Flow control in the presence of shocks Enrique Zuazua BCAM - Basque - - PowerPoint PPT Presentation

Flow control in the presence of shocks

Enrique Zuazua

BCAM - Basque Center for Applied Mathematics & Ikerbasque Bilbao, Basque Country, Spain zuazua@bcamath.org http://www.bcamath.org/zuazua/

PDE’s, Dispersion, Scattering theory and Control theory, Monastir, Tunisia, June 2013

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 1 / 59

Outline

1 Introduction: Motivation and examples 2 Optimal shape design in aeronautics 3 Shocks: Some remedies 4 Related topics

Several space dimensions (R. Lecaros) Large time asymptotics (L. Ignat & A. Pozo) Steady state models (M. Ersoy, E. feireisl & E. Z.) Viscous models Flux identification

5 Perspectives

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 2 / 59

Intro

Outline

1 Introduction: Motivation and examples 2 Optimal shape design in aeronautics 3 Shocks: Some remedies 4 Related topics

Several space dimensions (R. Lecaros) Large time asymptotics (L. Ignat & A. Pozo) Steady state models (M. Ersoy, E. feireisl & E. Z.) Viscous models Flux identification

5 Perspectives

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 3 / 59

Intro

Motivation Control problems for PDE are interesting for at least two reasons: They emerge in most real applications. PDE as the models of Continuum and Quantum Mechanics. Furthermore, in real world, there is something to be optimized, controlled, optimally shaped, etc. Answering to these control problems often requires a deep understanding of the underlying dynamics and a better master of the standard PDE models. Surprisingly enough, this has led to an important ensemble of new tools and results and some fascinating problems are still widely open. Furthermore, these kind of techniques are of application in some other fields, such as inverse problems theory and parameter identification issues.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 4 / 59

Intro

The same can be said about the numerical approximation aspects of these problems. Classical numerical analysis techniques do not suffice. Furthermore, from a control theoretical viewpoint this raises interesting issues about passing to the limit from finite to infinite space dimensions. Discrete versus continuous approaches.... In this talk we discuss the impact of shock discontinuities in solutions when addressing control problems for some toy models in Fluid Mechanics.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 5 / 59

Intro

Some relevant applications Noise reduction in cavities and vehicles. “Acoustic noise reduction” versus “active versus passive controllers”. Laser control in quantum mechanical and molecular systems. Seismic waves, earthquakes. Flexible structures. Environment: The Thames Barrier. Optimal shape design in aeronautics. Human cardiovascular system.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 6 / 59

Intro

Introduction A complete analysis of these issues involves, at least: Partial Differential Equations: Models describing motion in the various fields of Mechanics: Elasticity, Fluids,... Numerical Analysis: Allowing to discretize these models so that solutions may be approximated algorithmically. Optimal Design: Design of shapes to enhance the desired properties (bridges, dams, airplanes,..) Control: Automatic and active control of processes to guarantee their best possible behavior and dynamics. Parameter identification, inverse problems.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 7 / 59

Intro

Computer simulation → far beyond the fields in which its use is justified (consistency + stability ≡ convergence). The risk: To end up getting numerical data whose validity....

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 8 / 59

Intro

Is this difficulty solvable in practice? Solvable for problems with well known data by means of post-processing. Much harder for inverse, design and control problems,,,, In those cases the obtained final numerical results and simulations may simply mean nothing.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 9 / 59

Shape design in aeronautics

Outline

1 Introduction: Motivation and examples 2 Optimal shape design in aeronautics 3 Shocks: Some remedies 4 Related topics

Several space dimensions (R. Lecaros) Large time asymptotics (L. Ignat & A. Pozo) Steady state models (M. Ersoy, E. feireisl & E. Z.) Viscous models Flux identification

5 Perspectives

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 10 / 59

Shape design in aeronautics

Shape design in aeronautics

Optimal shape design in aeronautics. Two aspects: Shocks. Oscillations.

Optimal shape ∼ Active control. The shape of the cavity or airfoil controls the surrounding flow of air.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 11 / 59

Shape design in aeronautics

Optimal shape design in aeronautics Aeronautics: to simulate and optimize complex processes is indispensable. Long tradition: J. L. Lions, A. Jameson,... However, this needs an immense computational effort. For practical optimization problems, in which at least 100 design variables are to be considered, current methodological approaches applied in industry will need more than a year to obtain an optimized aircraft.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 12 / 59

Shape design in aeronautics

Francisco Palacios, Stanford University

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 13 / 59

Shape design in aeronautics

Mathematical problem formulation Minimize J(Ω∗) = min

Ω∈Cad

J(Ω) Cad = class of admissible domains. J = cost functional (drag reduction, lift maximization, exploitation cost,

• verall cost over the life cycle of the aircraft, benefit maximization, etc).

