This paper deals with the input-output stability of a class of nonlinear distributed systems defined by their Laplace-Green’s functions, or similarly from a practical point of view, by their distributed transfer functions. The time dependent nonlinear feedback element is distributed and bounded by two limiting gains which depend explicitly upon the distributed parameter. These systems are disturbed by a state and space dependent Gaussian noise which is added to the input of their linear components. This noise depends explicitly upon the output of the system via its own nonlinear feedback gain. Some input-output stability criteria are stated, which can be considered as being stochastic distributed versions of the circle criterion available for deterministic lumped parameter systems. They involve the stochastic mean square norm and they are expressed in term of the relative positions, in the complex plane, of a circle which depends upon the nonlinearities and the variance of the noise on the one hand; and a locus which may be interpreted as being the Nyquist locus of the linear part on the other hand.

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