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.