In addition to developing ISS theorems, equipped with gain estimates with respect to external disturbances, the authors develop small-gain stability theorems for systems involving PDEs. A variety of system combinations are considered:

• PDEs (of either class) with static maps;
  
• PDEs (again, of either class) with ODEs;
  
• PDEs of the same class (parabolic with parabolic and hyperbolic with hyperbolic); and
  
• feedback loops of PDEs of different classes (parabolic with hyperbolic).
  

In addition to stability results (including ISS), the text develops existence and uniqueness theory for all systems that are considered. Many of these results answer for the first time the existence and uniqueness problems for many problems that have dominated the PDE control literature of the last two decades, including—for PDEs that include non-local terms—backstepping control designs which result in non-local boundary conditions.