Null controllability of Kolmogorov-type equations

Abstract

We study the null controllability of Kolmogorov-type equations $\partial _t f + v^\gamma \partial _x f - \partial _v^2 f = u(t,x,v) 1_{\omega }(x,v)$ in a rectangle $\Omega$, under an additive control supported in an open subset $\omega$ of $\Omega$. For $\gamma =1$, with periodic-type boundary conditions, we prove that null controllability holds in any positive time, with any control support $\omega$. This improves the previous result by Beauchard and Zuazua (Ann Ins H Poincaré Anal Non Linéaire 26:1793–1815, 2009), in which the control support was a horizontal strip. With Dirichlet boundary conditions and a horizontal strip as control support, we prove that null controllability holds in any positive time if $\gamma =1$ or if $\gamma =2$ and $\omega$ contains the segment $\{v=0\}$, and only in large time if $\gamma =2$ and $\omega$ does not contain the segment $\{v=0\}$. Our approach, inspired from Benabdallah et al. (C R Math Acad Sci Paris 344(6):357–362, 2007), Lebeau and Robbiano (Commun Partial Differ Equ 20:335–356, 1995), is based on two key ingredients: the observability of the Fourier components of the solution of the adjoint system, uniformly with respect to the frequency, and the explicit exponential decay rate of these Fourier components.

Appendix: Proof of Proposition 1

Let $a^{\prime }, b^{\prime }$ be such that $-1<a<a^{\prime }<b^{\prime }<b<1$. All the computations of the proof will be made assuming, first, $g \in H^1(0,T;L^2(-1,1)) \cap L^2(0,T;H^2 \cap H^1_0(-1,1))$. Then, the conclusion of Proposition 1 will follow by a density argument.

Consider the weight function

$$\begin{aligned} \alpha (t,v):=\frac{M \beta (v)}{t(T-t)}\, ,\quad (t,v) \in (0,T) \times \mathbb R , \end{aligned}$$

(35)

where $\beta \in C^2([-1,1])$ satisfies

$$\begin{aligned}&\beta \geqslant 1\quad \text{ on }\quad (-1,1),\end{aligned}$$

(36)

$$\begin{aligned}&|\beta ^{\prime }|>0 \quad \text{ on }\quad [-1,a^{\prime }] \cup [b^{\prime },1],\end{aligned}$$

(37)

$$\begin{aligned}&\beta ^{\prime }(1)>0,\quad \beta ^{\prime }(-1)<0,\end{aligned}$$

(38)

$$\begin{aligned}&\beta ^{\prime \prime } <0\quad \text{ on } \quad [-1,a^{\prime }] \cup [b^{\prime },1] \end{aligned}$$

(39)

and $M = M(T,n,\beta )>0$ will be chosen later on. We also introduce the function

$$\begin{aligned} z(t,v):=g(t,v) e^{-\alpha (t,v)}, \end{aligned}$$

(40)

that satisfies

$$\begin{aligned} e^{-\alpha } \mathcal P _n g = P_{1}z + P_{2} z + P_{3} z, \end{aligned}$$

(41)

where

$$\begin{aligned} P_{1}z:= - \frac{\partial ^{2} z}{\partial v^{2}} +(\alpha _{t}-\alpha _{v}^{2}) z ,\, P_{2}z:= \frac{\partial z}{\partial t} - 2 \alpha _{v} \frac{\partial z}{\partial v} + i n v^\gamma z,\, P_{3}z:= - \alpha _{vv} z. \end{aligned}$$

(42)

We develop the classical proof (see [25]), taking the $L^{2}(Q)$-norm in the identity (41), then developing the double product which leads to

$$\begin{aligned} \int \limits _{Q} \left( \mathfrak{R }[ P_{1}z \overline{P_{2}z}] - \frac{1}{2} |P_{3}z|^{2} \right) \mathrm{d}v \mathrm{d}t \leqslant \int \limits _{Q} | e^{-\alpha } \mathcal{P }_n g |^2 \mathrm{d}v \mathrm{d}t, \end{aligned}$$

