Home > shape optimization > Shape Optimization Course Day 3

Shape Optimization Course Day 3

Speaker — Giuseppe Buttazzo

(for courses given in the first two days see the Shape Optimization page)

Optimal control problems are minimum problems which describe the behavior of systems that can be modified by the action of an operator. Many problems in applied sciences can be modeled by means of optimal control problems. There are two kinds of variables involved: one which describes the state of the system, and which cannot be modified directly by the operator it is called the state variable; the second one is under the direct control of the operator it is called the control variable.

The operator modifies the state of the system indirectly, acting directly on the control variable; only these variables may act on the system through a link between the control variable and the state variable, usually called state equation. The operator, acting directly on controls and indirectly on states through the state equation must achieve a goal usually written as a minimization of a functional which depends on the state and the control variables.

The typical example of an optimal control problem is driving a car: the driver cannot act directly on the position of the vehicle, but he can act directly only on the controls which are in this case the accelerator, the brake and the steering wheel. The state of the car is described by its position and velocity, which depend on the controls chosen by the driver, but which are not directly controlled by him. The state equations are, in this case, usual equations of mechanics which to a given choice of acceleration and steering angle associate the position and the velocity of the car (this is a simplified model, but we could take into account the specifications of the engine, of the aerodinamics of the car, technological constraints, etc.). Finally, the driver wants to achieve a goal, for instance minimize the fuel consumption to run on a given path. The optimal control problem consists in choosing the best driving strategy to minimize the fuel consumption.

Accodring to the facts stated above the ingredients of an optimal control problem are:

  • (i) a space of states {Y};
  • (ii) a set of controls {U};
  • (iii) the set {\mathcal{A}} of admissible pairs, i.e. the subset of pairs {(u,y) \in U\times Y} such that {y} is linked to {u} through the state equation;
  • (iv) a cost functional {J : U \times Y \rightarrow \overline{\Bbb{R}}}

The optimal control problem then takes the form of a minimization problem and can be written as

\displaystyle \min \{ J(u,y) : (u,y) \in \mathcal{A}\}.

In our case we are especially interested in the study of shape optimization problems in which the control variable runs over classes of domains. This gives rise to problems, since we have to consider a framework general enough to include cases when the control variable does not belong to a space with linear topological structure.

One important matter in the study of this kind of problems is the choice of topology on {y} and {U}, because this is crucial when studying the existence of solutions to such optimal problems. This is related to the use of direct methods of the calculus of variations which require suitable lower semicontinuity and compactness assumptions.

Consider an open domain {D \subset \Bbb{R}^N}. Suppose we have an isolated system modeled as follows:

\displaystyle \left\{\begin{array}{rll} -\Delta u &=f & \text{ in } D \\ \frac{\partial u }{\partial n} &=0 & \text{ on }\partial D \text{ (isolated system)} \end{array}\right.

Since this system does not have a unique solution (if we have a solution then adding a constant yields another solution), we cannot predict what is happening in the system. So we would like to introduce some controls in the form of some refrigerators for example, modeled by some compact set {K \subset D}.

Then we get the following problem with mixed boundary conditions:

\displaystyle \left\{\begin{array}{rll} -\Delta u &=f & \text{ in } D\setminus K \\ u &=0 & \text{ on } K \text{ (refrigerators)} \\ \frac{\partial u }{\partial n} &=0 & \text{ on }\partial D \text{ (isolated system)} \end{array}\right.

which has a unique solution depending on {K}.

As noted in the system, consider an isolated system, i.e. there is no temperature exchange on the boundary of {D}, and there are some refrigerators {K} inside {D}. This system has a unique solution {u_K} (we index about {K}, since {K} is our shape variable which we can control). The goal is to get the temperature {u_K} as close as we can to a desired temperature {\overline{u}}, which can be expressed, for example in minimizing a functional of the form {\displaystyle \int_\Omega |u_k-\overline u|^2}.

