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We present a new theorem on the differential inequality $u^{(m)} \leq w (u)$ . Next, we apply this result to obtain existence theorems for the equation $x^{(m)} = f (t, x)$ .

Oszillationsmethoden für Systeme gewöhnlicher Differentialgleichungen.

Joachim Weidmann (1971)

Mathematische Zeitschrift

Output feedback regulation for large-scale uncertain nonlinear systems with time delays

Shutang Liu, Weiyong Yu, Fangfang Zhang (2015)

Kybernetika

This paper is concerned with the problem of global state regulation by output feedback for large-scale uncertain nonlinear systems with time delays in the states and inputs. The systems are assumed to be bounded by a more general form than a class of feedforward systems satisfying a linear growth condition in the unmeasurable states multiplying by unknown growth rates and continuous functions of the inputs or delayed inputs. Using the dynamic gain scaling technique and choosing the appropriate Lyapunov-Krasovskii...

Overstability and resonance

Augustin Fruchard, Reinhard Schäfke (2003)

Annales de l’institut Fourier

We consider a singularity perturbed nonlinear differential equation $ε u^{'} = f (x) u + + ε P (x, u, ε)$ which we suppose real analytic for $x$ near some interval $[a, b]$ and small $| u |$ , $| ε |$ . We furthermore suppose that 0 is a turning point, namely that $x f (x)$ is positive if $x \neq 0$ . We prove that the existence of nicely behaved (as $ϵ \to 0$ ) local (at $x = 0$ ) or global, real analytic or $C^{\infty}$ solutions is equivalent to the existence of a formal series solution $\sum u_{n} (x) ε^{n}$ with $u_{n}$ analytic at $x = 0$ . The main tool of a proof is a new “principle of analytic continuation” for such “overstable” solutions....

Oб инвариантности характеристических показателей линейных систем при экспоненциально убывающих возмущениях

Nikolai Alekseevich Izobov, O. P. Stepanovich (1990)

Archivum Mathematicum

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