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arXiv · 2509.08471

Hierarchical exact controllability for a parabolic equation with Hardy potential

Haiyang Lin·Bo You
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Abstract

The main objective of this paper is to study the hierarchical exact controllability for a parabolic equation with Hardy potential by Stackelberg-Nash strategy. In linear case, we employ Lax-Milgram theorem to prove the existence of an associated Nash equilibrium pair corresponding to a bi-objective optimal control problem for each leader, which is responsible for an exact controllability property. Then the observability inequality of a coupled parabolic system is established by using global Carleman inequalities, which results in the existence of a leader that drives the controlled system exactly to any prescribed trajectory. In semilinear case, we first prove the well-posedness of the coupled parabolic system to obtain the existence of Nash quasi-equilibrium pair and show that Nash quasi-equilibrium is equivalent to Nash equilibrium. Based on these results, we establish the existence of a leader that drives the controlled system exactly to a prescribed (but arbitrary) trajectory by Leray-Schauder fixed point theorem.

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Haiyang Lin, Bo You. 2025-09-10. Hierarchical exact controllability for a parabolic equation with Hardy potential. https://arxiv.org/abs/2509.08471

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