Transformation Operators and Modified Sobolev Spaces in Controllability Problems on a Half-Axis
DOI:
https://doi.org/10.15407/mag12.01.017Анотація
Розглянуто керовану систему $w_{tt}=\frac{1}{\rho}(kw_x)_x+\gamma w$, $w_x(0,t)=u(t)$, $x>0$, $t\in(0,T)$, у спеціальних модифікованих просторах соболєвського типу. Тут $\rho,k,\gamma$ - задані на $[0,+\infty)$ функції; $u\in L^\infty(0,\infty)$ - керування; $T>0$ - константа. Зростання розподілів із цих просторів залежить від зростання $\rho$ та $k$. За допомогою деяких операторів перетворення показано, що ця керована система відтворює властивості керованості допоміжної системи $z_{tt}=z_{\xi\xi}-q^2z$, $z_\xi(0,t)=v(t)$, $\xi>0$, $t\in(0,T)$, і навпаки. Тут $q\geq 0$ - константа, а $v\in L^\infty(0,\infty)$ - керування. Одержано необхідні та достатні умови $L^\infty$-керованості та наближеної $L^\infty$-керованості з уже відомих умов для допоміжної системи.Mathematics Subject Classification: 93B05, 35B37, 35L05.
Ключові слова:
хвильове рівняння, піввісь, проблема керованості, оператор перетворення, модифікований простір соболєвського типуПосилання
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