Two results on the control of fluids

Two results on the control of fluids Enrique Fernández-Cara [email protected] Dept. EDAN and IMUS, University of Sevilla, Spain Abstract This talk is devoted to recall and comment two results recently obtained concerning the control of viscous fluids. In general terms, we fix an initial state and we try to find internal or boundary data such that an associated solution vanishes at a prescribed time. Among others, we will consider fluids modeled by coupled systems of the Boussinesq kind. It will be seen that, under some circumstances, the systems are controllable or “quasi-controllable” in an appropriate sense. We will also take a look at some minimal time control problems and will present several theoeretical and numerical results. Also, several open problems will be mentioned. 1. Introduction Control theory is a scientific discipline that, roughly speaking, try to find out “how can we act on systems”. It is multidisciplinary and strongly motivated by real-world applications and, of course, involves mathematics, physics, engineering and other disciplines. In fact, control problems and their analysis have been considered since ancient times. Among other examples, we can mention • The irrigation systems (Mesopotamia, since 6,000 B.C.), where the aim was to make reach and stay water regularly in the region. • The Roman aqueducts (from II Century B.C. to IV Century A.C.), with a similar goal. • The steam engine (about 1700), an invention that changed the world: the first device able to convert heat into motion in a controlled way. At present, control theory is applied to problems coming from many different fields: • From structural mechanics, where control techniques serve to stabilize dangerous and/or undesirable vibrations. • From the autonomous car driving sector, where very interesting multi-objective problems appear naturally. • From epidemics and pandemics studies, concerning (for instance) the optimization of vaccination and quarantine strategies. • From population dynamics, where the objective is to design optimal feeding, tranporting or spreading processes. • Frombiomedical sciences, wherecontrolis usually orientedtotherapy. Thisapplies to cancer, diabetes, alzheimer, etc. In fluid mechanics, control problems are also very relevant. They are found when one tries to reduce as much as possible turbulence effects, to ensure fluid transportation or design optimal parachutes, wind tunnels and airfcrafts. In this work, we will be motivated by the control of the Navier-Stokes and the Boussinesq systems. Recall that these PDEs model the behavior of incompressible homogeneous Newtonian fluid respectively insensitive and subject to heat effects. In the boundary controlled case, they read as follows: �u𝑡+(u⋅∇)u−𝜈Δu+∇𝑝=0,∇⋅u=0,(x,𝑡)∈Ω×(0,𝑇), u=f1𝛾,(x,𝑡)∈𝜕Ω×(0,𝑇), +… French-German-Spanish Conference on Optimization Gijón, June 18-21, 2024 (pp. 55-62) FGS2024 55 ISBN 978-84-10135-30-7 and ⎧ ⎨ ⎩ u𝑡+(u⋅∇)u−𝜈Δu+∇𝑝=𝜃k,∇⋅u=0,(x,𝑡)∈Ω×(0,𝑇), 𝜃𝑡+u⋅∇𝜃−𝜅Δ𝜃=0,(x,𝑡)∈Ω×(0,𝑇), u=f1𝛾,𝜃=𝛽1𝛾,(x,𝑡)∈𝜕Ω×(0,𝑇), +… where Ω⊂R𝑁is a bounded open connected domain with Lipschitz-continuous bounndary, 𝑇>0,u,𝑝and 𝜃 respectively denote the velocity field, pressure and temperature of the fluid, fand 𝛽are the controls, 𝛾is a (small) part of 𝜕Ω,𝜈and 𝜅are positive constants, kis a constant vector and the dots contain conditions at 𝑡=0. As shown below, the choice of fand 𝛽can be motivated by many different reasons. 