Consider now the system with Dirichlet boundary conditions (the domain has a ‘refrigerated’ boundary):

\displaystyle \left\{\begin{array}{rll} -\Delta u &=1 & \text{ in } D \\ u &=0 & \text{ on }\partial D \end{array}\right.

This system has a unique solution, but maybe the behavior of the solution it is not what we ‘desire’ so we introduce some refrigerators inside. Think for example of nuclear reactors (this is just fictive, things are more complicated there). If we only refrigerate the walls of the room where the reaction takes place, we probably wouldn’t be able to control the temperature very well, and that is not good. Then we obtain the following system

\displaystyle \left\{\begin{array}{rll} -\Delta u &=1 & \text{ in } D \\ u&=0 & \text{ on }K\\ u &=0 & \text{ on }\partial D \end{array}\right.

And here we arrive at another example where the optimal shape {K} does not exist. The reason is that the optimal solution tends to split into many small parts. For example consider the same functional {J(K)=\int_D |u_K-c|dx}, assume that {K} is a solution and {K} is regular, such that there exists a ball {B_\varepsilon} of radius {\varepsilon} in {D\setminus K}. Then we can see that by considering {K'=K \cup B_\varepsilon} we obtain a better configuration. Indeed, we have:

\displaystyle \begin{array}{lclll} K & \longrightarrow & \int_K |u_K-c|^2dx & +\int_{B_\varepsilon}c^2 dx & +\int_{D \setminus (K \cup B_\varepsilon)}c^2 dx \\ K \cup B_\varepsilon & \longrightarrow & \int_K |u_K-c|^2dx & +\int_{B_\varepsilon}|u_\varepsilon -c|^2 dx &+\int_{D \setminus (K \cup B_\varepsilon)}c^2 dx \end{array}

If {K} is optimal then we must have

\displaystyle \int_{B_\varepsilon} |u_\varepsilon-c|^2dx \geq \int_{B_\varepsilon} c^2 dx,

and {u_\varepsilon} can be explicitly computed

\displaystyle u_\varepsilon(x)= \frac{\varepsilon^2 - |x-x_0|^2}{2n}

where {n} is the dimension of the euclidean space containing {D}. Thus we would have

\displaystyle \int_{B_\varepsilon}u_\varepsilon^2(x)dx \geq 2c\int_{B_\varepsilon}u_\varepsilon(x)dx

for every {\varepsilon>0}. Note that for {c} very big we may choose {K=\emptyset} (no refrigerators), but for {c} small taking {\varepsilon} small will eventually contradict this inequality. Therefore no regular solution can exist in this case. Note that for {c} very big we may choose {K=\emptyset} (no refrigerators).

A few words on capacity. Intuitively, if we have an open set {E \subset D \Bbb{R}^n} then we solve the Laplace equation

\displaystyle \begin{cases} -\Delta u=0 &\text{ in }D\\ u=1 & \text{ on }E \\ u=0 & \text{ on }\partial D \end{cases}

and define {\text{cap}_D(E)=\displaystyle \int_D |\nabla u|^2}. For example in two dimensions the capacity of a point is zero, but the capacity of a segment is non-zero. In three dimensions the capacity of a segment is zero, but the capacity of a {2}-dimensional regular surface is non-zero. In a way, capacity is more sensible as the Lebesgue measure, and it turns out to be the right tool when working with Sobolev spaces.

In 1982 Cioranescu and Murat gave the following example. Suppose we have the following problem

\displaystyle \begin{cases} -\Delta u=f & \text{ in }\Omega_\varepsilon \\ u=0 & \text{ on }\partial D \end{cases}

where {\Omega_\varepsilon =D \setminus \bigcup_i B_i(R_\varepsilon)} and denote by {u_\varepsilon} the solution. It can be proved easily that {u_\varepsilon} is bounded in {H_0^1(D)} and therefore it has a weakly convergent subsequence to a limit {u^*}. Denote by {u} the solution of the above problem when {\Omega_\varepsilon=D}. Then if the radii {r_\varepsilon} get too small then the holes dissappear, and {u^*=u}, if the radii {r_\varepsilon} are too big then {u^*=0}, but if we are in the critical case (in between the other two cases) then {u^*} is not a solution of the same system as above, but it satisfies an equation of the type

\displaystyle \begin{cases} -\Delta u+cu=f & \text{ in }\Omega_\varepsilon \\ u=0 & \text{ on }\partial D \end{cases}

For more details see for example Henrot, Pierre, Variation et Optimizsation de Formes, Proposition 3.2.11 and Exercise 3.8.

