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Gustave
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I want to ask about the observability inequality for the heat equation ( internal control), we consider the backward heat equation with Dirichlet boundary conditions:

\begin{array}{c} \varphi _{t}+\Delta \varphi =0\text{ in }\Omega \times (0,T) \\ \varphi =0\text{ on } \partial \text{}\Omega \times (0,T)\text{} \\ \varphi (T)=\varphi _{T}\text{ }% \end{array} The observability inequality in some books is given by: $$ \left\Vert \varphi (0)\right\Vert ^{2}\geq c\int_{\omega }\int_{0}^{T}\varphi (x,t)dtdx $$$$ \left\Vert \varphi (0)\right\Vert ^{2}\leq c\int_{\omega }\int_{0}^{T}\varphi (x,t)dtdx $$ and some times $$\left\Vert \varphi (T)\right\Vert ^{2}\geq c\int_{\omega }\int_{0}^{T}\varphi (x,t)dtdx$$$$\left\Vert \varphi (T)\right\Vert ^{2}\leq c\int_{\omega }\int_{0}^{T}\varphi (x,t)dtdx$$ and $$\int_{\omega }\int_{T/4}^{3T/4}\varphi (x,t)dtdx\geq c\int_{\omega }\int_{0}^{T}\varphi (x,t)dtdx$$$$\int_{\omega }\int_{T/4}^{3T/4}\varphi (x,t)dtdx\leq c\int_{\omega }\int_{0}^{T}\varphi (x,t)dtdx$$ To passe from the first to the second I guess that they made the substitution $s=T-t$, however, for the third it comes from the golobal Carleman estimates but I can not see the equivalence between them. Thank you.

I want to ask about the observability inequality for the heat equation ( internal control), we consider the backward heat equation with Dirichlet boundary conditions:

\begin{array}{c} \varphi _{t}+\Delta \varphi =0\text{ in }\Omega \times (0,T) \\ \varphi =0\text{ on } \partial \text{}\Omega \times (0,T)\text{} \\ \varphi (T)=\varphi _{T}\text{ }% \end{array} The observability inequality in some books is given by: $$ \left\Vert \varphi (0)\right\Vert ^{2}\geq c\int_{\omega }\int_{0}^{T}\varphi (x,t)dtdx $$ and some times $$\left\Vert \varphi (T)\right\Vert ^{2}\geq c\int_{\omega }\int_{0}^{T}\varphi (x,t)dtdx$$ and $$\int_{\omega }\int_{T/4}^{3T/4}\varphi (x,t)dtdx\geq c\int_{\omega }\int_{0}^{T}\varphi (x,t)dtdx$$ To passe from the first to the second I guess that they made the substitution $s=T-t$, however, for the third it comes from the golobal Carleman estimates but I can not see the equivalence between them. Thank you.

I want to ask about the observability inequality for the heat equation ( internal control), we consider the backward heat equation with Dirichlet boundary conditions:

\begin{array}{c} \varphi _{t}+\Delta \varphi =0\text{ in }\Omega \times (0,T) \\ \varphi =0\text{ on } \partial \text{}\Omega \times (0,T)\text{} \\ \varphi (T)=\varphi _{T}\text{ }% \end{array} The observability inequality in some books is given by: $$ \left\Vert \varphi (0)\right\Vert ^{2}\leq c\int_{\omega }\int_{0}^{T}\varphi (x,t)dtdx $$ and some times $$\left\Vert \varphi (T)\right\Vert ^{2}\leq c\int_{\omega }\int_{0}^{T}\varphi (x,t)dtdx$$ and $$\int_{\omega }\int_{T/4}^{3T/4}\varphi (x,t)dtdx\leq c\int_{\omega }\int_{0}^{T}\varphi (x,t)dtdx$$ To passe from the first to the second I guess that they made the substitution $s=T-t$, however, for the third it comes from the golobal Carleman estimates but I can not see the equivalence between them. Thank you.

