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Controllability of the Strongly Damped Wave Equation with Impulses and Delay

Hugo Leiva — 2017

Nonautonomous Dynamical Systems

Evading fixed point theorems we prove the interior approximate controllability of the following semilinear strongly damped wave equation with impulses and delay [...] in the space Z1/2 = D((−Δ)1/2)×L2(Ω),where r > 0 is the delay, Γ = (0, τ)×Ω, ∂Γ = (0, τ) × ∂Ω, Γr = [−r, 0] × Ω, (ϕ,ψ) ∈ C([−r, 0]; Z1/2), k = 1, 2, . . . , p, Ω is a bounded domain in ℝℕ(ℕ ≥ 1), ω is an open nonempty subset of , 1 ω denotes the characteristic function of the set ω, the distributed control u ∈ L2(0, τ; U), with...

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

Hugo Leiva — 2015

Nonautonomous Dynamical Systems

In this paper we prove the interior approximate controllability of the following Semilinear Heat Equation with Impulses and Delay [...] where 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 that [...] Under this...

A Representation Theorem for ϕ -Bounded Variation of Functions in the Sense of Riesz

Wadie Aziz;  Hugo Leiva;  Nelson Merentes;  Beata Rzepka — 2010

Commentationes Mathematicae

In this paper we extend the well known Riesz lemma to the class of bounded ϕ -variation functions in the sense of Riesz defined on a rectangle I a b ⊂ ℝ 2 . This concept was introduced in [2], where the authors proved that the space B V ϕ R ( I a b ; ℝ of such functions is a Banach Algebra. Moreover, they characterized also the Nemytskii operator acting in this space. Thus our result creates a continuation of the paper [2].

On the Hammerstein equation in the space of functions of bounded ϕ -variation in the plane

Luis Azócar;  Hugo Leiva;  Jesús Matute;  Nelson Merentes — 2013

Archivum Mathematicum

In this paper we study existence and uniqueness of solutions for the Hammerstein equation u ( x ) = v ( x ) + λ ∫ I a b K ( x , y ) f ( y , u ( y ) ) d y , x ∈ I a b : = [ a 1 , b 1 ] × [ a 2 , b 2 ] , in the space B V ϕ ℝ ( I a b ) of function of bounded total ϕ - variation in the sense of Riesz, where λ ∈ ℝ , K : I a b × I a b → ℝ and f : I a b × ℝ → ℝ are suitable functions.

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