OPTIMAL CONTROL OF THE TWO-DIMENSIONAL EVOLUTIONARY NAVIER--STOKES EQUATIONS WITH MEASURE

In this paper, we consider an optimal control problem for the two-dimensional evolutionary Navier--Stokes system. Looking for sparsity, we take controls as functions of time taking values in a space of Borel measures. The cost functional does not involve directly the control but we assume some constraints on them. We prove the well-posedness of the control problem and derive necessary and sufficient conditions for local optimality of the controls.

Our motivation for the analysis of measure-valued controls is two-fold.On the one hand there it is the genuine interest in low-order regularity of the controls, on the other hand it relates to their sparsity promoting structure.Indeed, it has been observed and analyzed in much previous work that the optimal controls are typically zero over subsets of the domain, whereas they would simply be ``small,"" but not zero, if they would be replaced by a control in a Hilbert space, for example.We refer, exemplarily to the work in [6,8,23], which treats these phenomena for equations of diffusion type as well as for wave equations.In these papers the sparsity promoting terms is part of the cost, whereas in [14] the measure valued term appears as a constraint like in U ad above.It should also be mentioned that in case the measure-valued setting is replaced by an L 1 formulation together with L 2 constraints or penalties, again sparsity phenomena occur, but the optimal controls are, of course, functions in this case rather than measures [11,21].
In the literature, the optimal control of the Navier--Stokes equations has received much attention; we refer exemplarily to [1,4,16,17,22,30], and the monograph [20] and the survey [7].The controls are always considered as functions in these contributions.Apparently the only work on measure valued optimal controls in the case of the Navier--Stokes equations is [13] which treats the stationary case.
For evolutionary Navier--Stokes equations with forcing functions of low regularity, allowing for measure-valued forcing, very little analysis has been carried out even for the state equation by itself.We are only aware of [26], where the right-hand side in (1.1) is chosen in W 1,\infty (I; W - 1,p (\Omega )), with W - 1,p (\Omega ) = \bigotim d i=1 W - 1,p (\Omega ), d \in \{ 2, 3\} , and p \in ( d 2 , 2].It is mentioned there that likely the result is not optimal.In our previous work [15] we have obtained the necessary well-posedness results for (1.1) which are required for the study of optimal control problems.Thus the current work is the first one which considers optimal control for evolutionary Navier--Stokes equations with measure-valued controls.
When formulating optimal control problems some restrictions on the class of admissible controls are essential to guarantee existence of minimizers to be obtained by the standard method of the calculus of variations.Such restrictions are also well motivated by applications.One possible choice consists in adding a properly chosen control cost to the cost-functional J in (P).In our case it could be a term of the form \beta q \int T 0 \| u(t)\| q \bfM (\omega ) dt, where \beta is a positive weight.For technical reasons q = 2 seems not to be possible, since it does not imply sufficient temporal regularity on the class of admissible controls.From the analytical point of view it would suffice to take q > 4. But we prefer to rather work with pointwise constraints in time.In this way we arrive at the class U ad and the problem formulation chosen in (P).This choice of temporal pointwise constraints also poses new challenges in deriving both necessary and sufficient second order optimality conditions, regardless of the measure-valued norm in space.
Structure of paper.In the following section, well-posed results on the state equation relevant for the remainder of the paper are summarized.Here we can rely on results from [15].Existence of solutions to (P) and first-order optimality conditions are the contents of section 3. Necessary and sufficient second-order optimality conditions will be given in section 4.This requires further detailed analysis of the state equations and its linearization in function spaces of low regularity.
We finish this section proving the a continuity result for G.
Therefore, y satisfies (2.1) and, hence, y = y \bfu .Since every convergent subsequence of \{ y \bfu k \} \infty k=1 converges to the same limit y \bfu , we conclude that the whole sequence converges as claimed in the theorem to y \bfu .
3. Existence of solutions of (P) and first-order optimality conditions.We start this section by proving the existence of solutions for the control problem (P).Then, we show the differentiability of the cost functional and deduce the firstorder necessary optimality conditions.From these conditions we infer the sparsity properties of the stationary controls.
Before stating the optimality conditions satisfied by a solution of (P), we analyze the differentiability of the cost functional.
Next, we prove the first order necessary optimality conditions.Since (P) is not a convex problem, it is convenient to discuss necessary optimality conditions in the context of local solutions.Here, we say that \= u is a local solution of (P) if there exists a neighborhood \scrA of \= u in L \infty (I; M(\omega )) such that J(\= u) \leq J(u) for all u \in \scrA .If the inequality is strict for all u \in \scrA with u \not = \= u, we say that \= u is a strict local solution.We will also consider local solutions in the L q (I; W - 1,p (\Omega )) topology.Let us observe that the continuous embedding L \infty (I; M(\omega )) \subset L q (I; W - 1,p (\Omega )) implies that any local solution in the L q (I; W - 1,p (\Omega )) topology is also a local solution in the L \infty (I; M(\omega )) topology.
for i = 1, 2 and almost every point t \in I, where \= is the Jordan decomposition of the measure \= u i (t).
Next we define the Lagrangian function associated with the control problem (P).To this end, first we consider the functional j : M (\omega ) -\rightar [0, \infty ) given by j(u) = \| u\| M (\omega ) .This is a convex and Lipschitz functional having directional derivatives j \prime (u; v) for all u, v \in M (\omega ).To give an expression for the derivative j \prime (u; v) we consider the Lebesgue decomposition of v with respect to | u| : v = v a + v s with dv a = g v d| u| , where v a and v s are the absolutely continuous and singular parts of v with respect to | u| , and g v \in L 1 (| u| ) is the Radon--Nikodym derivative of v with respect to | u| .We can also write du = g u d| u| where g u is a measurable function such that | g u (x)| = 1 for all x \in \omega .Actually, g u is the Radon--Nikodym derivative of u with respect to | u| .The reader is referred, for instance, to [25,Chapter 6] for these issues.Now we have the following result taken from [12,Proposition 3.3].
Given u, v \in L \infty (I; M(\omega )), we denote by g vi (t) the Radon--Nikodym derivative of v i (t) with respect to | u i (t)| and v is (t) the singular part of v i (t) with respect to | u i (t)| .Then, g vi : \omega \times I -\rightar \BbbR is a measurable function.

