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.