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Erschienen in:
2020 | OriginalPaper | Buchkapitel
11. Existence and Stability for Semilinear Differential Equations
verfasst von : Ruth Curtain, Hans Zwart
Erschienen in: Introduction to Infinite-Dimensional Systems Theory
Verlag: Springer New York
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Abstract
In the previous chapter we treated linear systems. In this chapter we study the semilinear differential equation \(\dot{z}(t) = A z(t) + f(z(t))\), with A the infinitesimal generator of strongly continuous semigroup. We do this for two cases, the first one in which we assume that f is Lipschitz continuous on the state space, and the second one, where we assume that A generates a holomorphic semigroup and f is Lipschitz continuous on the domain of a fractional power of A, i.e., \(D(A^{\alpha })\), \(\alpha \in (0,1)\). Since for non-linear systems stability is studied via Lyapunov function, this is treated in detail. Among others, an infinite-dimensional version of LaSalle’s invariance theorem is proved. The chapter ends with a set of 25 exercises and a notes and references section.