Optimal shape and location of sensors. actuators in PDE models

Transcription

1 Modeling Optimal shape and location of sensors or actuators in PDE models Emmanuel Tre lat1 1 Sorbonne Universite (Paris 6), Laboratoire J.-L. Lions Fondation Sciences Mathe matiques de Paris Works with Yannick Privat and Enrique Zuazua E. Tre lat Optimal shape and location of sensors

2 Modeling What is the best shape and placement of sensors? - Reduce the cost of instruments. - Maximize the efficiency of reconstruction and estimations. E. Tre lat Optimal shape and location of sensors

3 The observed system may be described by: wave equation tt y = y or Schrödinger equation i t y = y general parabolic equations t y = Ay (e.g., heat or Stokes equations) in some domain Ω, with either Dirichlet, Neumann, mixed, or Robin boundary conditions For instance, when dealing with the heat equation: What is the optimal shape and placement of a thermometer?

4 Modeling Waves propagating in a cavity: tt y y = 0 Observable y(t, ) ω y(t, ) Ω = 0 Observability inequality Let T > 0. The observability constant CT (ω) is the largest nonnegative constant such that Z TZ y solution CT (ω) k(y(0, ), t y(0, ))k2l2 H 1 6 y(t, x) 2 dx dt 0 ω The system is said observable on [0, T ] if CT (ω) > 0 (otherwise, CT (ω) = 0). E. Tre lat Optimal shape and location of sensors

5 Modeling Waves propagating in a cavity: tt y y = 0 Observable y(t, ) ω y(t, ) Ω = 0 Observability inequality Let T > 0. The observability constant CT (ω) is the largest nonnegative constant such that Z TZ y solution CT (ω) k(y(0, ), t y(0, ))k2l2 H 1 6 y(t, x) 2 dx dt 0 ω Bardos Lebeau Rauch (SICON 1992): Observability holds if the pair (ω, T ) satisfies the Geometric Control Condition (GCC) in Ω: Every ray of geometrical optics that propagates in Ω and is reflected on its boundary Ω intersects ω in time less than T. (recent extension to time-varying domains: Le Rousseau Lebeau Terpolilli Tre lat, APDE 2017) E. Tre lat Optimal shape and location of sensors

6 Modeling Waves propagating in a cavity: tt y y = 0 Observable y(t, ) ω y(t, ) Ω = 0 Observability inequality Let T > 0. The observability constant CT (ω) is the largest nonnegative constant such that Z TZ y solution CT (ω) k(y(0, ), t y(0, ))k2l2 H 1 6 y(t, x) 2 dx dt 0 Q: What is the best possible subdomain ω of fixed given measure? ω (say, ω = L Ω with 0 < L < 1) N.B.: we want to optimize not only the placement but also the shape of ω, over all possible measurable subsets. (they do not have a prescribed shape, they are not necessarily BV, etc) E. Tre lat Optimal shape and location of sensors

7 The model Observability inequality T y solution C T (ω) (y(0, ), t y(0, )) 2 L 2 H 1 y(t, x) 2 dx dt 0 ω Let L (0, 1) and T > 0 fixed. It is a priori natural to model the problem as: sup C T (ω) ω Ω ω =L Ω with { T 0 ω C T (ω) = inf y(t, } x) 2 dx dt (y(0, ), t y(0, )) 2 (y(0, ), t y(0, )) L 2 (Ω) H 1 (Ω) \ {(0, 0)} L 2 H 1 BUT...

8 The model Observability inequality T y solution C T (ω) (y(0, ), t y(0, )) 2 L 2 H 1 y(t, x) 2 dx dt 0 ω Let L (0, 1) and T > 0 fixed. It is a priori natural to model the problem as: BUT: sup C T (ω) ω Ω ω =L Ω 1 Theoretical difficulty due to crossed terms in the spectral expansion (cf Ingham inequalities: Jaffard Tucsnak Zuazua, JFAA 1997). 2 In practice: many experiments, many measures. This deterministic constant is pessimistic: it gives an account for the worst case. Optimize shape and location of sensors in average, over a large number of measurements. Define an averaged observability inequality.

