## Lyapunov pairs for continuous perturbations of nonlinear evolutions.(English)Zbl 1173.37009

For the abstract Cauchy problem $y'(t)\in Ay(t) + F(y(t))\;,\;y(0)=\xi$ with $$A:D(A)\subseteq X\mapsto X$$ a (possibly) multivalued $$m$$-dissipative operator, $$X$$ being a real Banach space, $$\emptyset\neq M\subset \overline{D(A)}$$ and $$F:M\mapsto X$$ a function. The pair $$V,g:X\mapsto R$$ is a Liapunov pair of the above equation if $$\mathrm{dom} (V)\subseteq M$$ and the dissipativeness inequality holds $V(y(t)) + \int_0^t g(y(\tau))d\tau \leq V(\xi)\;,\;0\leq t\leq T$ The necessary and sufficient condition for $$(V,g)$$ to be a Liapunov pair is $\underline{D}^AV(\xi)F(\xi)+g(\xi)\leq 0\;,\;\xi\in \mathrm{dom}(V)$ where $$\underline{D}^AV(\xi)(v)$$ is the $$A$$-contingent derivative of $$V$$ at $$\xi\in \mathrm{dom}(V)$$ in the direction $$v\in X$$.

### MSC:

 37B25 Stability of topological dynamical systems 47J35 Nonlinear evolution equations 93C25 Control/observation systems in abstract spaces 93D30 Lyapunov and storage functions
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### References:

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