For this kind of problem we can define a convergence of domains in the following way: say that {\Omega_\varepsilon} {\gamma}-converges to {\Omega} if {u_\varepsilon \rightharpoonup u} where {u_\varepsilon} and {u} are the solutions of the corresponding systems. It turns out that this topology defined on {\{\Omega \subset D\}} is metrizable, but not compact. (see the example above) In 1987 Dal Maso and Mosco proved that the compactification of the above metric space under the {\gamma}-convergence is the space of capacitary measures, i.e. Borel measures with {\mu(E)=0} for every {E} with {\text{cap}(E)=0}.

This space has many good properties, and gives the existence of optimal shapes in this case. As it was seen in the above example, there are problems in which finding an optimal shape is impossible if we do not relax the problem a little bit. The problem is that the relaxed optimum is not really a shape, it is an imaginary object which in practice does not exist. Still, deriving some asymptotic behavior for the minimizing sequence can be of great use, because if we cannot achieve the minimum we can get as close as we want to it, and sometimes this is the best we can do.

Another great class of shape optimization problems is Spectral Optimization. We know that the resolvent operator which associates to {f} the solution of the problem

\displaystyle \begin{cases} -\Delta u=f & \text{ in }\Omega \\ u=0 & \text{ on }\partial \Omega \end{cases}

is compact, and so it has a discrete spectrum

\displaystyle \lambda_1(\Omega)\leq \lambda_2(\Omega),... \rightarrow \infty.

Denote by {\lambda(\Omega)} the sequence of eigenvalues of the Laplace operator on {\Omega}. Then spectral optimization problem try to find the optimal shapes for functionals of the form

\displaystyle J(\Omega)= \phi(\lambda(\Omega)),

for example {J(\Omega)=\lambda_1(\Omega)}. The problems can be stated as

\displaystyle \min\{\phi(\lambda(\Omega)) : |\Omega|\leq m\}.

We know that the shape which minimizes {\lambda_1(\Omega)} is the ball (Faber-Krahn), the shape which minimizes {\lambda_2} is the union of two equal balls (Szego). Still there are many open problems in this domain; for example it is not known what the optimal shape for {\lambda_3} is even in dimension {2} where is conjectured to be a ball. The best shape for {\lambda_3} in dimension {3} is most likely not a ball, and for dimension greater than {3} the shape is unknown. Still there are some very general existence results like the following

(Buttazzo, Dal Maso) Suppose we have a function {\phi :\Bbb{R}^p \rightarrow \Bbb{R}} which is lower semicontinuous and non-decreasing with respect to every variable. Then the problem

\displaystyle \min_{\Omega \in \mathcal{A}_c} \phi (\lambda_1(\Omega),..,\lambda_p(\Omega))

has a solution where {\mathcal{A}_c= \{ \Omega \subset D : \Omega \text{ quasi-open}, |\Omega|=c\}} and {D} is a fixed bounded open set.

(Bucur, Buttazzo, Figuerido) Let {\phi : \overline{\Bbb{R}}^2 \rightarrow \Bbb{R}} be a lower semicontinuous function, {D} a given set and {c} a given constant. Then the problem

\displaystyle \min\{ \phi(\lambda_1(\Omega),\lambda_2(\Omega)) : \Omega \subset D, |\Omega|=c \}

has a solution.

A good reference for spectral optimization problem is A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators where many other results are presented, as well as many open problems.

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