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The Cost of Null Controllability for a Backward Stochastic Degenerate Parabolic Equation in the Vanishing Viscosity Limit

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Abstract

In this paper, we investigate the cost of null controllability for a backward stochastic degenerate parabolic equation with a boundary control in the vanishing viscosity limit. Firstly we obtain a new Carleman estimate for the adjoint stochastic equation. Combining this Carleman estimate and an exponential dissipation estimate, we obtain the uniform null controllability for large control time. When the waiting time tends to 0, we prove that the cost of null controllability increases to infinity.

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Acknowledgements

The authors thank all the anonymous referees for their valuable suggestions and comments, which make the paper much improved.

Funding

This work is supported by NSFC (No.12171248)

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Authors and Affiliations

  1. School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, People’s Republic of China

    Qun Chen & Bin Wu

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  1. Qun Chen
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  2. Bin Wu
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Correspondence to Bin Wu.

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Appendix

Appendix

This appendix is devoted to proving Lemma 2.1.

Proof of Lemma 2.1

Notice that \(\theta =e^l\), \(l=s\varphi \) and \(p=e^\zeta \theta z\). It is easy to verify that

$$\begin{aligned} e^\zeta \theta \textrm{d}z=\textrm{d}p-l_t p\textrm{d}t, \end{aligned}$$
(A.1)

and

$$\begin{aligned} \varepsilon e^\zeta \theta (x^\alpha z_x)_x\textrm{d}t=&\left[ \varepsilon e^\zeta (e^{-\zeta } x^\alpha p_x)_x-\varepsilon e^\zeta (e^{-\zeta } x^\alpha l_x)_x p-2\varepsilon x^\alpha l_x p_x\right. \nonumber \\&\left. +\varepsilon x^\alpha l_x^2 p-\varepsilon e^\zeta \theta (x^\alpha \zeta _x z)_x\right] \textrm{d}t. \end{aligned}$$
(A.2)

Noting that \(x^\alpha \zeta _x=-\frac{M}{\varepsilon } \), we then obtain

$$\begin{aligned}&e^\zeta \theta \left[ \textrm{d}z-\varepsilon (x^\alpha z_x)_x\textrm{d}t+M z_x\textrm{d}t\right] \nonumber \\&\quad = \textrm{d}p+\varepsilon e^\zeta (e^{-\zeta } x^\alpha l_x)_x p\textrm{d}t+2\varepsilon x^\alpha l_x p_x\textrm{d}t\nonumber \\&\qquad +\left[ -\varepsilon e^\zeta (e^{-\zeta } x^\alpha p_x)_x-\varepsilon x^\alpha l_x^2 p-l_t p\right] \textrm{d}t. \end{aligned}$$
(A.3)

Therefore,

$$\begin{aligned}&\theta \left[ -\varepsilon e^\zeta (e^{-\zeta } x^\alpha p_x)_x-\varepsilon x^\alpha l_x^2 p-l_t p\right] \left[ \textrm{d}z-\varepsilon (x^\alpha z_x)_x\textrm{d}t+Mz_x\textrm{d}t\right] \nonumber \\&\quad = e^{-\zeta }\left[ -\varepsilon e^\zeta (e^{-\zeta } x^\alpha p_x)_x-\varepsilon x^\alpha l_x^2 p-l_t p\right] \textrm{d}p\nonumber \\&\qquad +\varepsilon \left[ -\varepsilon e^\zeta (e^{-\zeta } x^\alpha p_x)_x-\varepsilon x^\alpha l_x^2 p-l_t p\right] (e^{-\zeta } x^\alpha l_x)_x p\textrm{d}t\nonumber \\&\qquad +2\varepsilon e^{-\zeta }\left[ -\varepsilon e^\zeta (e^{-\zeta } x^\alpha p_x)_x-\varepsilon x^\alpha l_x^2 p-l_t p\right] x^\alpha l_x p_x\textrm{d}t\nonumber \\&\qquad +e^{-\zeta }\left[ -\varepsilon e^\zeta (e^{-\zeta } x^\alpha p_x)_x-\varepsilon x^\alpha l_x^2 p-l_t p\right] ^2\textrm{d}t. \end{aligned}$$
(A.4)

Now we deal with the first three terms on the right-hand side of (A.4) one by one. For the first term, applying Itô’s formula, we have

$$\begin{aligned}&e^{-\zeta }\left[ -\varepsilon e^\zeta (e^{-\zeta }x^\alpha p_x)_x-\varepsilon x^\alpha l_x^2 p-l_t p\right] \textrm{d}p\nonumber \\&\quad = -(\varepsilon e^{-\zeta }x^\alpha p_x\textrm{d}p )_x+\textrm{d}\left( \frac{1}{2}\varepsilon e^{-\zeta }x^\alpha p_x^2\right) -\frac{1}{2}\varepsilon e^{-\zeta }x^\alpha (\textrm{d}p_x)^2\nonumber \\&\qquad -\textrm{d}\left( \frac{1}{2}\varepsilon e^{-\zeta } x^\alpha l_x^2 p^2\right) +\varepsilon e^{-\zeta } x^\alpha l_x l_{xt} p^2\textrm{d}t+\frac{1}{2}\varepsilon e^{-\zeta } x^\alpha l_x^2 (\textrm{d}p)^2\nonumber \\&\qquad -\textrm{d}\left( \frac{1}{2} e^{-\zeta } l_t p^2\right) +\frac{1}{2} e^{-\zeta } l_{tt} p^2\textrm{d}t+\frac{1}{2} e^{-\zeta } l_t(\textrm{d} p)^2. \end{aligned}$$
(A.5)

