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On the constrained and unconstrained controllability of semilinear Hilfer fractional systems

Sikora Beata

Archives of Control Sciences

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2023

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Vol. 33, No. 1

155--178

EN

In the paper finite-dimensional semilinear dynamical control systems described by fractional-order state equations with the Hilfer fractional derivative are discussed. The formula for a solution of the considered systems is presented and derived using the Laplace transform. Bounded nonlinear function f depending on a state and controls is used. New sufficient conditions for controllability without constraints are formulated and proved using Rothe’s fixed point theorem and the generalized Darbo fixed point theorem. Moreover, the stability property is used to formulate constrained controllability criteria. An illustrative example is presented to give the reader an idea of the theoretical results obtained. A transient process in an electrical circuit described by a system of Hilfer type fractional differential equations is proposed as a possible application of the study.

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Controllability of time varying semilinear non-instantaneous impulsive systems with delay, and nonlocal conditions

Cabada Dalia, Garcia Katherine, Guevara Cristi, Leiva Hugo

Archives of Control Sciences

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2022

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Vol. 32, No. 2

335--357

EN

In this paper we prove the exact controllability of a time varying semilinear system considering non-instantaneous impulses, delay, and nonlocal conditions occurring simultaneously. It is done by using the Rothe’s fixed point theorem together with some sub-linear conditions on the nonlinear term, the impulsive functions, and the function describing the nonlocal conditions. Furthermore, a control steering the semilinear system from an initial state to a final state is exhibited.

3

Approximate controllability of the impulsive semilinear heat equation

Leiva H., Merentes N.

Journal of Mathematics and Applications

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2015

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Vol. 38

85--104

EN

In this paper we apply Rothe's Fixed Point Theorem to prove the interior approximate controllability of the following semilinear impulsive Heat Equation [...] where k = 1, 2, . . . , p, Ω is a bounded domain in [...] is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ω, the distributed control u belongs to [...]. Under this condition we prove the following statement: For all open nonempty subsets ω of Ω the system is approximately controllable on [0, τ]. Moreover, we could exhibit a sequence of controls steering the nonlinear system from an initial state z0 to an ϵ neighborhood of the nal state z1 at time τ > 0.

4

Approximate controllability of semilinear impulsive strongly damped wave equation

Larez H., Leiva H., Rebaza J., Rios A.

Journal of Applied Analysis

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2015

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Vol. 21, nr 1

45--57

EN

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 Z1/2 = D((-Δ)1/2) × L2(Ω), where Ω is a bounded domain in Rn (n ≥ 1). Under some conditions we prove the following statement: For all open nonempty subsets ω of Ω the system is approximately controllable on [0, τ]. Moreover, we exhibit a sequence of controls steering the nonlinear system from an initial state z0 to a neighborhood of the final state z1 at time τ > 0.