2. The null controllability problem for the Boussinesq PDEs We will first deal with the nonlinear system �Boussinesq for (u,𝑝,𝜃),(x,𝑡)∈Ω×(0,𝑇), u=f1𝛾,𝜃=𝛽1𝛾,(x,𝑡)∈𝜕Ω×(0,𝑇), u(⋅,0)=u0,𝜃(⋅,0)=𝜃0. Let nstand for the unit outwards normal vector on 𝜕Ωand let us intoduce the Hilbert spaces 𝐻∶={v∈𝐿2(Ω)𝑁∶∇⋅v=0,v⋅n|𝜕Ω =0} and 𝑉∶={v∈𝐻1 0(Ω)𝑁∶∇⋅v=0,v|𝜕Ω =0}, respectively endowed with the norms of 𝐿2(Ω)𝑁and 𝐻1 0(Ω)𝑁. The problem considered in this section is the following: For any given u0∈𝐻and 𝜃0∈𝐿2(Ω), find fand ℎin appropriate spaces and an associated solution such that u(⋅,𝑇)=0and 𝜃(⋅,𝑇)=0. (2.1) ThatthisproblemcanbesolvedwasconjecturedbyJ.-L.LionsfortheNavier-StokesPDEsin1990, see[15]. Since then, many partial (positive) results have been obtained, see among others [1–4,6,7,9,11–14]. However, the conjecture is open. Even more, whether or not the following approximative version is solvable is also unknown: For any given u0∈𝐻and 𝜃0∈𝐿2(Ω)and any 𝜀>0, find f𝜀and ℎ𝜀in appropriate spaces and an associated solution such that ‖(u,𝜃)(⋅,𝑇)‖≤𝜀 (here and henceforth, ‖⋅‖stands for the usual 𝐿2norm in Ω). We will assume that 𝑁=3,Ωis a cube and 𝛾is the complement of a face and we will recall a result that proves the conjecture in a different “approximate” sense. More precisely, the following result holds: Theorem 2.1 Let (u0,𝜃0)∈𝐻×𝐿2(Ω)be given. There exist a family of “ghost” right hand sides {(F𝜖,𝐺𝜖)} with (F𝜀,𝐺𝜀)→(0,0)in an appropriate (large) space as 𝜀→0such that, for any 𝜀there exist controls (f𝜀,ℎ𝜀) and associated solutions to the nonhomogeneous systems ⎧ ⎨ ⎩ u𝑡−𝜈Δu+(u⋅∇)u+∇𝑝=𝜃k+F𝜖, ∇⋅u=0,(x,𝑡)∈Ω×(0,𝑇), 𝜃𝑡−𝜅Δ𝜃+u⋅∇𝜃=𝐺𝜖,(x,𝑡)∈Ω×(0,𝑇), u=f𝜀1𝛾, 𝜃=𝛽𝜀1𝛾,(x,𝑡)∈𝜕Ω×(0,𝑇), +… such that (2.1) holds. Sketch of the proof: The complete proof is given in [13] (resp. [7]) in the case of the Navier-Stokes (resp. Boussinesq) system. For example, let us assume that Ω=(−2,2)3and 𝛾is the complement to the face {𝑥1=−2}(see Fig. 1) and let 𝜀>0be given. The proof consists of four steps: Two results on the control of fluids 56 Fig. 1 The control acts on 𝛾, the complement of the left vertical face. Step 1: Take 𝑇1close to 𝑇and zero controls in [0,𝑇1]and let the system evolve from 𝑡=0to 𝑡=𝑇1. We can assume that (u1,𝜃1)∶=(u,𝜃)(⋅,𝑇1)∈(𝑉∩𝐻2(Ω)3)×𝐻2(Ω). Thus, in this step, the controls and the ghost vanish. Step 2: Take 𝑇2in (𝑇1,𝑇)and a ghost (F𝜖,𝐺𝜖)in [𝑇1,𝑇2]such that (u2,𝜃2)∶=(u,𝜃)(⋅,𝑇2)is regular enough and compactly supported in Ω. In this step, the control is zero and it can be assumed that, for some norm ‖(F𝜖,𝐺𝜖)‖∗is 𝑂(𝜀)for some appropriate norm ‖⋅‖∗. Step 3: Now, we introduce 𝑇3in (𝑇2,𝑇)and a new ghost (F𝜖,𝐺𝜖)in [𝑇2,𝑇3]such that, at time 𝑇3, the state (U,Θ)∶=(u,𝜃)(⋅,𝑇3)is of the form ((𝜙,0,𝜓),𝜁)(⋅,𝑇3)for functions 𝜙,𝜓and 𝜁that only depend on 𝑥2 and 𝑡. This is obviously the main step and, again, we can assume that leads to a small ghost, that is, ‖(F𝜖,𝐺𝜖)‖∗= 𝑂(𝜀). Step 4: Finally, in [𝑇3,𝑇]we control a 1D heat PDE coupled to a 1D, 2×2parabolic system. More precisely, we consider the system ⎧ ⎨ ⎩𝜙𝑡−𝜈𝜙𝑥2,𝑥2=0,(𝑥2,𝑡)∈(−2,2)×(0,𝑇), 𝜓𝑡−𝜈𝜓𝑥2,𝑥2=−𝜁, 𝜁𝑡−𝜅𝜁𝑥2,𝑥2=0,(𝑥2,𝑡)∈(−2,2)×(0,𝑇), 𝜙|𝑥2=−2 =𝜓|𝑥2=−2 =𝜁|𝑥2=−2 =0,𝜙|𝑥2=2 =𝑘(𝑡),𝜓|𝑥2=2 =ℓ(𝑡),𝜁|𝑥2=2 =𝑞(𝑡),𝑡∈(0,𝑇), +… where we choose the boundary controls 𝑘,ℓand 𝑞such that ((𝜙,0,𝜓),𝜁)(⋅,𝑇)=(0,0). This can be done in view of the results in [8]. We see that, in this last step, the ghost is zero. It is clear that, after this construction, the proof is done; the previous argument is illustrated in Fig. 2.  