(43)

where $Q:=(0,T) \times (-1,1)$. After computations (see [5] for details, we get

$$\begin{aligned}&\int \limits _{Q} |z|^{2} \left\{ \!-\! \frac{1}{2} (\alpha _{t}-\alpha _{v}^{2})_{t} \!+\! \left[ (\alpha _{t}-\alpha _{v}^{2}) \alpha _{v} \right] _{v} \!-\! \frac{1}{2}\alpha _{vv}^{2} \right\} \mathrm{d}v \mathrm{d}t \nonumber \\&\quad \!+\! \int \limits _{Q} \left\{ n \gamma v^{\gamma -1} \mathfrak I \left( \overline{z} \frac{\partial z}{\partial v} \right) - \alpha _{vv} \Big | \frac{\partial z}{\partial v} \Big |^{2} \alpha _{vv} \right\} \mathrm{d}v\mathrm{d}t \leqslant \int \limits _{Q} | e^{-\alpha } \mathcal P _n g |^2 \mathrm{d}v \mathrm{d}t. \end{aligned}$$

(44)

Now, in the left hand side of (44), we separate the terms on $(0,T) \times (a^{\prime },b^{\prime })$ and those on $(0,T)\times [(-1,a^{\prime }) \cup (b^{\prime },1)]$. One has

$$\begin{aligned}&-\alpha _{vv}(t,v) \geqslant \frac{C_{1} M}{t(T-t)} \qquad \forall v \in [-1,a^{\prime }] \cup [b^{\prime },1], \ t \in (0,T),\nonumber \\&|\alpha _{vv}(t,v)| \leqslant \frac{C_{2} M}{t(T-t)} \qquad \forall v \in [a^{\prime }b^{\prime }], \ t \in (0,T), \end{aligned}$$

(45)

where $C_{1}=C_1(\beta ):=\min \{ -\beta ^{\prime \prime }(x) ; x \in [-1,a^{\prime }] \cup [b^{\prime },1] \}$ is positive thanks to the assumption (39) and $C_{2}=C_2(\beta ):=\sup \{ |\beta ^{\prime \prime }(x)| ; x \in [a^{\prime },b^{\prime }] \}$. Moreover,

$$\begin{aligned} \begin{array}{l} \displaystyle -\frac{1}{2}(\alpha _{t}-\alpha _{v}^{2})_{t} + [ (\alpha _{t}-\alpha _{v}^{2}) \alpha _{v} ]_{v} - \frac{1}{2} \alpha _{vv}^{2} = \frac{1}{(t(T-t))^3} \left\{ M\beta (3Tt - T^{2} -3t^{2}) \right. \nonumber \\ \qquad \left. \displaystyle + M^{2} \left[ (2t-T) (\beta ^{\prime \prime } \beta +2\beta ^{\prime 2}) - \frac{t(T-t)\beta ^{\prime \prime ^2}}{2} \right] - 3M^{3} \beta ^{\prime \prime } \beta ^{\prime 2} \right\} . \end{array} \end{aligned}$$

Hence, owing to (37) and (39), there exist $m_1=m_1(\beta )>0\,C_{3}=C_{3}(\beta )>0$ and $C_{4}=C_{4}(\beta )>0$ such that for every $M \geqslant M_1$ and $t \in (0,T)$,

$$\begin{aligned}&\displaystyle -\frac{1}{2}(\alpha _{t}-\alpha _{v}^{2})_{t} \displaystyle + \left[ (\alpha _{t}-\alpha _{v}^{2}) \alpha _{v}\right] _{v} \displaystyle - \frac{1}{2} \alpha _{vv}^{2} \geqslant \frac{C_{3}M^{3}}{[t(T-t)]^{3}}\quad \forall v \in [0,a^{\prime }] \cup [b^{\prime },1],\nonumber \\&\quad \displaystyle \Big | -\frac{1}{2}(\alpha _{t}-\alpha _{v}^{2})_{t} + \left[ (\alpha _{t}-\alpha _{v}^{2}) \alpha _{v}\right] _{v} - \frac{1}{2} \alpha _{vv}^{2} \Big | \leqslant \frac{C_{4}M^{3}}{[t(T-t)]^{3}}\quad \forall v \in [a^{\prime },b^{\prime }]\, \end{aligned}$$

