ◽  
Nonempty Subset ◽  
Nonlinear Functions ◽  
Semilinear Heat Equation ◽  
Approximately Controllable

AbstractIn this paper we prove the interior approximate controllability of the following Semilinear Heat Equation with Impulses and Delaywhere Ω is a bounded domain in RN(N ≥ 1), φ : [−r, 0] × Ω → ℝ is a continuous function, ! is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ! and the distributed control u be- longs to L2([0, τ]; L2(Ω; )). Here r ≥ 0 is the delay and the nonlinear functions f , Ik : [0, τ] × ℝ × ℝ → ℝ are smooth enough, such thatUnder this condition we prove the following statement: For all open nonempty subset ! of Ω the system is approximately controllable on [0, τ], for all τ > 0.

scholarly journals Boundary Controllability of a Pseudoparabolic Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Qiang Tao ◽  
Hang Gao ◽  
Zheng-an Yao
Keyword(s):  
Explicit Solution ◽  
Approximate Controllability ◽  
Null Controllability ◽  
Pseudoparabolic Equation ◽  
Controllability Problem ◽  
Boundary Controllability ◽  
Boundary Controls ◽  
Controllability Property

We deal with the controllability problem for the pseudoparabolic equation by means of boundary controls. Due to the unusual spectrum of this kind of equations, we prove that the null controllability property is false. Furthermore, by the explicit solution, we show that the approximate controllability holds.

scholarly journals Finite approximate controllability for semilinear heat equations in noncylindrical domains

2004 ◽  
Vol 76 (3) ◽  
pp. 475-487
Author(s):  
Silvano B. de Menezes ◽  
Juan Limaco ◽  
Luis A. Medeiros
Keyword(s):  
Fixed Point ◽  
Heat Equation ◽  
Approximate Controllability ◽  
State Equation ◽  
Heat Equations ◽  
Semilinear Heat Equation ◽  
Fixed Point Result ◽  
Linearized Problem

We investigate finite approximate controllability for semilinear heat equation in noncylindrical domains. First we study the linearized problem and then by an application of the fixed point result of Leray-Schauder we obtain the finite approximate controllability for the semilinear state equation.

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