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Controllability of nonautonomous measure driven integrodifferential evolution equations with nonlocal conditions

Mamadou Niang, Mamadou Pathe LY, Abdoul Aziz Ndiaye, Abdoul Aziz Ndiaye, Mamadou Abdoul Diop

TL;DR

The paper addresses exact controllability of semilinear measure-driven integrodifferential equations with nonlocal initial conditions in Banach spaces. By employing Grimmer's resolvent framework, evolution systems, the measure of noncompactness, and the Mönch fixed point theorem, it derives sufficient controllability conditions that do not rely on compactness of the linear part. A mild-solution representation is used to connect controls to terminal nonlocal targets, and the main result provides explicit smallness inequalities ensuring controllability on a fixed interval. An illustrative PDE example demonstrates applicability to parabolic-type dynamics with measure-driven impulses and nonlocal data, underscoring the method's relevance to non-smooth/measure-driven phenomena and potential extensions to delays and random effects.

Abstract

This research delves into the exact controllability of semilinear measure-driven integrodifferential systems in nonlocal settings. We provide sufficient controllability requirements using the measure of noncompactness and the Mönch fixed point theorem without making any assumptions about how compact the evolution system is in relation to the linear part of the measure system. Here, we obtain results that both generalize and improve upon many prior findings.

Controllability of nonautonomous measure driven integrodifferential evolution equations with nonlocal conditions

TL;DR

The paper addresses exact controllability of semilinear measure-driven integrodifferential equations with nonlocal initial conditions in Banach spaces. By employing Grimmer's resolvent framework, evolution systems, the measure of noncompactness, and the Mönch fixed point theorem, it derives sufficient controllability conditions that do not rely on compactness of the linear part. A mild-solution representation is used to connect controls to terminal nonlocal targets, and the main result provides explicit smallness inequalities ensuring controllability on a fixed interval. An illustrative PDE example demonstrates applicability to parabolic-type dynamics with measure-driven impulses and nonlocal data, underscoring the method's relevance to non-smooth/measure-driven phenomena and potential extensions to delays and random effects.

Abstract

This research delves into the exact controllability of semilinear measure-driven integrodifferential systems in nonlocal settings. We provide sufficient controllability requirements using the measure of noncompactness and the Mönch fixed point theorem without making any assumptions about how compact the evolution system is in relation to the linear part of the measure system. Here, we obtain results that both generalize and improve upon many prior findings.
Paper Structure (8 sections, 15 theorems, 49 equations)

This paper contains 8 sections, 15 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Integrodifferential equations
  4. Norm-continuity of the resolvent operators
  5. Equi-regulated functions and measure of noncompactness
  6. Main results
  7. Example
  8. Conclusion

Key Result

Lemma 2.1

Grimmer If $\mathcal{A}(t)\equiv \mathcal{A}$ and $\Delta(t,s)\equiv \Delta(t-s)$ and there exists a resolvent operator $\mathsf{R}(t,s)$, then $\mathsf{R}(t,s)=\mathsf{R}(t-s,0)$.

Theorems & Definitions (26)

  • Definition 1
  • Lemma 2.1
  • Definition 2
  • Remark 2.2
  • Lemma 2.3
  • Definition 3
  • Theorem 1
  • Definition 4
  • Lemma 2.4
  • Definition 5
  • ...and 16 more