(46)

where

$$\begin{aligned} M_1=M_{1}(T,\beta ):=m_1(\beta )(T+T^2). \end{aligned}$$

(47)

Using (44), (45) and (46), we deduce, for every $M \geqslant M_{1}$,

$$\begin{aligned}&\displaystyle \int \limits _{0}^{T} \int \limits _{(-1,a^{\prime })\cup (b^{\prime },1)} \frac{C_{1} M}{t(T-t)} \Big | \frac{\partial z}{\partial v} \Big |^{2} \mathrm{d}v\mathrm{d}t\nonumber \\&\qquad + \displaystyle \int \limits _{0}^{T} \int \limits _{(-1,a^{\prime })\cup (b^{\prime },1)} \left[ \frac{C_{3} M^{3}}{(t(T-t))^{3}} |z|^{2} - |n| \gamma |v|^{\gamma -1} \mathfrak I \left( \frac{\partial z}{\partial v} \overline{z} \right) \right] \mathrm{d}v \mathrm{d}t\nonumber \\&\qquad \displaystyle \leqslant \int \limits _{0}^{T} \int \limits _{a^{\prime }}^{b^{\prime }} \left[ \frac{C_{2} M}{t(T-t)} \Big | \frac{\partial z}{\partial x} \Big |^{2} + \frac{C_{4} M^{3}}{(t(T-t))^{3}} |z|^{2} + |n| \gamma v^{\gamma -1} \mathfrak I \left( \frac{\partial z}{\partial v} \overline{z} \right) \right] \mathrm{d}v \mathrm{d}t\nonumber \\&\qquad \displaystyle + \int \limits _{Q} | e^{-\alpha } \mathcal P _n g |^2 \mathrm{d}v \mathrm{d}t. \end{aligned}$$

(48)

Let

$$\begin{aligned} M_2=M_2(T,\beta ):=\frac{T^2 \sqrt{|n| \gamma }}{4 \root 4 \of {C_1 C_3}}. \end{aligned}$$

(49)

When $M \geqslant M_2$, we have

$$\begin{aligned} \begin{array}{ll} \Big |n \gamma v^{\gamma -1} \mathfrak I \left( \frac{\partial z}{\partial v} \overline{z} \right) \Big | &{} \leqslant \frac{1}{2} \frac{ C_{3} M^{3} }{ (t(T-t))^{3} } |z|^{2} + \frac{1}{2} \frac{ (t(T-t))^{3} }{ C_{3} M^{3} } \gamma ^2 n^{2} \Big | \frac{\partial z}{\partial v} \Big |^{2} \\ &{} \leqslant \frac{1}{2} \frac{C_{3} M^{3}}{(t(T-t))^{3}} |z|^{2} + \frac{C_{1} M}{2 t(T-t)} |\frac{\partial z}{\partial v}|^{2}, \end{array} \end{aligned}$$

(50)

because

$$\begin{aligned} \begin{array}{ll} \frac{1}{2} \frac{ (t(T-t))^{3} }{ C_{3} M^{3} } \gamma ^2 n^{2} &{} = \frac{C_{1} M}{2 t(T-t)} \frac{(t(T-t))^{4} \gamma ^2 n^{2} }{C_{1}C_{3}M^{4} }\nonumber \\ &{} \leqslant \frac{C_{1} M}{2t(T-t)} \frac{ (T^{2}/4)^{4} \gamma ^2 n^{2} }{ C_{1}C_{3} M^{4} }\nonumber \\ &{} = \frac{C_{1} M}{2 t(T-t)} \frac{M_{2}^{4}}{M^{4}} \end{array} \end{aligned}$$

