Control of partial differential equations/Examples of control systems modeled by PDE's
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A transport equation
This is the simplest control system modeled by PDE's. It is the following one \[\tag{1} y_t+y_x=0, \, x\in (0,L), \]
\[\tag{2} y(t,0)=u(t), \]
where, at time \(t\in (0,T)\ ,\) the control is \(u(t)\in \mathbb{R}\ ,\) the state is \(y(t,\cdot):(0,L) \rightarrow \mathbb{R}\) and \(L\) is a given positive real number. For the Cauchy problem associated to (1)-(2), see at this link. For the controllability of the control system (1)-(2), see at this link.
A Korteweg-de Vries equation
Let\(L>0\) be given. Our Korteweg-de Vries control system is \[\tag{3} y_t+y_x+y_{xxx}+yy_x=0, \, t\in(0,T), \, x\in (0,L), \]
\[\tag{4} y(t,0)=y(t,L)=0,\, y_x(t,L)=u(t),\, t\in(0,T), \]
where, at time \(t\in [0,T]\ ,\) the control is \(u(t)\in \mathbb{R}\) and the state is \(y(t,\cdot):(0,L)\mapsto \mathbb{R}\ .\) Equation (3) is a Korteweg-de Vries equation, which serves to model various physical phenomena, for example, the propagation of small amplitude long water waves in a uniform channel (see, e.g., Section 4.4, pages 155--157 in (Lokenath Debnath, 1994) or Section 13.11 in (Gerald Whitham, 1974)). Let us recall that, as pointed out in (Jerry Bona and Ragnar Winther, 1983) the term \(y_x\) in (3) has to be added to model the water waves when \(x\)denotes the spatial coordinate in a fixed frame. It is also interesting to consider the linearized control system around the trajectory \((\bar y ,\bar u):=(0,0)\ ,\) i.e. the following linear control system: \[\tag{5} y_t+y_x+y_{xxx}=0, \, x\in (0,L), \]
\[\tag{6} y(t,0)=y(t,L)=0,\, y_x(t,L)=u(t), \]
where, at time \(t\ ,\) the control is \(u(t)\in \mathbb{R}\) and the state is \(y(t,\cdot):(0,L)\mapsto \mathbb{R}\ .\) For the Cauchy problem associated to (5)-(6), see at this link. For the controllability of the control system (5)-(6), see at this link. For the Cauchy problem and the controllability for (3)-(4), see at this link and at this link.
A heat equation
Let \(\Omega\) be a nonempty bounded open set of \(\mathbb{R}^l\) and let \(\omega\) be a nonempty open subset of \(\Omega\ .\) We consider the following linear control system: \[ y_t-\Delta y = u(t,x), \, x\in \Omega, \] \[ y=0 \text{ on } (0,T)\times \partial \Omega, \] where, at time \(t\ ,\) the state is \(y(t,\cdot):\Omega \rightarrow \mathbb{R}\) and the control is \(u(t,\cdot) : \Omega \rightarrow \mathbb{R}\ .\) We require that \[ u(\cdot,x)=0, \, x\in \Omega\setminus \omega. \] Hence we consider the case of "internal control" (in contrast with the above examples where the control was on the boundary of the domain). For the Cauchy problem associated to this linear control system, see at this link. For the controllability of this linear control system, see at this link.
