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Remarks on finite-approximate controllability of impulsive evolution systems via resolvent-like operator in Hilbert spaces

Javad A. Asadzade, Nazim I. Mahmudov

TL;DR

The paper addresses finite-approximate controllability ($F_A$-controllability) for impulsive evolution systems in Hilbert spaces by introducing a resolvent-like operator to handle linear dynamics and employing Schauder fixed-point theory to extend results to semilinear systems. It provides explicit finite-approximating controls, characterizes controllability through a Gramian-based positivity condition, and establishes equivalence with resolvent-limit properties. The results are then applied to a heat equation, illustrating how the abstract theory governs impulsive, infinite-dimensional systems and enabling finite-time steering to within a prescribed tolerance. Overall, the work advances $F_A$-controllability theory for impulsive, infinite-dimensional systems and demonstrates practical applicability to PDEs like the heat equation.

Abstract

In this manuscript, we examine impulsive evolution systems in Hilbert spaces. Using a resolvent-like operator, we first establish the finite-approximate controllability for linear systems. Subsequently, by applying the Schauder fixed-point theorem (SFPT), we prove the existence of a solution and demonstrate the finite-approximate controllability of semilinear impulsive systems in Hilbert spaces. Finally, we extend these results to a broader application, specifically to the heat equation.

Remarks on finite-approximate controllability of impulsive evolution systems via resolvent-like operator in Hilbert spaces

TL;DR

The paper addresses finite-approximate controllability (-controllability) for impulsive evolution systems in Hilbert spaces by introducing a resolvent-like operator to handle linear dynamics and employing Schauder fixed-point theory to extend results to semilinear systems. It provides explicit finite-approximating controls, characterizes controllability through a Gramian-based positivity condition, and establishes equivalence with resolvent-limit properties. The results are then applied to a heat equation, illustrating how the abstract theory governs impulsive, infinite-dimensional systems and enabling finite-time steering to within a prescribed tolerance. Overall, the work advances -controllability theory for impulsive, infinite-dimensional systems and demonstrates practical applicability to PDEs like the heat equation.

Abstract

In this manuscript, we examine impulsive evolution systems in Hilbert spaces. Using a resolvent-like operator, we first establish the finite-approximate controllability for linear systems. Subsequently, by applying the Schauder fixed-point theorem (SFPT), we prove the existence of a solution and demonstrate the finite-approximate controllability of semilinear impulsive systems in Hilbert spaces. Finally, we extend these results to a broader application, specifically to the heat equation.
Paper Structure (6 sections, 6 theorems, 89 equations)

This paper contains 6 sections, 6 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $F_{A}$ contrability of linear system
  4. $F_{A}$ contrability of semilinear system
  5. Application
  6. Conclusion

Key Result

Lemma 2.1

Mahmudov1 Let $(H,\Vert \cdot\Vert)$ be a Hilbert space. If $\Gamma: H\to H$ is a linear non-neqative operator, then the operator $\alpha(\mathcal{I}-\pi_{D})+\Gamma: H\to H$ is invertible and where $\delta=\min\{\langle \pi_{D}\Gamma\pi_{D}\varphi, \varphi\rangle: \Vert \pi_{D} \varphi \Vert=1\}$. Moreover, if $\Gamma: H\to H$ is a linear positive operator, then

Theorems & Definitions (14)

  • Definition 2.1
  • Lemma 2.1
  • Lemma 3.1
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.1
  • Theorem 4.1
  • proof
  • Remark 4.1
  • ...and 4 more