J depends on Ω through u(Ω), solution of the PDE (elasticity, Fluid Mechanics,...). The domains under consideration are often complex. Geometric and parametrization issues play a key role.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 14 / 59

Shape design in aeronautics

The dependence of the functional

• n the domain, through the

solution of the PDE is complex as

• well. J it is far from being a nice

convex function. Analytical difficulties: Lack of good existence, uniqueness, and continuous dependence theory for the PDE. Lack of convexity of the functional. Lack of compactness within the class of relevant domains...

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 15 / 59

Shape design in aeronautics

In practice Descent algorithm (gradient based method) on a discrete version of the problem: The domains Ω have been discretized (finite element mesh) The PDE has been replaced by a numerical scheme, The functional J has been replaced by a discrete version.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 16 / 59

Shape design in aeronautics

Note however that computing gradients, in practice, may be very hard.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 17 / 59

Shape design in aeronautics

We end up with a discrete optimization problem of huge dimensions. Divergence of numerical iterative algorithms may be hard to detect. [Boundary control of vibrations, E. Z. SIAM Review, 2005] Can we guarantee this kind of pathologies do not arise in realistic problems

• f optimal shape design in aeronautics?

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 18 / 59

Shape design in aeronautics

Two approaches: Discrete: Discretization + gradient Advantages: Discrete clouds of values. No shocks. Automatic differentiation, ... Drawbacks:

”Invisible” geometry. Scheme dependent.

Continuous: Continuous gradient + discretization. Advantages: Simpler computations. Solver independent. Shock detection. Drawbacks:

Yields approximate gradients. Subtle if shocks.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 19 / 59

Shocks: A remedy

Outline

1 Introduction: Motivation and examples 2 Optimal shape design in aeronautics 3 Shocks: Some remedies 4 Related topics

Several space dimensions (R. Lecaros) Large time asymptotics (L. Ignat & A. Pozo) Steady state models (M. Ersoy, E. feireisl & E. Z.) Viscous models Flux identification

5 Perspectives

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 20 / 59

Shocks: A remedy

The relevant models in aeronautics (Fluid Mechanics): Navier-Stokes equations; Euler equations; Turbulent models: Reynolds-Averaged Navier-Stokes (RANS), Spalart-Allmaras Turbulence Model, k − ε model; .... Burgers equation (as a 1 − d theoretical laboratory).

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 21 / 59

Shocks: A remedy

Solutions may develop shocks or quasi-shock configurations. For shock solutions, classical calculus fails; For quasi-shock solutions the sensitivity is so large that classical sensitivity clalculus is meaningless.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 22 / 59

Shocks: A remedy Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 23 / 59

Shocks: A remedy

Burgers equation Viscous version: ∂u ∂t − ν ∂2u ∂x2 + u ∂u ∂x = 0. Inviscid one: ∂u ∂t + u ∂u ∂x = 0.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 24 / 59

Shocks: A remedy

In the inviscid case, the simple and “natural” rule ∂u ∂t + u ∂u ∂x = 0 → ∂δu ∂t + δu ∂u ∂x + u ∂δu ∂x = 0 breaks down in the presence of shocks δu = discontinuous, ∂u

∂x = Dirac delta ⇒ δu ∂u ∂x ????

The difficulty may be overcame with a suitable notion of measure valued weak solution using Volpert’s definition of conservative products and duality theory (Bouchut-James, Godlewski-Raviart,...)

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 25 / 59

Shocks: A remedy

A new viewpoint: Solution = Solution + shock location. Then the pair (u, ϕ) solves:            ∂tu + ∂x(u2 2 ) = 0, in Q− ∪ Q+, ϕ′(t)[u]ϕ(t) =

• u2/2
• ϕ(t) ,

t ∈ (0, T), ϕ(0) = ϕ0, u(x, 0) = u0(x), in {x < ϕ0} ∪ {x > ϕ0}.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 26 / 59

Shocks: A remedy

The corresponding linearized system is:                    ∂tδu + ∂x(uδu) = 0, in Q− ∪ Q+, δϕ′(t)[u]ϕ(t) + δϕ(t)

• ϕ′(t)[ux]ϕ(t) − [uxu]ϕ(t)
• +ϕ′(t)[δu]ϕ(t) − [uδu]ϕ(t) = 0,

in (0, T), δu(x, 0) = δu0, in {x < ϕ0} ∪ {x > ϕ0}, δϕ(0) = δϕ0, Majda (1983), Bressan-Marson (1995), Godlewski-Raviart (1999), Bouchut-James (1998), Giles-Pierce (2001), Bardos-Pironneau (2002), Ulbrich (2003), ... None seems to provide a clear-cut recipe about how to proceed within an

• ptimization loop.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 27 / 59

Shocks: A remedy

A new method for optimization A new method for optimization: Splitting + alternating descent algorithm.

