# Control of partial differential equations/Examples of control systems modeled by linear PDE's

## 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 $\tag{1} y_t+y_x=0,\, t\in (0,T),\, x\in (0,L),$

$\tag{2} y(t,0)=u(t),\, t\in(0,T),$

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 $\tag{3} (Bu)z=u z(0), \, \forall u \in \mathbb{R}, \forall z\in D(A^*).$

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: $\tag{4} \forall T>0, \exists C_T>0 \text{ such that } \int_0^T\|B^*S(t)^* z\|_{U}^2dt \leqslant C_T \|z\|^2_H, \, \forall z\in D(A^*).$

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 $\tag{5} \int_0^Tz(t,0)^2dt \leqslant C_T \int_0^L z^0(x)^2 dx.$

Let us prove this inequality for$$C_T:=1\ .$$ We have $\tag{6} z_t=z_x, \, t\in(0,T), \, x\in (0,L),$

$\tag{7} z(t,L)=0, \, t\in (0,T),$

$\tag{8} z(0,x)=z^0(x), \, x\in (0,L).$

We multiply (6) by $$z$$ and integrate on $$[0,T]\times[0,L]\ .$$ Using (7), (8) and integrations by parts, we get $\tag{9} \int_0^Tz(t,0)^2dt= \int_0^Lz^0(x)^2dx - \int_0^Lz(T,x)^2dx\leqslant \int_0^Lz^0(x)^2dx,$

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

In fact, as one can easily check, the solution to the following Cauchy problem $\tag{10} y_t+y_x=0, \, t\in(0,T),\, x\in (0,L),$

$\tag{11} y(t,0)=u(t),\,t\in(0,T),$

$\tag{12} y(0,x)=y^0(x), \, x\in (0,L),$

where $$T>0\ ,$$ $$y^0\in L^2(0,L)$$ and $$u\in L^2(0,T)$$ are given data, is $\tag{13} y(t,x)=y^0(x-t), \, \forall (t,x) \in [0,T]\times(0,L) \text{ such that } t\leqslant x,$

$\tag{14} y(t,x)=u(t-x), \, \forall (t,x) \in [0,T]\times(0,L)\text{ such that } t > x.$

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

## 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 $\tag{15} y_t+y_x+y_{xxx}=0,\, t\in (0,T), \, x\in (0,L),$

$\tag{16} y(t,0)=y(t,L)=0,\, y_x(t,L)=u(t), \, t\in (0,T),$

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 $\tag{17} (Bu)z=u z_x(L), \, \forall u \in \mathbb{R}, \forall z\in D(A^*).$

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 $\tag{18} z(t,\cdot)=S(t)^*z^0.$

The regularity property (4) is equivalent to $\tag{19} \int_0^T|z_x(t,L)|^2dt \leqslant C_T \int_0^L |z^0(x)|^2 dx.$

From (18), one has $\tag{20} z_t-z_x-z_{xxx}=0 \text{ in } C^0([0,+\infty); L^2(0,L)),$

$\tag{21} z(t,0)=z_x(t,0)=z(t,L)=0, \, t \in [0,+\infty),$

$\tag{22} z(0,x)=z^0(x), \, x \in [0,L].$

We multiply (20) by $$z$$ and integrate on $$(0,T)\times(0,L)\ .$$ Using (21), (22) and simple integrations by parts one gets $\tag{23} \int_0^T |z_x(t,L)|^2 dt=\int_0^L|z^0(x)|^2dx- \int_0^L|z(T,x)|^2dx\leqslant \int_0^L|z^0(x)|^2dx,$

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

## 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 $\tag{24} y_t-\Delta y = u(t,x),\, t\in (0,T), \, x\in \Omega,$

$\tag{25} y=0 \text{ on } (0,T)\times \partial \Omega,$

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 $\tag{26} u(\cdot,x)=0, \, x\in \Omega\setminus \omega.$

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 $\tag{27} D(A)=H^1_0(\Omega)\cap H^2(\Omega).$

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 $\tag{28} D(A) \text{ is dense in }L^2(\Omega),$

$\tag{29} A \text{ is closed.}$

Moreover, $\tag{30} (Ay,y)_{H}=-\int_\Omega |\nabla y |^2 dx,\, \forall y \in D(A).$

Let $$A^*$$ be the adjoint of$$A\ .$$ One easily checks that $\tag{31} A^*=A.$

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.