From now on, we take

$$\begin{aligned} M = M(T,n,\beta ):= \mathcal C _2 \max \{ T+T^2 ; \sqrt{|n|} T^2 \}\, \end{aligned}$$

(51)

where

$$\begin{aligned} \mathcal C _2=\mathcal C _2(\beta ):=\max \left\{ m_1 ; \frac{1}{4 \root 4 \of {C_1 C_3}} \right\} \end{aligned}$$

so that $M \geqslant M_1$ and $M_2$ (see (47) and (49)). We have

$$\begin{aligned}&\int \limits _{0}^{T} \int \limits _{(0-1,a^{\prime }) \cup (b^{\prime },1)} \left( \frac{C_{1} M}{2t(T-t)} \Big | \frac{\partial z}{\partial v} \Big |^{2} + \frac{C_{3} M^{3}}{2 (t(T-t))^{3}} |z|^{2} \right) \mathrm{d}v \mathrm{d}t\nonumber \\&\quad \leqslant \int \limits _{0}^{T} \int \limits _{a^{\prime }}^{b^{\prime }} \left( \frac{C_{2}^{\prime } M}{t(T-t)} \Big | \frac{\partial z}{\partial v} \Big |^{2}+ \frac{C_{6} M^{3}}{(t(T-t))^{3}} |z|^{2} \right) \mathrm{d}v \mathrm{d}t\nonumber \\&\quad + \int \limits _{Q} | e^{-\alpha } \mathcal P _n g |^2 \mathrm{d}v \mathrm{d}t, \end{aligned}$$

(52)

where $C_6=C_6(\beta ):=C_4+C_3/2,\,C_2^{\prime }=C_2(\beta )=C_2+C_1/2$. Since for every $\epsilon >0$

$$\begin{aligned}&\!\!\!\!\!\!\!\!\frac{C_{1} M}{2 t(T-t)} \Big | \frac{\partial g}{\partial x} - \alpha _x g \Big |^{2} +\frac{C_{3} M^{3}}{2 (t(T-t))^{3}} |g|^{2}\nonumber \\&\!\!\!\!\!\!\!\!\quad \geqslant \left( 1-\frac{1}{1+\epsilon } \right) \frac{C_{1} M}{2t(T-t)} \Big | \frac{\partial g}{\partial x} \Big |^{2} + \frac{M^3}{2(t(T-t))^3}\left( C_{3} - \epsilon C_1 (\beta ^{\prime })^2 \right) |g|^2\, \end{aligned}$$

(53)

and choosing

$$\begin{aligned} \epsilon =\epsilon (\beta ):=\frac{C_3}{2 C_1\Vert \beta ^{\prime }\Vert _\infty ^2}\, , \end{aligned}$$

from (52), (53) and (40) we deduce that

$$\begin{aligned}&\int \limits _{0}^{T} \int \limits _{(0,a^{\prime }) \cup (b^{\prime },1)} \left( \frac{C_{7} M}{t(T-t)} \Big | \frac{\partial g}{\partial v} \Big |^{2} + \frac{C_{3} M^{3}|g|^{2}}{4(t(T-t))^{3}} \right) e^{-2\alpha } \mathrm{d}v \mathrm{d}t\nonumber \\&\quad \leqslant \int \limits _{Q} | e^{-\alpha } \mathcal P _n g |^2 \mathrm{d}v \mathrm{d}t + \int \limits _{0}^{T} \int \limits _{a^{\prime }}^{b^{\prime }} \left( \frac{C_{9} M^{3} |g|^{2}}{(t(T-t))^{3}} + \frac{C_{8} M}{t(T-t)} \Big | \frac{\partial g}{\partial v}\Big |^{2} \right) e^{-2\alpha } \mathrm{d}v \mathrm{d}t\, ,\nonumber \\ \end{aligned}$$

