Examples of control systems modeled by linear PDE's

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< Control of partial differential equations

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Curator: Dr. Jean-Michel Coron, University Paris 6, Paris, France

A transport equation

We return to the transport equation presented in the Section Examples of control systems modeled by PDE's. Let $L>0$ . The linear control system we study is

(1)

$y_t+y_x=0,\, t\in (0,T),\, x\in (0,L),$

(2)

$y(t,0)=u(t),\, t\in(0,T),$

where, at time $t$ , the control is $u(t)\in \mathbb{R}$ and the state is $y(t,\cdot):(0,L)\rightarrow \mathbb{R}$ .

One can put this linear control system in the general framework detailed at this link in the following way. For the Hilbert space $H$ , we take $H:=L^2(0,L)$ . For the operator $A: D(A)\rightarrow H$ we take

$D(A):=\{f\in H^1(0,L);\, f(0)=0\},$

$Af:=-f_x,\, \forall f\in D(A).$

Then $D(A)$ is dense in $L^2(0,L)$ , $A$ is closed. Moreover

$(Af,f)_{L^2(0,L)}=-\frac{1}{2}f(L)^2, \, \forall f \in D(A),$

showing that $A$ is dissipative. The adjoint $A^*$ of $A$ is defined by

$D(A^*):=\{f\in H^1(0,L);\, f(L)=0\},$

$A^*f:=f_x,\, \forall f\in D(A^*).$

As the operator $A$ , the operator $A^*$ is also dissipative. Hence, by the Lumer-Phillips theorem, the operator $A$ is the infinitesimal generator of a strongly continuous semigroup $S(t),\, t\in[0,+\infty)$ , of continuous linear operators on $H$ .

For the Hilbert space $U$ , we take $U:=\mathbb{R}$ . The operator $B:\mathbb{R}\rightarrow D(A^*)'$ is defined by

(3)

$(Bu)z=u z(0), \, \forall u \in \mathbb{R}, \forall z\in D(A^*).$

Note that $B^*:D(A^*)\rightarrow \mathbb{R}$ is defined by

$B^*z=z(0), \, \forall z \in D(A^*).$

Let us deal with the regularity property:

(4)

$\forall T>0, \exists C_T>0 \text{ such that } \int_0^T\|B^*S(t)^* z\|_{U}^2dt \leqslant C_T \|z\|^2_H, \, \forall z\in D(A^*).$

Let $z^0\in D(A^*)$ . Let

$z\in C^0([0,T]; D(A^*))\cap C^1([0,T];L^2(0,L))$

be defined by $z(t,\cdot)=S(t)^*z^0$ . Inequality (4) is equivalent to

(5)

$\int_0^Tz(t,0)^2dt \leqslant C_T \int_0^L z^0(x)^2 dx.$

Let us prove this inequality for $C_T:=1$ . We have

(6)

$z_t=z_x, \, t\in(0,T), \, x\in (0,L),$

(7)

$z(t,L)=0, \, t\in (0,T),$

(8)

$z(0,x)=z^0(x), \, x\in (0,L).$

We multiply (6) by $z$ and integrate on $[0,T]\times[0,L]$ . Using (7), (8) and integrations by parts, we get

(9)

$\int_0^Tz(t,0)^2dt= \int_0^Lz^0(x)^2dx - \int_0^Lz(T,x)^2dx\leqslant \int_0^Lz^0(x)^2dx,$

which shows that (5) holds for $C_T:=1$ .

In fact, as one can easily check, the solution to the following Cauchy problem

(10)

$y_t+y_x=0, \, t\in(0,T),\, x\in (0,L),$

(11)

$y(t,0)=u(t),\,t\in(0,T),$

(12)

$y(0,x)=y^0(x), \, x\in (0,L),$

where $T>0$ , $y^0\in L^2(0,L)$ and $u\in L^2(0,T)$ are given data, is

(13)

$y(t,x)=y^0(x-t), \, \forall (t,x) \in [0,T]\times(0,L) \text{ such that } t\leqslant x,$

(14)

$y(t,x)=u(t-x), \, \forall (t,x) \in [0,T]\times(0,L)\text{ such that } t > x.$

For the controllability of the linear control system (1)-(2), see at this link.

[edit]

A linear Korteweg-de Vries equation

We return to the linear Korteweg-de Vries equation already mentioned at this link in the Section Examples of control systems modeled by PDE's. Let $L>0$ . The linear control system we study is

(15)

$y_t+y_x+y_{xxx}=0,\, t\in (0,T), \, x\in (0,L),$

(16)

$y(t,0)=y(t,L)=0,\, y_x(t,L)=u(t), \, t\in (0,T),$

where, at time $t$ , the control is $u(t)\in \mathbb{R}$ and the state is $y(t,\cdot):(0,L)\rightarrow \mathbb{R}$ .

One can put this linear control system in the general framework detailed at this link in the following way. For the Hilbert space $H$ , we take $H=L^2(0,L)$ . For the operator $A: D(A)\rightarrow H$ , we take

$D(A):=\{f\in H^3(0,L);\, f(0)=f(L)=f_x(L)=0\},$

$Af:=-f_x-f_{xxx},\, \forall f\in D(A).$

Then $D(A)$ is dense in $L^2(0,L)$ , $A$ is closed. Simple integrations by parts give

$(Af,f)_{L^2(0,L)}= -\frac{1}{2}f_x(0)^2, \, \forall f\in L^2(0,L),$

which shows that $A$ is dissipative. The adjoint $A^*$ of $A$ is defined by

$D(A^*):=\{f\in H^3(0,L);\, f(0)=f(L)=f_x(0)=0\},$

$A^*f:=f_x+f_{xxx},\, \forall f\in D(A^*).$

As $A$ , the operator $A^*$ is also dissipative. Hence, by the Lumer-Phillips theorem, the operator $A$ is the infinitesimal generator of a strongly continuous semigroup $S(t),\, t\in[0,+\infty)$ , of continuous linear operators on $L^2(0,L)$ .

