학술논문

Some averaging results for ordinary differential inclusions.
Document Type
Journal
Author
Bourada, Amel (DZ-UDL-LM) AMS Author Profile; Guen, Rahma (DZ-UDL-LM) AMS Author Profile; Lakrib, Mustapha (DZ-UDL-LM) AMS Author Profile; Yadi, Karim (DZ-ABBT-SDA) AMS Author Profile
Source
Discussiones Mathematicae. Differential Inclusions, Control and Optimization (Discuss. Math. Differ. Incl. Control Optim.) (20150101), 35, no. 1, 47-63. ISSN: 1509-9407 (print).eISSN: 2084-0365.
Subject
34 Ordinary differential equations -- 34A General theory
  34A60 Differential inclusions
Language
English
ISSN
20840365
Abstract
In this paper, the authors study the possibility of applying the fullaveraging method to ordinary differential inclusions of the form$$\dot x(t) \in F\left(\frac t\varepsilon , x(t)\right),$$where $\varepsilon > 0$ denotes the small perturbation parameter, $t\in [0,L]$ the time variable and $F$ a multifunction with values thatare nonempty compact convex subsets of ${\Bbb R}^d$. They make thefollowing assumptions on the multivalued mapping $F(t,x)$:\roster\item mapping$F(\cdot, x)$ is measurable for all $x \in U$;\item thereexist a locally Lebesgue integrable function $b \: {\Bbb R}_+ \to {\BbbR}_+$ such that $ {\|F(t,x)\|\leq b(t)} $ for all $x\in U$ and $t\in{\Bbb R}_+$, and $\lim_{T\to\infty}\frac 1T\int_0^T b(t)dt =B>0$;\item thereexists a limit, in the sense of the Hausdorff metric $\rho$,$$\lim\limits_{T\to\infty}\frac 1T\int\limits _0^T F(t,x)dt=\overline{F}(x)$$for all $x \in U$;\item thereexists a constant $\lambda > 0$ such that$$\rho\left (\int\limits _{t_1}^{t_2}F(t,u(t))dt, \int\limits_{t_1}^{t_2}F(t,v(t))dt\right)\leq \lambda \int\limits _{t_1}^{t_2}|u(t)-v(t)|dt,$$for any continuous functions $u, v \: {\Bbb R}_+ \to U$ and any real$t_1, t_2 \in {\Bbb R}_+$, $t_1 \leq t_2$.\endroster