具有无界算子的进化方程的全局可解性

IF 0.6 Q3 MATHEMATICS
A. Chernov
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引用次数: 2

摘要

设$X$是Hilbert空间,$U$是Banach空间,$G\colon X\to X$是一个线性算子,使得算子$B_\lambda=\lambda I-G$是具有某种(任意给定)$\lambda\in\mathbb{R}$的极大单调。对于与控制的半线性演化方程相关的柯西问题\[x^\prime(t)=Gx(t)+f\bigl( t,x(t),u(t)\bigr),\quad t\in[0;T];\quad x(0)=x_0\in X,\],其中$u=u(t)\colon[0;T]\to U$为控制,$x(t)$为未知函数,值在$X$,我们根据空间$\mathbb{R}$中与某常微分方程相关的柯西问题的全局可解性,证明了完全(关于一组允许控制)全局可解性。在弱连续函数空间$\mathbb{C}_w\bigl([0;T];X\bigr)$中寻找弱意义上的解$x$。事实上,对于有界算子$G$,我们推广了前人证明的类似结果。这种推广的本质在于,算子$B_\lambda$的假设性质使我们有可能用有界线性算子构造Yosida近似,从而将所需的估计从“有界”推广到“无界”情况。作为例子,我们考虑与热方程和波动方程相关的初边值问题。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On totally global solvability of evolutionary equation with unbounded operator
Let $X$ be a Hilbert space, $U$ be a Banach space, $G\colon X\to X$ be a linear operator such that the operator $B_\lambda=\lambda I-G$ is maximal monotone with some (arbitrary given) $\lambda\in\mathbb{R}$. For the Cauchy problem associated with controlled semilinear evolutionary equation as follows \[x^\prime(t)=Gx(t)+f\bigl( t,x(t),u(t)\bigr),\quad t\in[0;T];\quad x(0)=x_0\in X,\] where $u=u(t)\colon[0;T]\to U$ is a control, $x(t)$ is unknown function with values in $X$, we prove the totally (with respect to a set of admissible controls) global solvability subject to global solvability of the Cauchy problem associated with some ordinary differential equation in the space $\mathbb{R}$. Solution $x$ is treated in weak sense and is sought in the space $\mathbb{C}_w\bigl([0;T];X\bigr)$ of weakly continuous functions. In fact, we generalize a similar result having been proved by the author formerly for the case of bounded operator $G$. The essence of this generalization consists in that postulated properties of the operator $B_\lambda$ give us the possibility to construct Yosida approximations for it by bounded linear operators and thus to extend required estimates from “bounded” to “unbounded” case. As examples, we consider initial boundary value problems associated with the heat equation and the wave equation.
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来源期刊
CiteScore
1.20
自引率
40.00%
发文量
27
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