Some remarks and related open questions are in order: • The argument in the proof is also applicable to a domain with a flat piece of boundary, see for instance Fig. 3. It would be interesting to deduce an extension to other boundaries, maybe taking into account an appropriate variable change. • It is possible to improve the result in several directions. Thus, only two scalar controls are needed in Step 4 and this means that Theorem 2.1 remains true with two components of f𝜀equal to zero; also, the control regioncan becomposed ofonlythreefaces; on theother hand, it isfoundin [5]that a similar result can be deduced in dimension two in a rectangle with ghosts (F𝜖,𝐺𝜖)that converge much better to zero, etc. Enrique Fernández Cara 57 Fig. 2 Main idea: drive (u, 𝜃) at 𝑡 = 𝑇 3exactly to (U, Θ). • It is unknown whether the same result can be proved for the so called “full” Boussinesq system, where the heat PDE is 𝜃𝑡+u⋅∇𝜃−𝜅Δ𝜃=𝜈𝐷u∶∇u. • If we view this result as a preliminary step in the proof of null controllability, it becomes clear that what remains to do is to get an estimate of (h𝜀,𝛽𝜀)somewhere independent of 𝜀. But this is at present unknown. • On the other hand, it would be interesting to figure out whether this result implies approximate controllability in the sense indicated above. Again, this is an open question. Fig. 3 The control acts on 𝛾, the complement of the flat boundary. 3. A minimal time control problem In thissection, wewill be concernedwith acontrol problem where thegoal isto findthe minimal timeneeded to drive the system near to a desired state. Two results on the control of fluids 58 For simplicity, we will consider the Navier-Stokes PDEs with controls in the right hand side. Thus, let Ω⊂ R𝑁be a bounded connected open set with (for instance) Lipschitz-continuous boundary, let 𝜔⊂⊂Ωbe a (small) open set and consider the controlled system �u𝑡+(u⋅∇)u−𝜈Δu+∇𝑝=f1𝜔,∇⋅u=0,(x,𝑡)∈Ω×(0,𝑇), u=0,(x,𝑡)∈𝜕Ω×(0,𝑇), u(⋅,0)=u0.(3.1) The problem is the following: Given u0,u𝑇∈𝐻and 𝛿>0, find the minimal time 𝑇>0satisfying ‖u(⋅,𝑇)−u𝑇‖≤𝛿, (3.2) where uis, together with some 𝑝, a solution to (3.1) for some f. For general u0and u𝑇in 𝐻, the existence of times 𝑇such that (3.2) holds is unknown. But, even if they exist, it is not clear at all that a minimal time can be found. These considerations justify the following approximated (or penalized) version of the problem: �Minimize Φ(𝑇,f)∶=𝑇2 2+𝑏2�𝜔×(0,+∞) |f|2𝑑𝑥𝑑𝑡 Subject to: (𝑇,f)∈ℋ𝑎𝑑,(3.3) where 𝑏>0and ℋ𝑎𝑑 ∶={(𝑇,f)∶𝑇≥0,f∈𝐿2(𝜔×(0,+∞))𝑁,‖u(⋅,𝑇)−u𝑇‖=𝛿}. Thus, it can be a good strategy to solve (3.3) for any 𝑏>0, then take 𝑏→0+and see what happens. Notethat, inseveralparticularsituations, theassumptionℋ𝑎𝑑 ≠∅canbe asserted; forinstance, if‖u𝑇‖< 𝛿this is the case. In a work in collaboration with I. Marın-Gayte (see [10]), a set of results have been obtained for (3.3): • First, it is proved that, if ℋ𝑎𝑑 ≠∅, there exist optimal couples (𝑇,f). Fig. 4 A numerical test. The domain, the control region and the mesh. Number of points: 1287. Enrique Fernández Cara 59 • Then, it is found that, if (𝑇,f)is optimal and an associated uis regular enough, a suitable optimality system is satisfied. More precisely, the following coupled