论文标题
Hilfer时交叉微分方程的近似和平均可控性能
Approximate and mean approximate controllability properties for Hilfer time-fractional differential equations
论文作者
论文摘要
我们研究了与所谓的Hilfer类型的时间差异导数和非负性自我ADXHIXHIXHIXEXHING OPITOR $ A_B $具有$ l^2（ω）$的$ l^2（ω）$的近似类型的时间差异衍生物和非负性自我偶发算子$ a_b $相关的分数部分差分方程的近似和近似可控性能。更准确地说，我们表明，如果$ 0 \leν\ le 1 $，$ 0 <μ\ le 1 $和$ω\ subset \ subset \ mathbb r^n $是一个有界的开放集，则系统$ \ mathbb d_t^{μ，n $，num，num，ushbb d_t^{μ，n $， \ mbox {in} \; ω\ times（0，t），\，\，（\ Mathbb i_t^{（1-ν）（1-μ）} u）（\ cdot，0）= u_0 \ mbox {in} \ emum，$ the $ t> 0 $ t> 0 $，$ u_0 \ in l^2（$ undy）y y y lnont oction y y y lnonry opery y y y y opery y y y.chormy y ynony obley ynony obley ynony $ $ $ $ $ $ $。此外，如果操作员$ a_b $具有唯一的延续属性，则系统也大致可控制。运算符$ a_b $可以是在$ l^2（ω）$中实现的对称，非负椭圆形的二阶二阶操作员，或带有dirichlet或robin边界条件的$ l^2（ω）$在$ l^2（ω）$中实现，分数laplace laplace运算符$（ - δ） $ \ mathbb r^n \setMinusΩ$，或非局部罗宾外部条件，$ \ mathcal n^su+βu= 0 $ = 0 $ in $ \ mathbb r^n \ setMinus \overlineΩ$。
We study the approximate and mean approximate controllability properties of fractional partial differential equations associated with the so-called Hilfer type time-fractional derivative and a non-negative selfadjoint operator $A_B$ with a compact resolvent on $L^2(Ω)$, where $Ω\subset\mathbb{R}^N$ ($N\ge 1$) is a bounded open set. More precisely, we show that if $0\leν\le 1$, $0<μ\le 1$ and $Ω\subset\mathbb R^N$ is a bounded open set, then the system $$\mathbb D_t^{μ,ν} u+A_Bu=f|_ω\;\; \mbox{ in }\; Ω\times (0,T),\,\, (\mathbb I_t^{(1-ν)(1-μ)}u)(\cdot,0)=u_0 \mbox{ in }\;Ω,$$ is approximately controllable for any $T>0$, $u_0\in L^2(Ω)$ and any non-empty open set $ω\subsetΩ$. In addition, if the operator $A_B$ has the unique continuation property, then the system is also mean approximately controllable. The operator $A_B$ can be the realization in $L^2(Ω)$ of a symmetric, non-negative uniformly elliptic second order operator with Dirichlet or Robin boundary conditions, or the realization in $L^2(Ω)$ of the fractional Laplace operator $(-Δ)^s$ ($0<s<1$) with the Dirichlet exterior condition, $u=0$ in $\mathbb R^N\setminusΩ$, or the nonlocal Robin exterior condition, $\mathcal N^su+βu=0$ in $\mathbb R^N\setminus\overlineΩ$.
