On the stability of a class of non-linear continuous systems†

Abstract

A procedure to find the existence and domain of asymptotic stability for a class of systems described by a set of non-linear partial differential equations is presented in the context of semigroup theory.

No a priori bound is imposed on the non-linearity which is assumed to be a quad ratic (bilinear) or multilinear form.

The linearized system can be non-self-adjoint and need not be asymptotically stable equation by equation.

The solution of the stability problem is carried out. in two steps :

(a) the development of an L₂-like norm that bounds above the sup norm, a requirement to obtain a useful estimate of the non-linear term, and

(a)the computation of an estimate of the L₂-likc norm of the semigroup generated by the linearized system in terms of its usual L₂ norm.

Notes

Work supported by the National Science Foundation, NSF-GK-23091.

Additional information

Notes on contributors

FERNANDO V. WATSON§

COPPE/UFRJ, CP 1191, Rio de Janeiro, GB, Brazil.