system must hold for 𝑇,f,u,𝑝, a multiplier 𝜆 and the adjoint variables zand 𝑞: ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ Classical OS for given 𝜆and 𝑇: ⎡⎢⎢⎢⎣ Navier-Stokes for (u,𝑝)and f Ajoint system for (z,𝑞)and u … Pontryagin for fand z:f=−1 𝜆𝑏z�𝜔×(0,𝑇) Additional conditions for 𝜆and 𝑇: �‖u(⋅,𝑇)−u𝑇‖=𝛿 𝑇=−𝜆(u𝑡(⋅,𝑇),u(⋅,𝑇)−u𝑇) Fig. 5 A numerical test. The target 𝜃𝑇. For the proof, the Dubovitsky-Milyutin principle can be used. The argument reads as follows: –First, we note that, if (𝑇,f)is optimal, there cannot exist descent directions for Φat the same time admissible for the imposed constraints. Consequently, the intersection of the associated cones of directions must be empty. –By duality, we find that a nontrivial linear combination of dual directions must vanish. After a reformulation and a detailed analysis, we deduce that there exist 𝜆and (z,𝑞)satisfying the previous optimality system. • Finally, under the assumption ℋ𝑎𝑑 ≠∅, efficient numerical methods are exhibit for the computation of (an approximation to) an optimal (𝑇,f). Among others, the following strategy is proposed: 1. In a first (preliminar) step, for a lot of (𝜆,𝑇), compute the solution to OC problem ⎧ ⎪ ⎨ ⎪ ⎩ Navier-Stokes for (u,𝑝)and f Ajoint system for (z,𝑞)and u +… Pontryagin for fand z:f=−1 𝜆𝑏z�𝜔×(0,𝑇) 2. Then, solve the 2×2system �‖u(⋅,𝑇)−u𝑇‖=𝛿 𝑇=−𝜆(u𝑡(⋅,𝑇),u(⋅,𝑇)−u𝑇) This gives a (candidate to) solution 𝜆𝑏and 𝑇𝑏. Two results on the control of fluids 60 Fig. 6 A numerical test. Solving the equation 𝑇 = −𝜆(𝜃𝑡(⋅ , 𝑇), 𝜃(⋅ , 𝑇) − 𝜃𝑇). Computed values: 𝑇 = 13.8741,𝜇 = 199.374,𝜆 = 2.5142 × 105. In order to illustrate the method, let us give the results of an experiment for a simpler model. Specifically, we consider the problem �Minimize Ξ(𝑇,𝑓)∶=𝑇2 2+𝑏2�𝜔×(0,+∞) |𝑓|2𝑑𝑥𝑑𝑡, Subject to: (𝑇,𝑓)∈𝒦𝑎𝑑 (3.4) where 𝒦𝑎𝑑 ∶={(𝑇,𝑓)∶𝑇≥0,𝑓∈𝐿2(𝜔×(0,+∞)),‖𝜃(⋅,𝑇)−𝜃𝑇‖=𝛿} and, for each 𝑓∈𝐿2(𝜔×(0,+∞)), we denote by 𝜃the associated solution to �𝜃𝑡−Δ𝜃=𝑓1𝜔,(x,𝑡)∈Ω×(0,𝑇), 𝜃=0,(x,𝑡)∈𝜕Ω×(0,𝑇), 𝜃|𝑡=0 =0. (3.5) The domain, the control region and the mesh are depicted in Fig. 4. For the numerical solution of the state systems (3.5) we have used standard implcit Euler finite different approximations on time and ℙ1finite element techniques in space. We have taken 𝑏=100,𝛿=0.05and 𝜃𝑇(x)=𝑅2−|x|2 (see Fig. 5). The computation of 𝑇,𝜇and 𝜆=1/(𝑏(2𝜇−1))is explained in Fig. 6 and the computed final state 𝜃(⋅,𝑇) is given in Fig. 7. We will end this section with several comments and additional questions: • First, note that we can formulate and solve similar problems for the Boussinesq system, the variable density Navier-Stokes PDEs, boundary controlled models, etc. • Obviously, the most interesting question is what happens to the solution to (3.3) as 𝑏→0+? More precisely, whencanweensureuniformestimatesofthe solutions(𝑇,f)undertheassumptionℋ𝑎𝑑 ≠∅? • Finally, observe that an unexplored variant of the considered control problem consists of searching for the minimal time to escape from u0. Acknowledgements TheauthorhasbeenpartiallysupportedbygrantPID2020-114976GB-I00. ThisisfundedbyMICIU/AEI(Spain), with code 10.13039/501100011033. Enrique Fernández Cara 61 Fig. 7 A numerical test. The computed 𝜃(⋅ , 𝑇) with ‖𝜃(⋅ , 𝑇) − 𝜃𝑇‖ = 𝛿. References [1] F. W. Chaves-Silva, E. Fernandez-Cara, K. Le Balc’h, J. L. F. Machado, and D. A. Souza. 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