<p>The Keller–Segel system is a classical model in chemotaxis, widely used in biological and physical contexts, but also a challenging prototype for nonlinear PDE analysis. Our focus studies an insensitizing control problem for the nonlinear parabolic-parabolic Keller–Segel system, which models chemotactic behavior in biological systems. Our goal is to find a control that makes a certain functional of the solution insensitive to small perturbations in the initial data. We show that this problem is equivalent to achieving partial null controllability for a related cascade system that reflects the main structure of the original dynamics. Thanks to this equivalence, we focus on analyzing the controllability of the cascade system. We begin by studying the linearized version of the problem. Using a duality approach, along with carefully selected weighted estimates and energy techniques, we establish a suitable observability inequality. This key result enables us to move on to the nonlinear case. We address the nonlinear system through a local inverse mapping argument, relying on the continuity and differentiability of the control-to-state map in an appropriate functional framework, along with other key assumptions.</p>

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An Insensitizing Control Result for the Keller–Segel System

  • F. W. Chaves-Silva,
  • J. Prada

摘要

The Keller–Segel system is a classical model in chemotaxis, widely used in biological and physical contexts, but also a challenging prototype for nonlinear PDE analysis. Our focus studies an insensitizing control problem for the nonlinear parabolic-parabolic Keller–Segel system, which models chemotactic behavior in biological systems. Our goal is to find a control that makes a certain functional of the solution insensitive to small perturbations in the initial data. We show that this problem is equivalent to achieving partial null controllability for a related cascade system that reflects the main structure of the original dynamics. Thanks to this equivalence, we focus on analyzing the controllability of the cascade system. We begin by studying the linearized version of the problem. Using a duality approach, along with carefully selected weighted estimates and energy techniques, we establish a suitable observability inequality. This key result enables us to move on to the nonlinear case. We address the nonlinear system through a local inverse mapping argument, relying on the continuity and differentiability of the control-to-state map in an appropriate functional framework, along with other key assumptions.