9 Randomized observability constant Averaging over random solutions: Randomized observability inequality (wave equation) ( T C T,rand (ω) (y(0, ), y t (0, )) 2 L 2 H 1 E 0 ω ) y ν(t, x) 2 dx dt where + ( ) y ν(t, x) = β1,j ν a j e iλ j t + β2,j ν b j e iλ j t φ j (x) j=1 with β ν 1,j, βν 2,j i.i.d. random variables (e.g., Bernoulli, Gaussian) of mean 0 with (φ j ) j N Hilbert basis of eigenfunctions. (inspired from Burq Tzvetkov, Invent. Math. 2008) Randomization generates a full measure set of initial data does not regularize

10 Randomized observability constant Theorem (Privat Trélat Zuazua, ARMA 2015, JEMS 2016) with ω measurable C T,rand (χ ω) = T inf γ j N j (T ) φ j (x) 2 dx ω 1 for wave and Schrödinger equations γ j (T ) = for parabolic equations e 2Re(λ j )T 1 2Re(λ j ) with (φ j ) j N a fixed Hilbert basis of eigenfunctions of the underlying operator. Remark There holds C T,rand (χ ω) C T (χ ω). There are examples where the inequality is strict: in 1D: Ω = (0, π), T kπ. in multi-d: Ω stadium-shaped, ω containing the wings.

11 Randomized observability constant Theorem (Privat Trélat Zuazua, ARMA 2015, JEMS 2016) with ω measurable C T,rand (χ ω) = T inf γ j N j (T ) φ j (x) 2 dx ω 1 for wave and Schrödinger equations γ j (T ) = for parabolic equations e 2Re(λ j )T 1 2Re(λ j ) with (φ j ) j N a fixed Hilbert basis of eigenfunctions of the underlying operator. Conclusion: we model the problem as sup ω Ω ω =L Ω inf γ j N j (T ) φ j (x) 2 dx ω

12 A remark for fixed initial data Note that, if we maximize ω T 0 ω y(t, x) 2 dx dt for some fixed initial data then, using the bathtub principle (decreasing rearrangement): There always exists (at least) one optimal set ω (s.t. ω = L Ω ). The regularity of ω depends on the initial data: it may be a Cantor set of positive measure, even for C data. ϕ γ,t (Lπ) ϕ γ,t (Lπ) 0 α1 α2 α3 α4 π 0 Lπ π (Privat Trélat Zuazua, DCDS 2015)

13 A remark on the class of subdomains We search the best subdomain ω over all measurable subsets of Ω: no restriction. - Let A > 0 fixed. If we restrict the search to or or or {ω Ω ω = L Ω and P Ω (ω) A} (perimeter) {ω Ω ω = L Ω and χ ω BV (Ω) A} (total variation) {ω Ω ω = L Ω and ω satisfies the 1/A-cone property} ω ranges over some finite-dimensional (or compact ) prescribed set... then there always exists (at least) one optimal set ω. but then... - The complexity of ω may increase with A. - We want to know if there is a very best set (over all possible measurable subsets).

14 Related problems and existing results 1) What is the best domain for achieving HUM optimal control? y tt y = χ ωu 2) What is the best domain domain for stabilization (with localized damping)? y tt y = kχ ωy t Existing works by - Hébrard, Henrot: theoretical and numerical results in 1D for optimal stabilization. - Münch, Pedregal, Periago: numerical investigations (fixed initial data). - Cox, Freitas, Fahroo, Ito,...: variational formulations and numerics. - Frecker, Kubrusly, Malebranche, Kumar, Seinfeld,...: numerical investigations over a finite number of possible initial data. - Demetriou, Morris, Padula, Sigmund, Van de Wal,...: actuator placement (predefined set of possible candidates), Riccati approaches. - Armaou, Demetriou, Chen, Rowley, Morris, Yang, Kang, King, Xu,...: H 2 optimization, frequency methods, LQ criteria, Gramian approaches. -...