A water-tank control system
We consider a 1-D tank containing an inviscid incompressible irrotational fluid. The tank is subject to one-dimensional horizontal moves. We assume that the horizontal acceleration of the tank is small compared to the gravity constant and that the height of the fluid is small compared to the length of the tank. These physical considerations motivate the use of the Saint-Venant equations (Adhémar Saint-Venant, 1871) --also called shallow water equations-- to describe the motion of the fluid; see e.g. Section 4.2 in (Lokenath Debnath, 1994). Hence the considered dynamics equations are (see the paper (François Dubois, Nicolas Petit and Pierre Rouchon, 1999)) \[\tag{7} H_t\left(t,x\right)+(Hv)_x\left(t,x\right)=0, \, x\in[0,L], \]
\[\tag{8} v_t\left(t,x\right)+\left(gH+\frac{v^2}{2}\right)_x \left(t,x\right)=-u\left(t\right),\, x\in[0,L], \]
\[\tag{9} v(t,0)=v(t,L)=0, \]
\[\tag{10} \frac{\text{d}s}{\text{d}t}\left(t\right)=u\left(t\right), \]
\[\tag{11} \frac{\text{d}D}{\text{d}t}\left(t\right)=s\left(t\right), \]
where,
- \(L\) is the length of the 1-D tank,
- \(H\left(t,x\right)\) is the height of the fluid at time \(t\) and at the position \(x\in [0,L]\ ,\)
- \(v\left(t,x\right)\) is the horizontal water velocity of the fluid "in a referential attached to the tank" at time \(t\) and at the position \(x\in [0,L]\) (in the shallow water model, all the points on the same vertical have the same horizontal velocity),
- \(u\left(t\right)\) is the horizontal acceleration of the tank in the absolute referential,
- \(g\) is the gravity constant,
- \(s\)is the horizontal velocity of the tank,
- \(D\)is the horizontal displacement of the tank.
This is a control system where, at time \(t\ ,\)
- the state is \(Y(t)=(H(t,\cdot),v(t,\cdot),s(t),D(t))\ ,\)
- the control is \(u(t)\in \mathbb{R}\ .\)
For the controllability of this nonlinear control system, see at this link.
A Schrödinger equation
Let \(I=(-1,1)\) and let \(T>0\ .\) We consider the Schrödinger control system
\[\tag{12} \psi_{t}=i\psi_{xx} +i u(t) x \psi ,\, (t ,x) \in (0,T)\times I, \]
\[\tag{13} \psi(t,-1)=\psi(t,1)=0,\, t \in (0,T), \]
\[\tag{14} \dot{S}(t)=u(t),\, \dot{D}(t)=S(t) ,\, t \in (0,T). \]
This is a control system, where, at time \(t\in [0,T]\ ,\)
- the state is \((\psi(t,\cdot),S(t),D(t))\in L^2(I;\mathbb{C})\times \mathbb{R}\times\mathbb{R}\) with\(\int_{I}|\psi(t,x)|^{2}dx=1\ ,\)
- the control is \( u(t) \in \mathbb{R}\ .\)
This system has been introduced in (Pierre Rouchon,2003). It models a non-relativistic charged particle in a 1-D moving infinite square potential well. At time \(t\ ,\) \(\psi(t,\cdot)\) is the wave function of the particle in a frame attached to the potential well, \(S(t)\) is the speed of the potential well and \(D(t)\) is the displacement of the potential well. The control \(u(t)\) is the acceleration of the potential well at time \(t\ .\) For the controllability of linearized control systems associated to the control system (12)-(13)-(14), see at this link. For the controllability of the control system (12)-(13)-(14), see the papers (Karine Beauchard, 2005) and (Karine Beauchard and Jean-Michel Coron, 2006). For other related control models in quantum chemistry, let us mention the paper (Claude Le Bris, 2000) and the references therein.
Euler equations of incompressible fluids
Let us introduce some notations. Let \(l\in \{2,3\}\) and let \(\Omega\) be a bounded nonempty connected open subset of \(\mathbb{R}^l\) of class \(C^\infty\ .\) Let \(\Gamma_{0}\) be a nonempty open subset of \(\Gamma:= \partial \Omega\ .\) The set \(\Gamma_{0}\) is the part of the boundary \(\Gamma\) on which the control acts. The fluid that we consider is incompressible, so that the velocity field \(y\) satisfies \[ \mathrm{div}\ y = 0. \] On the part of the boundary \(\Gamma \setminus \Gamma_{0}\) where there is no control, the fluid does not cross the boundary: it satisfies \[\tag{15} y \cdot n = 0 \mbox{ on } \Gamma \setminus \Gamma_{0}, \]
