Approximate Internal Controllability and Synchronization for a System of Wave Equations With Coupled Robin Boundary Condition

In this paper, the approximate internal controllability and synchronization are studied for a system of wave equations with coupled Robin boundary conditions. The sufficient and necessary conditions for the approximate internal controllability are given. Based on this result, the approximate internal synchronization by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>p</mml:mi> </mml:math> -groups is established under the conditions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi mathvariant="bold">C</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> </mml:math> -compatibility. Furthermore, important properties of the approximate internal synchronization by groups under the minimal rank condition are given, such as the independence of approximately synchronizable states with respect to applied controls, the linear independence of the components of the approximately synchronizable state, and so on.