Stability and quadratic Lyapunov functions for nD systems

We discuss some ideas and preliminary results on the stability of nD systems described by linear constant coefficient PDE's. The stability concept used is Lscr
₂
-stability and Lscr
₂
-asymptotic stability, with time as a distinguished variable. For scalar equations, stability conditions are derived, including methods to make these conditions into LMI's in the system parameters. These conditions are interpreted in terms of Lyapunov functions for systems involving many independent variables. Several open problems for multivariable nD systems are formulated.