15 sup ω Ω ω =L Ω inf γ j N j (T ) φ j (x) 2 dx ω To solve the problem, we distinguish between: parabolic equations (e.g., heat, Stokes) wave or Schrödinger equations Remarks requires some knowledge on the asymptotic behavior of φ j 2 µ j = φ j 2 dx is a probability measure strong difference between γ j (T ) e λ j T (parabolic) and γ j (T ) = 1 (hyperbolic)

16 Parabolic equations t y = Ay We assume that Ω is piecewise C 1. Slight spectral assumptions on A. Satisfied for heat, Stokes, anomalous diffusions A = ( ) α with α > 1/2. Theorem (Privat Trélat Zuazua, ARMA 2015) There exists a unique optimal domain ω. Moreover, ω is open and semi-analytic; in particular, it has a finite number of connected components. Quite difficult proof, requiring in particular: Hartung minimax theorem; uniform lower estimates of φ j 2 by Apraiz Escauriaza Wang Zhang (JEMS 2014): φ j (x) 2 Ce C µ j j N ω (µ j : eigenvalues of ) with C > 0 uniform with respect to ω s.t. ω = L Ω. G. Buttazzo: Shape optimization wins. Algorithmic construction of the best observation set ω : to be followed (further). C T (χ ω ) < C T,rand (χ ω ).

17 Wave and Schrödinger equations Optimal value (Privat Trélat Zuazua, JEMS 2016) Under appropriate spectral assumptions: sup ω Ω ω =L Ω inf j N ω φ j (x) 2 dx = L Proof: 1) Convexification (relaxation). 2) No-gap (not obvious because not lsc: kind of homogenization arguments). G. Buttazzo: Homogenization wins. Main spectral assumption: QUE (Quantum Unique Ergodicity): the whole sequence φ j 2 dx dx Ω vaguely. True in 1D (indeed, φ j (x) = sin(jx) and sin 2 jx 1 ), but in multi-d? 2

18 Wave and Schrödinger equations Optimal value (Privat Trélat Zuazua, JEMS 2016) Under appropriate spectral assumptions: sup ω Ω ω =L Ω inf j N ω φ j (x) 2 dx = L Relationship to quantum chaos theory: what are the possible (weak) limits of the probability measures µ j = φ j 2 dx? (quantum limits, or semi-classical measures) See also Shnirelman theorem: ergodicity implies Quantum Ergodicity (QE; but possible gap to QUE!) If QUE fails, we may have scars QUE conjecture (negative curvature)

19 Modeling Wave and Schro dinger equations Optimal value (Privat Tre lat Zuazua, JEMS 2016) Under appropriate spectral assumptions: Z sup inf j N ω Ω ω =L Ω φj (x) 2 dx = L ω Remark: The above result holds true as well in the disk. Hence the spectral assumptions are not sharp. (proof: requires the knowledge of all quantum limits in the disk, Privat Hillairet Tre lat) µjk * δr =1 (this is one QL: whispering galleries) E. Tre lat Optimal shape and location of sensors

20 Wave and Schrödinger equations Optimal value (Privat Trélat Zuazua, JEMS 2016) Under appropriate spectral assumptions: sup ω Ω ω =L Ω inf j N ω φ j (x) 2 dx = L Supremum reached? Open problem in general. in 1D: reached L = 1/2 (infinite number of optimal sets) in 2D square: reached over Cartesian products L {1/4, 1/2, 3/4} Conjecture: Not reached for generic domains Ω and generic values of L. Construction of a maximizing sequence (kind of homogenization)

21 Spectral approximation Following Hébrard Henrot (SICON 2005), we consider the finite-dimensional spectral approximation: (P N ) sup ω Ω ω =L Ω min γ j (T ) φ j (x) 2 dx 1 j N ω Theorem Given any N N, the problem (P N ) has a unique solution ω N. Moreover, ω N is semi-analytic and thus has a finite number of connected components.