• C. Castro, F. Palacios, E. Z., M3AS, 2008.

Ingredients: The shock location is part of the state. State = Solution as a function + Geometric location of shocks. Alternate within the descent algorithm:

Shock location and smooth pieces of solutions should be treated differently; When dealing with smooth pieces most methods provide similar results; Shocks should be handeled by geometric tools, not only those based on the analytical solving of equations.

Lots to be done: Pattern detection, image processing, computational geometry,... to locate, deform shock locations,....

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 28 / 59

Shocks: A remedy

Alternating descent versus steepest descent Steepest descent: uk+1 = uk − ρ∇J(uk). Discrete version of continuous gradient systems u′(τ) = −∇J(u(τ)). Alternating descent: J = J(x, y), u = (x, y): uk+1/2 = uk − ρJx(uk); uk+1 = uk+1/2 − ρJy(uk). What’s the continuous analog? Does it correspond to a class of dynamical systems for which the stability is understood?

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 29 / 59

Shocks: A remedy

An example: Inverse design of initial data Consider J(u0) = 1 2 ∞

−∞

|u(x, T) − ud(x)|2dx. ud = step function. Gateaux derivative: δJ =

• {x<ϕ0}∪{x>ϕ0}

p(x, 0)δu0(x) dx + q(0)[u]ϕ0δϕ0, (p, q) = adjoint state                    −∂tp − u∂xp = 0, in Q− ∪ Q+, [p]Σ = 0, q(t) = p(ϕ(t), t), in t ∈ (0, T) q′(t) = 0, in t ∈ (0, T) p(x, T) = u(x, T) − ud, in {x < ϕ(T)} ∪ {x > ϕ(T)} q(T) =

1 2[(u(x,T)−ud)2]ϕ(T)

[u]ϕ(T)

.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 30 / 59

Shocks: A remedy

The gradient is twofold= variation of the profile + shock location. The adjoint system is the superposition of two systems = Linearized adjoint transport equation on both sides of the shock + Dirichlet boundary condition along the shock that propagates along characteristics and fills all the region not covered by the adjoint equations.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 31 / 59

Shocks: A remedy

State u and adjoint state p when u develops a shock:

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 32 / 59

Shocks: A remedy

Attention has to be paid to how the adjoint state p is extended within the region of influence of the shock. Otherwise, if applying directly the steepest descent algorithm we set u0 → u0 − ǫp(x, 0) then the number of shock discontinuities gets endless and unnecessarily multiplied during the iteration process.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 33 / 59

Shocks: A remedy

The discrete aproach Recall the continuous functional J(u0) = 1 2 ∞

−∞

|u(x, T) − ud(x)|2dx. The discrete version: J∆(u0

∆) = ∆x

2

∞

• j=−∞

(uN+1

j

− ud

j )2,

where u∆ = {uk

j } solves the 3-point conservative numerical approximation

scheme: un+1

j

= un

j − λ

• gn

j+1/2 − gn j−1/2

• = 0,

λ = ∆t ∆x , where, g is the numerical flux gn

j+1/2 = g(un j , un j+1), g(u, u) = u2/2.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 34 / 59

Shocks: A remedy

Examples of numerical fluxes gLF(u, v) = u2 + v2 4 − v − u 2λ , gEO(u, v) = u(u + |u|) 4 + v(v − |v|) 4 , gG(u, v) = minw∈[u,v] w2/2, if u ≤ v, maxw∈[u,v] w2/2, if u ≥ v.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 35 / 59

Shocks: A remedy

The Γ-convergence of discrete minimizers towards continuous ones is guaranteed for the schemes satisfying the so called one-sided Lipschitz condition (OSLC): un

j+1 − un j

∆x ≤ 1 n∆t , which is the discrete version of the Oleinick condition for the solutions of the continuous Burgers equations ux ≤ 1 t , which excludes non-admissible shocks and provides the needed compactness of families of bounded solutions. As proved by Brenier-Osher, 1 Godunov’s, Lax-Friedfrichs and Engquits-Osher schemes fulfil the OSLC condition.

