A Relaxation Theorem for Differential Inclusions with Applications to Stability Properties

Brian P. Ingalls; Eduardo D. Sontag; Yuan Wang

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A Relaxation Theorem for Differential Inclusions with Applications to Stability Properties

Brian P. Ingalls·Eduardo D. Sontag·Yuan Wang

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Abstract

The fundamental Filippov-Wazwski Relaxation Theorem states that the solution set of an initial value problem for a locally Lipschitz inclusion is dense in the solution set of the same initial value problem for the corresponding relaxation inclusion on compact intervals. In our recent work, a complementary result was provided for inclusions with finite dimensional state spaces which says that the approximation can be carried out over non-compact or infinite intervals provided one does not insist on the same initial values. This note extends the infinite-time relaxation theorem to the inclusions whose state spaces are Banach spaces. To illustrate the motivations for studying such approximation results, we briefly discuss a quick application of the result to output stability and uniform output stability properties.

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Brian P. Ingalls, Eduardo D. Sontag, Yuan Wang. 2002-06-24. A Relaxation Theorem for Differential Inclusions with Applications to Stability Properties. https://arxiv.org/abs/math/0206251

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