Genni Fragnelli

Motivated by the broad applications of wave propagation in non-uniform and time-varying environments, such as in acoustics, elasticity, and seismology, we investigate the controllability of degenerate wave equations with time-dependent wave speeds. In this work, we examine non-autonomous degenerate wave equations in a one-dimensional spatial domain, addressing both divergence and non-divergence forms. A control function is applied at the non-degeneracy boundary point, while Dirichlet or Neumann conditions are imposed at the degeneracy one. Using the generalized energy conservation law, we first establish boundary observability for the homogeneous problem, a key step in our analysis. Building on this, we prove the null-controllability of the non-autonomous degenerate wave systems. To achieve this, we construct solutions using the transposition method, which accommodates low regularity requirements, enabling us to handle weaker smoothness assumptions while maintaining rigorous control over the system’s behavior. We conclude by presenting some insightful observations and potential avenues for future work, which could further advance the understanding of this problem.

In this paper, we introduce a new model for beam-type equations, driven by a mixed operator which is the sum of the classical biharmonic operator with a regional fractional Laplacian. The novelty is the peridynamical approach: the Dirichlet conditions hold in an extended outer domain (the horizon), which describes well the fact that a floor in reinforce concrete is stuck in the wall. We will study the spectral properties of this operator and then study stationary and evolution problems associated to it.

In oceanic scenarios the Robin boundary condition proves valuable for modeling surface wave interactions with complex boundary layers. This work investigates the null-controllability of degenerate wave equations with time-dependent propagation speed, where Robin conditions are imposed at the degeneracy boundary point aiming to model wave interaction with dissipative surface layers like viscous films or elastic covers. We first establish boundary observability for the homogeneous problem through a generalized energy conservation law. We subsequently demonstrate null-controllability for the non-autonomous degenerate wave equation by employing the transposition method to construct solutions, which provides the flexibility to work with reduced regularity conditions while preserving precise control of the system dynamics. Finally, we discuss significant implications of these findings and outline directions for future research that could enhance our current understanding of such systems.

Motivated by the broad applications of wave propagation in non-uniform and time-varying environments, such as in acoustics, elasticity, and seismology, we investigate the controllability of degenerate wave equations with time-dependent wave speeds. In this work, we examine non-autonomous degenerate wave equations in a one-dimensional spatial domain, addressing both divergence and non-divergence forms. A control function is applied at the non-degeneracy boundary point, while Dirichlet or Neumann conditions are imposed at the degeneracy one. Using the generalized energy conservation law, we first establish boundary observability for the homogeneous problem, a key step in our analysis. Building on this, we prove the null-controllability of the non-autonomous degenerate wave systems. To achieve this, we construct solutions using the transposition method, which accommodates low regularity requirements, enabling us to handle weaker smoothness assumptions while maintaining rigorous control over the system's behavior. We conclude by presenting some insightful observations and potential avenues for future work, which could further advance the understanding of this problem.