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Controllability of the Strongly Damped Wave Equation with Impulses and Delay

100%

Leiva H.

Nonautonomous Dynamical Systems

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2017

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tom 4

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nr 1

31-39

EN

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 U = L2(Ω),η,γ, are positive numbers and f , Ik ∈ C([0, τ] × ℝ × ℝ; ℝ), k = 1, 2, 3, . . . , p. Under some conditions we prove the following statement: For all open nonempty subsets Ω of the system is approximately controllable on [0,τ]. Moreover, we exhibit a sequence of controls steering the nonlinear system from an initial state (ϕ (0), ψ(0)) to an ε-neighborhood of the final state z1 at time τ > 0.

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

100%

Leiva H.

Nonautonomous Dynamical Systems

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2015

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tom 2

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nr 1

EN

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 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.

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A Representation Theorem for \(\varphi\)-Bounded Variation of Functions in the Sense of Riesz

51%

Aziz W., Leiva H., Merentes N., Rzepka B.

Commentationes Mathematicae

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2010

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tom 50

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nr 2

EN

In this paper we extend the well known Riesz lemma to the class of bounded \(\varphi\)-variation functions in the sense of Riesz defined on a rectangle \(I_a^b\subset \mathbb{R}^2\). This concept was introduced in [2], where the authors proved that the space \(BV_\varphi^R (I_a^b;\mathbb{R}\) 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].

Ograniczanie wyników

2 Nonautonomous Dynamical Systems

1 Commentationes Mathematicae

3 Leiva H.

1 Aziz W.

1 Merentes N.

1 Rzepka B.

1 2017

1 2015

1 2010

Controllability of the Strongly Damped Wave Equation with Impulses and Delay

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

A Representation Theorem for \(\varphi\)-Bounded Variation of Functions in the Sense of Riesz