where \(n\) denotes the outward unit normal vector field on \(\Gamma\ .\) The control system of inviscid incompressible fluids is \[\tag{16} \frac{\partial y}{\partial t} + (y \cdot \nabla) y + \nabla p = 0 \mbox{ in } (0, T) \times \Omega, \]
\[\tag{17} \mathrm{div}\ y = 0 \mbox{ in } (0, T) \times \Omega , \]
\[\tag{18} y(t,x)\cdot n(x) =0, \, \forall (t,x) \in (0, T)\times(\Gamma \setminus \Gamma_{0}). \]
In (16) and throughout the whole article, for \(A:\Omega \rightarrow \mathbb{R}^l\) and \(B:\Omega \rightarrow \mathbb{R}^l\ ,\) \((A\cdot\nabla)B: \Omega \rightarrow \mathbb{R}^l\) is defined by \[ ((A\cdot\nabla)B)^k:=\sum_{j=1}^l A^j\frac{\partial B^k}{\partial x_j},\, \forall k\in \{1,\ldots,l\}. \] In the control system (16)-(17)-(18) the state at time \(t\in [0,T]\) is \(y(t,\cdot)\ .\)
Note that in this formulation of the control system associated to the Euler equations, the control does not appear explicitly. One can take, for example, \(y\cdot n\) on \(\Gamma\) with \(\int_{\Gamma}y\cdot n ds =0\) and
- If\(l=2\ ,\) \(\text{curl } y\) on \(\Gamma\) for the incoming flow (i.e. at the points \((t,x)\in [0,T]\times \Gamma\) such that \(y(t,\cdot)\cdot n(x)<0\))
- If\(l=3\ ,\) the tangential component of \(\text{curl } y\) on \(\Gamma\) for the incoming flow.
For the controllability of the control system (16)-(17)-(18), see this link.
In this section the incompressible fluid is now viscous. Equation (16) is replaced by \[\tag{19} \frac{\partial y}{\partial t} - \nu \Delta y + (y \cdot \nabla) y + \nabla p = 0 \mbox{ in } (0, T) \times \Omega, \]
where \(\nu>0\) is the viscosity of the fluid (a positive real number independent of \(y\ ;\) it depends only on the considered incompressible fluid). Since (19) contains a spatial second order partial differential term (namely the Laplacian \(\Delta y\) of \(y\)), the boundary condition (18) is no longer sufficient. One adds a "wall law". Two wall laws are classical
- The Stokes no-slip boundary condition:
\[\tag{20} y=0 \text{ on } (0,T)\times (\Gamma \setminus \Gamma_0), \]
which implies (18).
- The Navier slip boundary condition: besides (18), one also requires
\[\tag{21} \overline{\sigma} y \cdot \tau + (1 - \overline{\sigma}) \sum_{i=1,j=1}^{i=l,j=l} n^i\left( \frac{\partial y^i}{\partial x^j} + \frac{\partial y^j}{\partial x^i} \right)\tau^j = 0 \mbox{ on } (0,T)\times (\Gamma \setminus \Gamma_{0}),\, \forall \tau\in \mathcal{T}\Gamma, \]
where \(\overline{\sigma}\) is a constant in \([0, 1)\ .\) In (20), \(n=(n^1,\ldots,n^l)\ ,\) \(\tau =(\tau^1,\ldots,\tau^l)\) and \(\mathcal{T}\Gamma\) is the set of tangent vector fields on the boundary\(\Gamma\ .\) Note that the Stokes boundary condition (21) corresponds to the case \(\overline{\sigma} = 1\ .\)
For the control, one can simply take \(u(t,x):=y(t,x)\) for \((t,x)\in (0,T)\times (\Gamma\setminus \Gamma_0)\ .\) Of course, due to the smoothing effects of the Navier-Stokes equations, one cannot expect to move from a given state \(y^0\) to another given state \(y^1\) unless severe restrictions on the smoothness of \(y^1\ ,\) restrictions which are moreover not very explicit. In that case the good notion of controllability is the following: given a state \(y^0\) and a solution \((\hat y, \hat p) :[0,T]\times \Omega \rightarrow \mathbb{R}^l\times \mathbb{R}\) of the control system, does there exist a control \(u:[0,T]\times \Gamma_0 \rightarrow \mathbb{R}\) such that the solution \(( y, p) :[0,T]\times \Omega \rightarrow \mathbb{R}^l\times \mathbb{R}\) of the Navier-Stokes control system such that \(y(0,\cdot)=y^0\) satisfies \( y(T,\cdot)=\hat y(T,\cdot)\ ?\) For the controllability of this control system, see the references given at this link and at this link.