From the inequality
With (4.3) and Lebesgue's theorem we infer Below we shall argue that v k \rightar v in L q (I; M(\omega )).From (4.4) we get Moreover, from (3.19) we have that \partial\scrL \partial\bfu (\= u, \= \bfitpsi)v only depends on the singular part of v(t) with respect to \= u(t).Since the singular part of v k (t) is zero or equal to the singular part of v(t), we conclude that \partial\scrL \partial\bfu (\= u, \= \bfitpsi)v k = 0.This identity along with j \prime (\= u i (t); v k,i (t)) = 0 for t \in I + \gamma ,i , \= \varphi i (t) \equiv 0 on I \setminu I + \gamma ,i , and the equality Next we prove that \= u + \rho v k \in U ad for every \rho > 0 small enough.Indeed, first we observe that Let us take \rho k > 0 such that \| v\| L \infty (I;\bfM (\omega )) < 1 k .
For i = 1, 2, using (4.4), we deduce for t \in I \gamma ,i and \rho \leq \rho k < \gamma is fulfilled.Thus, we have that \= u + \rho v k \in U ad for every \rho < \rho k .Now, using that \= u is a local minimum of (P), J \prime (\= u)v k = 0 as proved before, and performing a Taylor expansion we get for k fixed and \rho small enough Dividing the expression by \rho 2 /2, using the fact that J : L q (I; M(\omega )) \rightar \BbbR is of class C \infty , and taking \rho \rightar 0 we infer J \prime \prime (\= u)v 2 k \geq 0. Now, using again Lebesgue's theorem it follows that for almost every t \in I Using these properties we easily obtain that v k \rightar v in L q (I; M(\omega )).Then, with Theorem 3.2 we can pass to the limit when k \rightar \infty in the above inequality and conclude that J \prime \prime (\= u)v 2 \geq 0. Finally, if v \in C \= \bfu \setminu L \infty (I; M(\omega )), then we take \{ v k \} \infty k=1 \subset L \infty (I; M(\omega )) defined as follows It is straightforward to check v k \in C \= \bfu \cap L \infty (I; M(\omega )) for every k \geq 1 and v k \rightar v in L q (I; M(\omega )).Hence, J \prime \prime (\= u)v 2 k \geq 0 holds for every k, and passing to the limit we obtain J \prime \prime (\= u)v 2 \geq 0.
Remark 4.3.Notice that in the proof of Theorem 3.2 the continuous embedding \scrY \subset L 4 (I; L 4 (\Omega )) was established.Moreover, by Theorem 2.2 we know that for q \geq 8 the regularity assumption on \= y is automatically satisfied with r = q if y 0 \in B 2,4 (\Omega ) + B p,q (\Omega ).If q \in (4, 8) the local regularity \= y in L r (0, t 0 , L 4 (\Omega )) for some t 0 \in (0, T ] can be established, but we were not able to prove global regularity in this case.The necessity of r > 4 is explicitly used in Lemmas 4.8 and 4.9.This assumption allows us to establish the C( \= I; C 1 ( \= \Omega )) regularity of the adjoint states and, consequently, to prove the crucial Lemma 4.11.Notice that Lemma 4.8 below follows from Theorem 2.3 if q \geq 8. Downloaded 06/19/21 to 176.83.85.252.Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/page/termsEDUARDO CASAS AND KARL KUNISCH Remark 4.4.Sufficient second-order optimality conditions are essential to prove stability of the optimal states with respect to perturbations in problem data; see, for instance, [13].They are also used for proving convergence rates of the optimal states in the context of numerical approximations of the control problem; see [10].