The second one can be rewritten as

$$\begin{aligned}&\varepsilon \left[ -\varepsilon e^\zeta (e^{-\zeta } x^\alpha p_x)_x-\varepsilon x^\alpha l_x^2 p-l_t p\right] (e^{-\zeta } x^\alpha l_x)_x p\textrm{d}t\nonumber \\&\quad = \left[ -\varepsilon ^2 (e^{-\zeta }x^\alpha l_x)_x x^\alpha p p_x \right] _x\textrm{d}t+\varepsilon ^2 e^{-\zeta } x^\alpha \left[ e^\zeta (e^{-\zeta }x^\alpha l_x)_x p\right] _x p_x\textrm{d}t \nonumber \\&\qquad -\varepsilon ^2 (e^{-\zeta } x^\alpha l_x)_x x^{\alpha } l_x^2 p^2\textrm{d}t-\varepsilon (e^{-\zeta } x^\alpha l_x)_x l_t p^2\textrm{d}t\nonumber \\&\quad = \left[ -\varepsilon ^2 (e^{-\zeta }x^\alpha l_x)_x x^\alpha p p_x \right] _x\textrm{d}t+\varepsilon ^2 (e^{-\zeta }x^\alpha l_x)_x x^\alpha p_x^2\textrm{d}t\nonumber \\&\qquad +\frac{1}{2} \varepsilon ^2e^{-\zeta }x^\alpha \left[ e^\zeta (e^{-\zeta }x^\alpha l_x)_x \right] _x (p^2)_x\textrm{d}t-\varepsilon ^2 (e^{-\zeta }x^\alpha l_x)_x x^{\alpha } l_x^2 p^2\textrm{d}t \nonumber \\&\qquad -\varepsilon (e^{-\zeta } x^\alpha l_x)_x l_t p^2\textrm{d}t\nonumber \\&\quad = \left[ -\varepsilon ^2 (e^{-\zeta }x^\alpha l_x)_x x^\alpha p p_x \right] _x\textrm{d}t+\varepsilon ^2 (e^{-\zeta }x^\alpha l_x)_x x^\alpha p_x^2\textrm{d}t\nonumber \\&\qquad +\left\{ \frac{1}{2}\varepsilon ^2 e^{-\zeta }x^\alpha \left[ e^\zeta (e^{-\zeta }x^\alpha l_x)_x \right] _x p^2\right\} _x\textrm{d}t-\frac{1}{2} \varepsilon ^2 \left\{ e^{-\zeta } x^\alpha \left[ e^\zeta (e^{-\zeta }x^\alpha l_x)_x \right] _x \right\} _x p^2 \textrm{d}t\nonumber \\&\qquad -\varepsilon ^2 (e^{-\zeta } x^\alpha l_x)_x x^{\alpha } l_x^2 p^2\textrm{d}t-\varepsilon (e^{-\zeta } x^\alpha l_x)_x l_t p^2\textrm{d}t. \end{aligned}$$
(A.6)

For the last one, we have

$$\begin{aligned}&2\varepsilon e^{-\zeta }\left[ -\varepsilon e^\zeta ( e^{-\zeta } x^\alpha p_x)_x-\varepsilon x^\alpha l_x^2 p-l_t p\right] x^\alpha l_x p_x\textrm{d}t\nonumber \\&\quad = -2\varepsilon ^2 \left[ (e^{-\zeta } x^\alpha )_x p_x+ e^{-\zeta }x^\alpha p_{xx}\right] x^\alpha l_x p_x\textrm{d}t-\varepsilon ^2 e^{-\zeta }x^{2\alpha } l_x^3 (p^2)_x\textrm{d}t\nonumber \\&\qquad -\varepsilon e^{-\zeta }x^\alpha l_xl_t (p^2)_x\textrm{d}t\nonumber \\&\quad = -\varepsilon ^2 (e^{-\zeta } x^\alpha )_x x^\alpha l_x p_x^2\textrm{d}t-\left( \varepsilon ^2 e^{-\zeta } x^{2\alpha } l_x p_x^2\right) _x\textrm{d}t+\varepsilon ^2 e^{-\zeta } x^\alpha (x^\alpha l_x)_x p_x^2\textrm{d}t\nonumber \\&\qquad -\left( \varepsilon ^2 e^{-\zeta } x^{2\alpha } l_x^3 p^2\right) _x\textrm{d}t+\varepsilon ^2 \left( e^{-\zeta } x^{2\alpha } l_x^3\right) _x p^2\textrm{d}t-(\varepsilon e^{-\zeta } x^\alpha l_xl_t p^2)_x \textrm{d}t\nonumber \\&\qquad +\varepsilon \left( e^{-\zeta } x^\alpha l_x l_t\right) _x p^2\textrm{d}t. \end{aligned}$$
(A.7)

Finally, by substituting (A.5)–(A.7) into (A.4), we arrive at the desired equality (2.4). This completes the proof. \(\square \)

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Chen, Q., Wu, B. The Cost of Null Controllability for a Backward Stochastic Degenerate Parabolic Equation in the Vanishing Viscosity Limit. Appl Math Optim 89, 20 (2024). https://doi.org/10.1007/s00245-023-10091-5

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