The authors attempt to introduce a decomposition of a very large system of simultaneous partial differential equations into blocks of subsystems. This is not an easy project, and starting with Lyapunov, von Neumann, and ending with the present large number of parallel computing gurus, outstanding numerical analysts and domain decomposition experts like Ronald Glowinski, one can still say that no light at the end of this tunnel is visible.
The authors follow Lyapunov and consider a system of second order PDE-s: \[ \partial Q(t,x)/ \partial t=\partial/ \partial x\bigl\{A(x) \partial Q(t,x)/ \partial x\bigr\} +BQ(t,x),\;0\leq x\leq 1 \] with zero boundary conditions, \(Q\in \mathbb{R}^n\); \(A\) are matrices of coefficients containing blocks of matrices of lower order. Matrices \(B\) also have entries describing some physical properties of the system. The dimension number \(n\) is very large. Now the system is rewritten as a collection of subsystems \[ \partial{\mathcal P}_i(t,x)/ \partial t=\partial/ \partial x\bigl \{A_{ii}(x) \partial{\mathcal P}_i (t,x)/ \partial x\bigr\} +B_{ii} {\mathcal P}_i (t,x)+ B_{ij(i\neq j)} {\mathcal P}_j (t,x), \] where \(B_{ij(i\neq j)}\) are the interconnecting system coefficients. Now for each subsystem the authors introduce a rather obvious Lyapunov functional \(W\), which is positive definite, while its derivative is negative along trajectories of the system. Thus stabilities of the connected subsystems can be studied, and hopefully some region of stability for the whole system may be established, subject to small perturbations. Let us hope that output from computer programs confirming the usefulness of this decomposition will soon be displayed.