Boundary controllability for the 1D Moore–Gibson–Thompson equation

The article delves into the boundary controllability of the 1D Moore–Gibson–Thompson equation, a third-order in time equation with internal damping originating from nonlinear acoustics. The study employs an operator-theoretical semigroup method and explores the implications of structural damping on controllability properties. The authors prove that the equation is not spectrally controllable, implying a lack of exact controllability, but demonstrate that it is approximately controllable. The research highlights the unique challenges posed by the equation’s internal damping and provides valuable insights into the control of high-intensity focused ultrasound applications. The findings contribute to the broader understanding of controllability in mathematical physics and have potential applications in biomedical problems associated with HIFU.