Frequency Domain Criteria for Stability of Systems Modeled by Certain Partial Differential Equations

Abstract

In recent years many of the methods used to investigate stability of lumped parameter systems have been extended to study the stability of distributed systems. In this work we employ an extension of the circle-criterion, cf. [1], to Hilbert space-valued systems, cf. [2], as well as some rather sophisticated results on elliptic boundary value problems, cf. [3]. The stimulus for the approach is the desire to have calculable criteria for stability. In particular, we study the “L ₂-stability” of boundary, initial-value problems of the form

$$\left. \begin{array}{l} \left[ {P\left( {{D_t}} \right) + Q\left( {{D_t}} \right)L\left( {x,{D_x}} \right)} \right]u + N(u) = 0{\rm{ }}in{\rm{ }}\left[ {0,\infty )} \right] \times D;\\ \left( {D_t^ku} \right)\left( {0,x} \right) = {f_k}(x),\,k = 0,...,l - 1;\\ {B_j}\left( {x,{D_x}} \right){\left. u \right|_{\partial D}} = 0,i = 1,...,m.\, \end{array} \right\}$$

Here L (x, D _x ) is an elliptic operator of order 2m in the bounded, open domain D ⊆ ℜⁿ, P and Q are polynomials of degrees p and q respectively,N is a nonlinearity subject to N(0) = 0, and B _j (x, D _x )are boundary operators which cover D (cf. [3] for details).