22 Modeling Wave and Schro dinger equations Parabolic equations The complexity of ω N is increasing with N. Spillover phenomenon: the best domain ω N for the N first modes is the worst possible for the N + 1 first modes. (Privat Tre lat Zuazua, JFAA 2013) Problem 2 (Dirichlet case): Optimal domain for N=2 and L= Problem 2 (Dirichlet case): Optimal domain for N=10 and L= Ω = (0, π) (satisfied, e.g., by ( )α with α > 1/2) The sequence of optimal sets ω N is stationary: N0 N > N0 ω N = ω N0 = ω with ω the optimal set for all modes. In particular, ω is semi-analytic and thus has a finite number of connected components. Problem 2 (Dirichlet case): Optimal domain for N=20 and L= Under a slight spectral assumption: Problem 2 (Dirichlet case): Optimal domain for N=5 and L= (e.g., heat, Stokes, anomalous diffusions) L = 0.2 4, 25, 100, 400 eigenmodes E. Tre lat Optimal shape and location of sensors

23 Modeling Wave and Schro dinger equations Parabolic equations The complexity of ω N is increasing with N. Spillover phenomenon: the best domain ω N for the N first modes is the worst possible for the N + 1 first modes. (Privat Tre lat Zuazua, JFAA 2013) Problem 2 (Dirichlet case): Optimal domain for N=1 and L=0.2 (e.g., heat, Stokes, anomalous diffusions) Under a slight spectral assumption: (satisfied, e.g., by ( )α with α > 1/2) The sequence of optimal sets ω N is stationary: Problem 2 (Dirichlet case): Optimal domain for N=5 and L=0.2 N0 N > N0 ω N = ω N0 = ω with ω the optimal set for all modes. In particular, ω is semi-analytic and thus has a finite number of connected components. Problem 2 (Dirichlet case): Optimal domain for N=10 and L=0.2 Ω = unit disk Problem 2 (Dirichlet case): Optimal domain for N=20 and L=0.2 L = 0.2 1, 25, 100, 400 eigenmodes E. Tre lat Optimal shape and location of sensors

24 Modeling Wave and Schro dinger equations Parabolic equations The complexity of ω N is increasing with N. Spillover phenomenon: the best domain ω N for the N first modes is the worst possible for the N + 1 first modes. (Privat Tre lat Zuazua, JFAA 2013) (e.g., heat, Stokes, anomalous diffusions) Under a slight spectral assumption: (satisfied, e.g., by ( )α with α > 1/2) The sequence of optimal sets ω N is stationary: N0 N > N0 ω N = ω N0 = ω with ω the optimal set for all modes. In particular, ω is semi-analytic and thus has a finite number of connected components. no fractal set! Ω = (0, π)2 1, 4, 9, 16, 25, 36 eigenmodes L = 0.2, T = 0.05 optimal thermometer in a square E. Tre lat Optimal shape and location of sensors

25 Conclusion and perspectives Similar results for optimal control or stabilization domains. Optimal design for boundary observability: sup inf γ ω =L Ω j N j (T ) ω 1 λ j φ j ν (role of Rellich function and identities) 2 dh n 1 Intimate relations between shape optimization and quantum chaos (quantum ergodicity properties). Strategies to avoid spillover? Discretization issues: do the numerical optimal designs converge to the continuous optimal design as the mesh size tends to 0? Y. Privat, E. Trélat, E. Zuazua, Optimal observation of the one-dimensional wave equation, J. Fourier Analysis Appl. (2013) Optimal location of controllers for the one-dimensional wave equation, Ann. Inst. H. Poincaré (2013) Complexity and regularity of maximal energy domains for the wave equation with fixed initial data, DCDS (2015) Optimal shape and location of sensors for parabolic equations with random initial data, ARMA (2015) Optimal observability of the multi-d wave and Schrödinger equations in quantum ergodic domains, JEMS (2016) Actuator design for parabolic distributed parameter systems with the moment method, SIAM J. Cont. Optim. (2017)

26 Conclusion and perspectives What can be said for the classical (deterministic) observability constant? A result for the wave observability constant (Humbert Privat Trélat, 2017): C lim T (ω) T + T = 1 2 min inf φ j (x) 2 dx, lim inf 1 T χ ω (γ(t)) dt j N ω T + {γ ray} T 0 }{{}}{{} Two quantities: spectral geometric (rays) randomized obs. constant optimize it?

