Abstract
This paper addresses the application of a multi-objective control problem to a parabolic equation. We present the Stackelberg-Nash strategy, which combines the concepts of controllability to trajectories with Nash equilibrium. Our assumption is that the system is influenced through a hierarchy of boundary controls, consisting of one main control (the leader) aiming to drive the solution to a specified target at a final time, and a pair of secondary controls (the followers) designed to minimize two prescribed cost functionals while adapting to the leader's objective. The main novelty of this work lies in the fact that all controls are located on the boundary. One of the primary challenges encountered is the derivation of suitable observability inequality of Carleman for an adjoint system with boundary coupling terms.
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