(54)

where $C_7=C_7(\beta ):=[1-1/(1+\epsilon )]C_1/2,\,C_{8}=C_{8}(\beta ):=2C_{2}^{\prime }$ and $C_{9}=C_{9}(\beta ):=C_{6}+2C_{2}^{\prime } \sup \{ \beta ^{\prime }(x)^{2} ; x \in [a^{\prime },b^{\prime }] \}$. Adding the same quantity in both sides, we get

$$\begin{aligned}&\int \limits _Q \left( \frac{C_{7} M}{t(T-t)} \Big | \frac{\partial g}{\partial v} \Big |^{2} + \frac{C_{3} M^{3}|g|^{2}}{4(t(T-t))^{3}} \right) e^{-2\alpha } \mathrm{d}v \mathrm{d}t \leqslant \int \limits _{Q} | e^{-\alpha } \mathcal P _n g |^2 \mathrm{d}v \mathrm{d}t\nonumber \\&\quad + \int \limits _{0}^{T} \int \limits _{a^{\prime }}^{b^{\prime }} \left( \frac{C_{11} M^{3} |g|^{2}}{(t(T-t))^{3}} + \frac{C_{10} M}{t(T-t)} \Big | \frac{\partial g}{\partial v}\Big |^{2} \right) e^{-2\alpha } \mathrm{d}v \mathrm{d}t, \end{aligned}$$

(55)

where $C_{10}=C_{10}(\beta ):=C_{8}+C_7$ and $C_{11}=C_{11}(\beta ):=C_{9}+C_3/4$. Thanks to a cut-off function $\rho$ such that

$$\begin{aligned} 0 \leqslant \rho \leqslant 1, \quad \rho \equiv 1 \text{ on } (a^{\prime },b^{\prime }), \quad \text{ Supp }(\rho ) \subset (a,b) \end{aligned}$$

it is classic to get

$$\begin{aligned}&\int \limits _0^T \int \limits _{a^{\prime }}^{b^{\prime }} \frac{C_{10} M}{t(T-t)} \Big | \frac{\partial g}{\partial v}\Big |^{2} e^{-2\alpha } \mathrm{d}v \mathrm{d}t\nonumber \\&\quad \leqslant \int \limits _Q |\mathcal P _n g|^2 e^{-2\alpha } \mathrm{d}v \mathrm{d}t + \int \limits _0^T \int \limits _a^b \frac{C_{12} M^3 |g|^2 e^{-2\alpha }}{(t(T-t))^3} \mathrm{d}v \mathrm{d}t \end{aligned}$$

for some constant $C_{12}=C_{12}(\beta )>0$. Combining (55) with the previous inequality, we get

$$\begin{aligned}&\int \limits _Q \left( \frac{C_{7} M}{t(T-t)} \Big | \frac{\partial g}{\partial v} \Big |^{2} + \frac{C_{3} M^{3}|g|^{2}}{4(t(T-t))^{3}} \right) e^{-2\alpha } \mathrm{d}v \mathrm{d}t\nonumber \\&\quad \leqslant \int \limits _{Q} 2 | e^{-\alpha } \mathcal{P }_n g |^2 \mathrm{d}v \mathrm{d}t + \int \limits _{0}^{T} \int \limits _{a}^{b} \frac{C_{13} M^{3} |g|^{2}}{(t(T-t))^{3}} e^{-2\alpha } \mathrm{d}v \mathrm{d}t\, , \end{aligned}$$

(56)

where $C_{13}=C_{13}(\beta ,\rho ):=C_{11}+C_{12}$. Then, the global Carleman estimates (10) holds with

$$\begin{aligned} \mathcal{C }_1=\mathcal{C }_1(\beta ):=\frac{\min \left\{ C_7 ; C_3 /4 \right\} }{\max \left\{ 2 ; C_{13} \right\} }. \end{aligned}$$