1Brenier, Y. and Osher, S. The Discrete One-Sided Lipschitz Condition for

Convex Scalar Conservation Laws, SIAM Journal on Numerical Analysis, 25 (1) (1988), 8-23.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 36 / 59

Shocks: A remedy

A new method: splitting+alternating descent Generalized tangent vectors (δu0, δϕ0) ∈ Tu0 s. t. δϕ0 = ϕ0

x− δu0 +

x+

ϕ0 δu0

[u]ϕ0. do not move the shock δϕ(T) = 0 and δJ =

• {x<x−}∪{x>x+}

p(x, 0)δu0(x) dx,

• −∂tp − u∂xp = 0,

in ˆ Q− ∪ ˆ Q+, p(x, T) = u(x, T) − ud, in {x < ϕ(T)} ∪ {x > ϕ(T)}. For those descent directions the adjoint state can be computed by “any numerical scheme”!

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 37 / 59

Shocks: A remedy

Analogously, if δu0 = 0, the profile of the solution does not change, δu(x, T) = 0 and δJ = − (u(x, T) − ud(x))2 2

• ϕ(T)

[u0]ϕ0 [u(·, T)]ϕ(T) δϕ0. This formula indicates whether the descent shock variation is left or right!

WE PROPOSE AN ALTERNATING STRATEGY FOR DESCENT

In each iteration of the descent algorithm do two steps: Step 1: Use variations that only care about the shock location Step 2: Use variations that do not move the shock and only affect the shape away from it.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 38 / 59

Shocks: A remedy

Splitting+Alternating wins!

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 39 / 59

Shocks: A remedy

Results obtained applying Engquist-Osher’s scheme and the one based on the complete adjoint system Splitting+Alternating method.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 40 / 59

Shocks: A remedy

Numerical schemes replace shocks by oscillations. The oscillations of the numerical solution introduce oscillations on the approximation of the functional J. Purely discrete methods increase the number of shocks by mimicking the phenoma of multiplication of shocks of classical continuous methods.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 41 / 59

Shocks: A remedy

We suggest to stay as close as possible to the true landscape if the functional to be minimized accepting, and even taking advantage of, its possible discontinuities.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 42 / 59

Shocks: A remedy

Sol y sombra!

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 43 / 59

Shocks: A remedy

Splitting+alternating is more efficient: It is faster. It does not increase the complexity. Rather independent of the numerical scheme. Extending these ideas and methods to more realistic multi-dimensional problems is a work in progress and much remains to be done. Numerical schemes for PDE + shock detection + shape, shock deformation + mesh adaptation,...

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 44 / 59

Related topics

Outline

1 Introduction: Motivation and examples 2 Optimal shape design in aeronautics 3 Shocks: Some remedies 4 Related topics

Several space dimensions (R. Lecaros) Large time asymptotics (L. Ignat & A. Pozo) Steady state models (M. Ersoy, E. feireisl & E. Z.) Viscous models Flux identification

5 Perspectives

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 45 / 59

Related topics Several space dimensions (R. Lecaros)

An inverse design problem for a 2 − d scalar conservation law solved by a purely discrete method, R. Lecaros.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 46 / 59

Related topics Several space dimensions (R. Lecaros)

An inverse design problem for a 2 − d scalar conservation law solved by the alternating descent method, R. Lecaros.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 47 / 59

Related topics Several space dimensions (R. Lecaros)

Comparison of the discrete and the alternating descent method, R. Lecaros.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 48 / 59

Related topics Large time asymptotics (L. Ignat & A. Pozo)

Some applications require solving inverse design in very large time intervals: Sonic boom for supersonic aircrafts. In those cases the large time behavior of the numerical scheme employed is essential. Consider the 1-D conservation law with or without viscosity:

• ut +
• u2

x = εuxx,

x ∈ R, t > 0. Then: If ε = 0, ||u(·, t) − N(·, t)||1 ≤ Ct− 1

2

If ε > 0, ||u(·, t) − uM(·, t)||1 → 0 uM is the constant sign self-similar solution of the Burgers equation (defined by the mass M of u0), while N is the so-called N-wave, defined as: N(x, t) :=

• 1

d

x

t − σ

• ,

if (pdt)

1 2 < x − σt < (qdt) 1 2

• therwise

p := −2 min

y∈R

y

∞

u0(x)dx, q := 2 max

y∈R

y

∞

u0(x)dx

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 49 / 59

Related topics Large time asymptotics (L. Ignat & A. Pozo) Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 50 / 59

Related topics Large time asymptotics (L. Ignat & A. Pozo)

Asymptotic behavior for numerical schemes A similar phenomenon occurs for solutions of numerical approximation schemes. Even when the goal is to approximate the hyperbolic inviscid dynamics, numerical schemes introduce artificial numerical viscosity. When t → ∞ this leads to an asymptotic behavior that may be purely hyperbolic or of parabolic nature. Solutions of the Lax-Friedrcihs scheme behave, as t → ∞, as the viscous continuous solutions with diffusivity ∆t/∆x. By the contrary, solutions of the Engquist-Osher and Godunov schemes behave in a hyperbolic manner.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 51 / 59

Related topics Large time asymptotics (L. Ignat & A. Pozo) Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 52 / 59

Related topics Steady state models (M. Ersoy, E. Feireisl & E. Z.)

