Abstract Rothe's fixed-point theorem is applied to prove the interior approximate controllability of a semilinear impulsive strongly damped wave equation with Dirichlet boundary conditions in the space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>Z</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>/</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mi>D</m:mi> <m:mrow> <m:mo>(</m:mo> <m:msup> <m:mrow> <m:mo>(</m:mo> <m:mo>-</m:mo> <m:mi>Δ</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mrow> <m:mn>1</m:mn> <m:mo>/</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mo>)</m:mo> </m:mrow> <m:mo>×</m:mo> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mi>Ω</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> ${Z_{1/2} = D((-\Delta )^{1/2}) \times L^{2}(\Omega )}$ , where Ω is a bounded domain in ℝ n ( n ≥ 1). Under some conditions we prove the following statement: For all open nonempty subsets ω of Ω the system is approximately controllable on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>τ</m:mi> <m:mo>]</m:mo> </m:mrow> </m:math> ${[0, \tau ]}$ . Moreover, we exhibit a sequence of controls steering the nonlinear system from an initial state z 0 to a neighborhood of the final state z 1 at time <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>τ</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> ${\tau &gt;0}$ .
Tópico:
Stability and Controllability of Differential Equations