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28 Remark: another way of arriving at the criterion (wave equation) Averaging in time: Time asymptotic observability inequality: C (χ ω) (y(0, ), y t (0, )) 2 1 T L 2 H 1 lim y(t, x) 2 dx dt, T + T 0 ω with C (χ ω) = inf lim 1 T0 ω y(t, x) 2 dx dt (y(0, ), yt T + T (y(0, ), y t (0, )) 2 (0, )) L 2 H 1 \ {(0, 0)} L 2 H 1. Theorem If the eigenvalues of g are simple then C (χ ω) = 1 2 inf φ j (x) 2 dx = 1 j N ω 2 J(χω). Remarks C (χ ω) 1 2 inf j N φ j (x) 2 dx. ω C T (χ ω) lim sup C (χ ω). There are examples where the inequality is strict. T + T

29 Remedies (wave and Schrödinger equations) 1. Existence of a maximizer Ensured if U L is replaced with any of the following choices: V L = {χ ω U L P Ω (ω) A} (perimeter) V L = {χ ω U L χ ω BV (Ω) A} (total variation) V L = {χ ω U L ω satisfies the 1/A-cone property} where A > 0 is fixed.

30 Remedies (wave and Schrödinger equations) 2. Weighted observability inequality ) T C T,σ (χ ω) ( (y(0, ), t y(0, )) 2L 2 H 1 + σ y(0, ) 2H 1 y(t, x) 2 dx dt 0 ω with a weight σ 0. Note that C T,σ (χ ω) C T (χ ω). By randomization: with σ j = λ2 j σ+λ 2 j. C T,σ,rand (χ ω) = T 2 inf σ j N j φ j (x) 2 dx ω

31 Remedies (wave and Schrödinger equations) Theorem Assume that L -QUE holds. If σ 1 < L < 1 then there exists N N such that sup χ ω U L inf j N j φ j 2 ω = max χ ω U L inf j φ j 2 1 j n ω σ 1 < L, for every n N. In particular there is a unique solution χ ω N. Moreover if M is analytic then ω N is semi-analytic and has a finite number of connected components. The condition σ 1 < L < 1 seems optimal (see numerical simulations). This result holds as well in any torus, or in the Euclidean n-dimensional square for Dirichlet or mixed Dirichlet-Neumann conditions.

32 Modeling L = 0.2 L = 0.4 L = 0.6 L = 0.9 E. Tre lat Optimal shape and location of sensors

33 Anomalous diffusion Anomalous diffusion equations, Dirichlet: t y + ( ) α y = 0 (α > 0 arbitrary) with a surprising result: In the square Ω = (0, π) 2, with the usual basis (products of sine): the optimal domain ω has a finite number of connected components, α > 0. In the disk Ω = {x R 2 x < 1}, with the usual basis (Bessel functions), the optimal domain ω is radial, and α > 1/2 ω = finite number of concentric rings (and d(ω, Ω) > 0) α < 1/2 ω = infinite number of concentric rings accumulating at Ω! (or α = 1/2 and T small enough) The proof is long and very technical. It uses in particular the knowledge of quantum limits in the disk. (L. Hillairet, Y. Privat, E.Trélat)

34 Modeling Ω = unit disk 1, 4, 9, 16, 25, 36 eigenmodes L = 0.2, T = 0.05, α = 1 E. Tre lat Optimal shape and location of sensors

35 Modeling Ω = unit disk 1, 4, 25, 100, 144, 225 eigenmodes L = 0.2, T = 0.05, α = 0.15 E. Tre lat Optimal shape and location of sensors

36 Comparison sup inf γ χ ω U L j N j (T ) φ j 2 ω square disk relaxed solution a = L relaxed solution a = L wave or Schrödinger ω for L { 1 4, 1 2, 3 4 } ω for L { 1 4, 1 2, 3 4 } otherwise (conjecture) otherwise (conjecture)!ω L α > 0!ω (radial) L α > 0 diffusion ( ) α #c.c.(ω) < + if α > 1/2 then #c.c.(ω) < + if α < 1/2 then #c.c.(ω) = +