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Gustave
Gustave
  • 617
  • 3
  • 8

I want to ask about the observability inequality for the heat equation ( internal control), we consider the backward heat equation with Dirichlet boundary conditions:

\begin{array}{c} \varphi _{t}+\Delta \varphi =0\text{ in }\Omega \times (0,T) \\ \varphi =0\text{ on } \partial \text{}\Omega \times (0,T)\text{} \\ \varphi (T)=\varphi _{T}\text{ }% \end{array} The observability inequality in some books is given by: $$ \left\Vert \varphi (0)\right\Vert ^{2}\geq c\int_{\omega }\int_{0}^{T}\varphi (x,t)dtdx $$ and some times $$\left\Vert \varphi (T)\right\Vert ^{2}\geq c\int_{\omega }\int_{0}^{T}\varphi (x,t)dtdx$$ and $$\int_{\omega }\int_{T/4}^{3T/4}\varphi (x,t)dtdx\geq c\int_{\omega }\int_{0}^{T}\varphi (x,t)dtdx$$ To passe from the firtfirst to the second I guess that they made the substitution $s=T-t$, however, for the thirthird it comes from the golobal Carleman estimates but I can not see the equivalence between them. Thank you.

I want to ask about the observability inequality for the heat equation ( internal control), we consider the backward heat equation with Dirichlet boundary conditions:

\begin{array}{c} \varphi _{t}+\Delta \varphi =0\text{ in }\Omega \times (0,T) \\ \varphi =0\text{ on } \partial \text{}\Omega \times (0,T)\text{} \\ \varphi (T)=\varphi _{T}\text{ }% \end{array} The observability inequality in some books is given by: $$ \left\Vert \varphi (0)\right\Vert ^{2}\geq c\int_{\omega }\int_{0}^{T}\varphi (x,t)dtdx $$ and some times $$\left\Vert \varphi (T)\right\Vert ^{2}\geq c\int_{\omega }\int_{0}^{T}\varphi (x,t)dtdx$$ and $$\int_{\omega }\int_{T/4}^{3T/4}\varphi (x,t)dtdx\geq c\int_{\omega }\int_{0}^{T}\varphi (x,t)dtdx$$ To passe from the firt to the second I guess that they made the substitution $s=T-t$, however, for the thir it comes from the golobal Carleman estimates but I can not see the equivalence between them. Thank you.

I want to ask about the observability inequality for the heat equation ( internal control), we consider the backward heat equation with Dirichlet boundary conditions:

\begin{array}{c} \varphi _{t}+\Delta \varphi =0\text{ in }\Omega \times (0,T) \\ \varphi =0\text{ on } \partial \text{}\Omega \times (0,T)\text{} \\ \varphi (T)=\varphi _{T}\text{ }% \end{array} The observability inequality in some books is given by: $$ \left\Vert \varphi (0)\right\Vert ^{2}\geq c\int_{\omega }\int_{0}^{T}\varphi (x,t)dtdx $$ and some times $$\left\Vert \varphi (T)\right\Vert ^{2}\geq c\int_{\omega }\int_{0}^{T}\varphi (x,t)dtdx$$ and $$\int_{\omega }\int_{T/4}^{3T/4}\varphi (x,t)dtdx\geq c\int_{\omega }\int_{0}^{T}\varphi (x,t)dtdx$$ To passe from the first to the second I guess that they made the substitution $s=T-t$, however, for the third it comes from the golobal Carleman estimates but I can not see the equivalence between them. Thank you.

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Gustave
Gustave
  • 617
  • 3
  • 8

Observability inequality for the heat equation

I want to ask about the observability inequality for the heat equation ( internal control), we consider the backward heat equation with Dirichlet boundary conditions:

\begin{array}{c} \varphi _{t}+\Delta \varphi =0\text{ in }\Omega \times (0,T) \\ \varphi =0\text{ on } \partial \text{}\Omega \times (0,T)\text{} \\ \varphi (T)=\varphi _{T}\text{ }% \end{array} The observability inequality in some books is given by: $$ \left\Vert \varphi (0)\right\Vert ^{2}\geq c\int_{\omega }\int_{0}^{T}\varphi (x,t)dtdx $$ and some times $$\left\Vert \varphi (T)\right\Vert ^{2}\geq c\int_{\omega }\int_{0}^{T}\varphi (x,t)dtdx$$ and $$\int_{\omega }\int_{T/4}^{3T/4}\varphi (x,t)dtdx\geq c\int_{\omega }\int_{0}^{T}\varphi (x,t)dtdx$$ To passe from the firt to the second I guess that they made the substitution $s=T-t$, however, for the thir it comes from the golobal Carleman estimates but I can not see the equivalence between them. Thank you.