We start with an informal presentation of the turnpike phenomenon for general dynamical optimal shape problems. Let . We consider the problem of determining a time-varying shape (viewed as a control, as in MR3350723 ) minimizing the cost functional
under the constraints
) may be a partial differential equation with various terminal and boundary conditions.
i.e., the problem of minimizing the instantaneous cost under the constraint of being an equilibrium of the control dynamics.
According to the well known turnpike phenomenon, one expects that, for large enough, optimal solutions of (1)-(2) remain most of the time “close” to an optimal (stationary) solution of the static problem (3).
The turnpike phenomenon was first observed and investigated by economists for discrete-time optimal control problems (see turnpikefirst ; 10.2307/1910955 ). There are several possible notions of turnpike properties, some of them being stronger than the others (see MR3362209 ). Exponential turnpike properties have been established in GruneSchallerSchiela ; MR3124890 ; MR3616131 ; TrelatZhangZuazua ; MR3271298 for the optimal triple resulting of the application of Pontryagin’s maximum principle, ensuring that the extremal solution (state, adjoint and control) remains exponentially close to an optimal solution of the corresponding static controlled problem, except at the beginning and at the end of the time interval, as soon as is large enough. This follows from hyperbolicity properties of the Hamiltonian flow. For discrete-time problems it has been shown in MR3217211 ; MR3654613 ; MR3782393 ; MR3470445 ; measureturnpikeTZ that exponential turnpike is closely related to strict dissipativity. Measure-turnpike is a weaker notion of turnpike, meaning that any optimal solution, along the time frame, remains close to an optimal static solution except along a subset of times of small Lebesgue measure. It has been proved in MR3654613 ; measureturnpikeTZ that measure-turnpike follows from strict dissipativity or from strong duality properties.
Applications of the turnpike property in practice are numerous. Indeed, the knowledge of a static optimal solution is a way to reduce significantly the complexity of the dynamical optimal control problem. For instance it has been shown in MR3271298 that the turnpike property gives a way to successfully initialize direct or indirect (shooting) methods in numerical optimal control, by initializing them with the optimal solution of the static problem. In shape design and despite of technological progress, it is easier to design pieces which do not evolve with time. Turnpike can legitimate such decisions for large-time evolving systems.
2 Shape turnpike for the heat equation
Throughout the paper, we denote by:
the Lebesgue measure of a measurable subset , ;
the scalar product in of ;
the -norm of ;
the indicator (or characteristic) function of;
the distance function to the set and the oriented distance function.
Let () be an open bounded Lipschitz domain. We consider a uniformly elliptic second-order operator
with , with , and its adjoint
(which is also uniformly elliptic, see (MR2597943, , Definition Chapter 6)), not depending on and with a constant of ellipticity (for written in nondivergence form), i.e.:
Moreover, is such that
where is the largest root of the polynomial with the Poincaré constant on . This assumption is used to ensure that an energy inequality is satisfied with constants not depending on the final time (see A for details).
Moreover, we assume throughout that satisfies the classical maximum principle (see (MR2597943, , sec. 6.4)) and that .
We define as the differential operator defined on the domain encoding Dirichlet conditions (when is or a convex polytop in , we have ). Let be the eigenelements of with an orthonormal eigenbasis of :
A typical example sartisfying all assumptions above is the Dirichlet Laplacian, which we will consider in our numerical simulations.
We recall that the Hausdorff distance between two compact subsets of is defined by
Let . We define the set of admissible shapes
Dynamical optimal shape design problem
Let and let be arbitrary. We consider the parabolic equation controlled by a (measurable) time-varying map of subdomains
Given and , we consider the dynamical optimal shape design problem of determining a measurable path of shapes that minimizes the cost functional
where is the solution of (5) corresponding to .
Static optimal shape design problem
Besides, for the same target function , we consider the following associated static shape design problem :
We are going to compare the solutions of and of when is large.
Given any measurable subset , we identify with its characteristic function and we identify with a subset of (as in MR2745777 ; MR3325779 ; MR3500831 ). Then, the convex closure of in weak star topology is
which is also weak star compact. We define the convexified (or relaxed) optimal control problem of determining a control minimizing the cost
under the constraints
The corresponding convexified static optimization problem is
Note that the control does not appear in the cost functionals of the above convexified control problems. Therefore the resulting optimal control problems are affine with respect to . Once we have proved that optimal solutions do exist, we expect that any minimizer will be an extremal point of the compact convex set , which is exactly : if this is true, then actually with . Here, as it is usual in shape optimization, the interest of passing by the convexified problem is to allow us to derive optimality conditions, and thus to characterize the optimal solution. It is anyway not always the case that the minimizer of the convexified problem is an extremal point of (i.e., a characteristic function): in this case, we speak of a relaxation phenomenon. Our analysis hereafter follows these guidelines.
Taking a minimizing sequence and by classical arguments of functional analysis (see, e.g., MR0271512 ), it is straightforward to prove existence of solutions and respectively of and of (see details in Section 3.1).
It can be noted that, when and with (for ) and , i.e., when (for ) and are characteristic functions of some subsets, then actually, and are optimal shapes, solutions respectively of and of .
Our next task is to apply necessary optimality conditions to optimal solutions of the convexified problems, and infer from these necessary conditions that, under appropriate assumptions, the optimal controls are indeed characteristic functions.
Necessary optimality conditions for
Necessary optimality conditions for
Similarly, applying (MR0271512, , Chapter 2, Theorem 1.4), for any optimal solution of there exists an adjoint state such that
Note that, if , then it follows from (15) that the static optimal control is actually the characteristic function of a shape and hence in that case we have existence of an optimal shape.