For the Hilbert space $U$ , we take $U:=\mathbb{R}$ . The operator $B:\mathbb{R}\rightarrow D(A^*)'$ is defined by

(17)

$(Bu)z=u z_x(L), \, \forall u \in \mathbb{R}, \forall z\in D(A^*).$

Note that $B^*:D(A^*)\rightarrow \mathbb{R}$ is defined by

$B^*z=z_x(L), \, \forall z \in D(A^*).$

Let us check the regularity property (4). Let $z^0\in D(A^*)$ . Let

$z\in C^0([0,+\infty); D(A^*))\cap C^1([0,+\infty);L^2(0,L))$

be defined by

(18)

$z(t,\cdot)=S(t)^*z^0.$

The regularity property (4) is equivalent to

(19)

$\int_0^T|z_x(t,L)|^2dt \leqslant C_T \int_0^L |z^0(x)|^2 dx.$

From (18), one has

(20)

$z_t-z_x-z_{xxx}=0 \text{ in } C^0([0,+\infty); L^2(0,L)),$

(21)

$z(t,0)=z_x(t,0)=z(t,L)=0, \, t \in [0,+\infty),$

(22)

$z(0,x)=z^0(x), \, x \in [0,L].$

We multiply (20) by $z$ and integrate on $(0,T)\times(0,L)$ . Using (21), (22) and simple integrations by parts one gets

(23)

$\int_0^T |z_x(t,L)|^2 dt=\int_0^L|z^0(x)|^2dx- \int_0^L|z(T,x)|^2dx\leqslant \int_0^L|z^0(x)|^2dx,$

which shows that (19) holds with $C_T:=1$ . For the controllability of the linear control system (15)-(16), see at this link.

[edit]

A heat equation

We return to the linear heat equation already considered at this link in the Section Examples of control systems modeled by PDE's. Let $\Omega$ be a non empty open subset of $\mathbb{R}^l$ and let $\omega$ be a non empty open subset of $\Omega$ . The linear heat equation considered in this section is

(24)

$y_t-\Delta y = u(t,x),\, t\in (0,T), \, x\in \Omega,$

(25)

$y=0 \text{ on } (0,T)\times \partial \Omega,$

where, at time $t \in [0,T]$ , the state is $y(t,\cdot) \in L^2(\Omega)$ and the control is $u(t,\cdot) \in L^2 (\Omega)$ . We require that

(26)

$u(\cdot,x)=0, \, x\in \Omega\setminus \omega.$

One can put this linear control system in the general framework detailed at this link in the following way. One chooses

$H:=L^2(\Omega),$

equipped with the usual scalar product. Let $A:D(A)\subset H\rightarrow H$ be the linear operator defined by

$D(A):=\left\{y\in H^1_0(\Omega);\, \Delta y \in L^2(\Omega) \right\},$

$Ay:=\Delta y \in H.$

Note that, if $\Omega$ is smooth enough (for example of class $C^2$ ), then

(27)

$D(A)=H^1_0(\Omega)\cap H^2(\Omega).$

However, without any regularity assumption on $\Omega$ , (27) is wrong in general (see in particular Theorem 2.4.3, page 57, in (Pierre Grisvard,1992)). One easily checks that

(28)

$D(A) \text{ is dense in }L^2(\Omega),$

(29)

$A \text{ is closed.}$

Moreover,

(30)

$(Ay,y)_{H}=-\int_\Omega |\nabla y |^2 dx,\, \forall y \in D(A).$

Let $A^*$ be the adjoint of $A$ . One easily checks that

(31)

$A^*=A.$

From the Lumer-Phillips theorem, (28), (29), (30) and (31), $A$ is the infinitesimal generator of a strongly continuous semigroup of linear contractions $S(t)$ , $t\in [0,+\infty)$ , on $H$ . For the Hilbert space $U$ we take $L^2(\omega)$ . The linear map $B\in \mathcal{L}(U;D(A^*)')$ is the map which is defined by

$(Bu)\varphi=\int_\omega u\varphi dx.$

Note that $B\in \mathcal{L}(U;H)$ . Hence the regularity property (4) is automatically satisfied. For the controllability of the linear control system (24)-(25), see at this link.

Invited by:	Dr. Jean-Jacques Slotine, Nonlinear Systems Lab, MIT, Cambridge, MA
Action editor:	Dr. Jean-Jacques Slotine, Nonlinear Systems Lab, MIT, Cambridge, MA
Assistant editor:	Mr. Leo Trottier, PhD student; University of California, San Diego
Assistant editor:	Dr. Elias August, ETH, Zurich, CH

Retrieved from "http://www.scholarpedia.org/article/Control_of_partial_differential_equations/Examples_of_control_systems_modeled_by_linear_PDE%27s"

Examples of control systems modeled by linear PDE's

From Scholarpedia

A transport equation

A linear Korteweg-de Vries equation

A heat equation

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