Steady state models (M. Ersoy, E. feireisl & E. Z.) Often in practice, optimal shapes and states are computed on the basis of steady state models. The relation between time evolution and steady state control is fairly clear in the context of linear systems. But a lot is still to be done to gain a good understanding in nonlinear models. ∂xf (v(x)) + v(x) = g(x), x ∈ (−, +∞), (1) ∂tu(t, x) + ∂xf (u(t, x)) + u(t, x) = g(x), u(0, x) = u0. (2)

• 2
• 1

1 2 3 2 4 6 8 10 T = 99.966

numerical solution v(x) source term g(x)

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 53 / 59

Related topics Viscous models

Viscous models Adjoint solutions for different viscous values of the viscosity parameter: ν = 0.5 (upper left), ν = 0.1 (upper right) and ν = 0.01 (lower left) and the exact adjoint solution (lower right).

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 54 / 59

Related topics Flux identification

Flux identification. ∂tu + ∂x(f (u)) = 0, in R × (0, T), u(x, 0) = u0(x), x ∈ R. This time the control is the nonlinearity f . It is actually an inverse problem.

• F. James and M. Sep´

ulveda, Convergence results for the flux identification in a scalar conservation law. SIAM J. Control Optim. 37(3) (1999) 869-891.

• C. Castro and E. Zuazua, Flux identification for 1-d scalar

conservation laws in the presence of shocks, Math. Comp. 80 (2011),

• no. 276, 2025

D2070.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 55 / 59

Perspectives

Outline

1 Introduction: Motivation and examples 2 Optimal shape design in aeronautics 3 Shocks: Some remedies 4 Related topics

Several space dimensions (R. Lecaros) Large time asymptotics (L. Ignat & A. Pozo) Steady state models (M. Ersoy, E. feireisl & E. Z.) Viscous models Flux identification

5 Perspectives

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 56 / 59

Perspectives

Analyze the Γ-convergence of discrete optimization problems towards the continuous ones. Further develop the alternating descent method in multi−d. Consider boundary value problems with optimal shape design problems in mind. Further analyze the suitability of the alternating descent method for viscous problems. Further analyze the viscous large time behavior for discrete schemes in multi-dimensional models. Analyze the proximity of time-evolution designs versus steady state

• nes.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 57 / 59

Perspectives

Some references

• C. Castro, C. Lozano, F. Palacios and E. Z., Systematic Continuous

Adjoint Approach to Viscous Aerodynamic Design on Unstructured Grids, AIAA Journal, 45 (9) (2007): 2125-2139.

• C. Castro, F. Palacios y E. Z., An alternating descent method for the
• ptimal control of the inviscid Burgers equation in the presence of

shocks, M3AS, 18 (3) (2008), 369-416.

• A. Baeza, C. Castro, F. Palacios and E. Z., 2-D Euler Shape Design
• n Nonregular Flows Using Adjoint Rankine - Hugoniot Relations,

AIAA Journal, 47, (3) (Mar. 2009), pp. 552-562.

• C. Castro and E. Z., Flux identification for 1-d scalar conservation

laws in the presence of shocks, Math. Comp. 80 (2011), no. 276, 2025–2070.

• A. Bueno-Orovio, C. Castro, F. Palacios and E. Z., Continuous adjoint

approach for the Spalart-Allmaras model in aerodynamic optimization, AIAA Journal, Vol. 50, No. 3, March 2012, pp. 631-646.

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 58 / 59

Perspectives

• F. Palacios, K. Duraisamy, J. J. Alonso and E. Z., Robust Grid

Adaptation for Efficient Uncertainty Quantification, AIAA Journal, 500 (7) (2012), 1538-1546.

• M. Ersoy, E. Feiresil and E. Z., Sensitivity analysis of 1 − d steady

forced scalar conservation laws, J. Differential Equations 254 (2013) 3817 - 3834.

• A. Porretta and E. Z., Long time versus steady state optimal control,

preprint. Thank you!

Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 59 / 59