2.3 Main results
Existence of optimal shapes
Proving existence of optimal shapes, solutions of and of , is not an easy task. We can find cases where there is no existence for a variant of in (henrot2005variation, , Sec. 4.2, Example 2): this is the relaxation phenomenon. Therefore, some assumptions are required on the target function to establish existence of optimal shapes.
We distinguish between Lagrange and Mayer cases.
(Mayer case): If is analytic hypoelliptic in then there exists a unique optimal shape , solution of .
(Lagrange case): Assuming that and that :
If or then there exist unique optimal shapes and , respectively, of and of .
There exists a function such that if , then there exists a unique optimal shape , solution of .
We recall that is said to be analytic hypoelliptic in the open set if any solution of with analytic in is also analytic in . Analytic hypoellipticity is satisfied for the second-order elliptic operator as soon as its coefficients are analytic in (for instance it is the case for the Dirichlet Laplacian, without any further assumption).
This result implies uniqueness of the optimal shapes. We deduce from (35) that we also have uniqueness of state and adjoint.
In what follows, we denote by
the optimal triple of and
the optimal triple of and
Integral turnpike in the Lagrange case
For (Lagrange case), there exists such that
Measure-turnpike in the Lagrange case
We say that satisfies the state-adjoint measure-turnpike property if for every there exists , independent of , such that
We refer to MR3155340 ; MR3654613 ; measureturnpikeTZ (and references therein) for similar definitions. Here, is the set of times along which the time optimal state-adjoint pair remains outside of an -neighborhood of the static optimal state-adjoint pair in topology.
We next recall the notion of dissipativity (see MR0527462 ).
We say that is strictly dissipative at an optimal stationary point of (6) with respect to the supply rate function
if there exists a storage function locally bounded and bounded below and a -class function such that, for any and any , the strict dissipation inequality
is satisfied for any pair solution of (5).
For (Lagrange case):
is strictly dissipative in the sense of Definition 5.
The state-adjoint pair satisfies the measure-turnpike property.
The exponential turnpike property is a stronger property and can be satisfied either by the state, by the adjoint or by the control or even by the three together.
For (Mayer case): For with boundary and there exist , and such that, for every ,
In the Lagrange case, based on the numerical simulations presented in Section 4 we conjecture the exponential turnpike property, i.e., given optimal triples and , there exist and independent of such that
for a.e. .
3.1 Proof of Theorem 1
We first show existence of an optimal shape, solution for and similarly for . We first see that the infimum exists. We take a minimizing sequence such that, for every , the pair satisfies (7) and . The sequence is bounded in , so, using (35), the sequence is bounded in . We show then, using (7), that the sequence is bounded in . We subtract a sequence always denoted by such that one can find a pair with
We deduce that
We get then by weak lower semi-continuity of and of the volume constraint, and by the Fatou Lemma that
hence an optimal control for , that we rather denote by (and for ).
We next proceed by proving existence of optimal shape designs.
1- We take (Mayer case). We consider an optimal triple of . Then it satisfies (12) and (14). It follows from the properties of the parabolic equation and from the assumption of analytic hypoellipticity that is analytic on and that all level sets have zero Lebesgue measure. We conclude that the optimal control satisfying (12)-(14) is such that
i.e., is a characteristic function. Hence, for a Mayer problem , existence of an optimal shape is proved.
2-(i) In the case (Lagrange case), we give the proof for the static problem . We suppose (we proceed similarly for ). Having in mind (12) and 15), we have . By contradiction, if , let us consider the solution of (8) with the control which is the same as verifying (15) except that ( if ) on . We have then (or if ). Then, by the maximum principle (see (MR2597943, , sec. 6.4)) and using the homogeneous Dirichlet condition, we get that the maximum (the minimum if ) of is reached on the boundary and hence (or if ). We deduce . This means that is an optimal control. We conclude by uniqueness.
We use a similar argument thanks to maximum principle for parabolic equations (see (MR2597943, , sec. 7.1.4)) for existence of an optimal shape solution of .
Given any and any such that , we have on .
Proof of Lemma 8..
A proof of a more general result can be found in (MR3793605, , Theorem 3.2). For completeness, we give here a short argument. denotes here the weak derivative of . We need first to show that for and for a function for which there exists such that , we have and . By the Meyer-Serrins theorem, we get a sequence such that in and pointwise too. We first get that . Then . We write
The first term tends to since in . As regards the second term, we use that pointwise and . By the Lebesgue dominated convergence and . Then, we consider and . We define
Note that . We deduce that for every . For we take the limit of when to get that
Since , we get on . We can find this Lemma in a weaker form in (MR3409135, , Theorem 6.3).∎
2-(ii) We assume that in with . Having in mind (12) and (15), we assume by contradiction that . By Lemma 8 and since and are differential operators, we have on . We infer that on , which contradicts . Hence and thus for some . Existence of solution for is proved.
Uniqueness of and of comes from the fact that the cost functionals of and are strictly convex whatever may be. Uniqueness of follows by application of the Poincaré inequality and uniqueness of follows from (36).
3.2 Proof of Theorem 3
For (Lagrange case), the cost is . We consider the triples and satisfying the optimality conditions (9), (10) and (12). Since is bounded at each time and by application of (36) to and we can find a constant depending only on such that
Setting , we have
By the Cauchy-Schwarz inequality we get a new constant such that
The two terms at the left-hand side are positive and using the inequality (35) with , we finally obtain
3.3 Proof of Theorem 6
(i) Strict dissipativity is established thanks to the storage function where is the optimal adjoint. Since , the storage function is locally bounded and bounded from below. Indeed, we consider an admissible pair satisfying (5), we multiply it by and we integrate over . Then we integrate in time on , we use the optimality conditions of static problem (12) and we get the strict dissipation inequality (16) with :