[PPT] - Dynamic control Greedy, turnpike, constraints and some applications PowerPoint Presentation

SLIDE 1

Dynamic control Greedy, turnpike, constraints and some applications

Enrique Zuazua1

DeustoTech-Bilbao & UAM-Madrid, Spain enrique.zuazua@deusto.es http://cmc.deusto.es

Marrakech, Avril 2018

1ERC Advanced Grant DYCON-Dynamic Control (http://cmc.deusto.es/dycon/)

and ANR Project ICON

E. Zuazua

Turnpike, constraints & greedy Marrakech, Avril 2018 1 / 50

SLIDE 2

Motivation

Outline

1 Motivation 2 Long time control: Numerics and turnpike

Motivation Long time numerics Turnpike

3 Constraints

Motivation

4 1-d constrained heat equation 5 Parameter depending control problems

Motivation Averaged control Greedy algorithms

6 Conclusion and open problems

E. Zuazua

Turnpike, constraints & greedy Marrakech, Avril 2018 6 / 50

SLIDE 7

Long time control: Numerics and turnpike Motivation

Sonic boom

Goal: the development of supersonic aircrafts, sufficiently quiet to be allowed to fly supersonically over land. The pressure signature created by the aircraft must be such that, when reaching ground, (a) it can barely be perceived by humans, and (b) it results in admissible disturbances to man-made structures. Juan J. Alonso and Michael R. Colonno, Multidisciplinary Optimization with Applications to Sonic-Boom Minimization, Annu. Rev. Fluid Mech. 2012, 44:505 – 526.

E. Zuazua

Turnpike, constraints & greedy Marrakech, Avril 2018 7 / 50

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Long time control: Numerics and turnpike Motivation

Many other examples in biomedicine, social sciences, economics, lead to natural questions of control in long time. Sustainable growth is a long-term challenge. And two key issues arise: Develop specific tools for long time control horizons. Build numerical schemes capable of reproducing accurately the control dynamics in long time intervals.

E. Zuazua

Turnpike, constraints & greedy Marrakech, Avril 2018 8 / 50

SLIDE 9

Long time control: Numerics and turnpike Long time numerics

Outline

1 Motivation 2 Long time control: Numerics and turnpike

Motivation Long time numerics Turnpike

3 Constraints

Motivation

4 1-d constrained heat equation 5 Parameter depending control problems

Motivation Averaged control Greedy algorithms

6 Conclusion and open problems

E. Zuazua

Turnpike, constraints & greedy Marrakech, Avril 2018 9 / 50

SLIDE 10

Long time control: Numerics and turnpike Long time numerics

Geometric/Symplectic integration Numerical integration of the pendulum (A. Marica)a

aHAIRER, E., LUBICH, Ch., WANNER, G.. Geometric Numerical

Integration. Structure-Preserving Algorithms for Ordinary Differential
Equations. 2nd ed. Berlin : Springer, 2006, 644 p.
E. Zuazua

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Long time control: Numerics and turnpike Long time numerics

Joint work with L. Ignat & A. Pozo, Math of Computation, 2014

Consider the 1-D conservation law with or without viscosity: ut + ⇥ u2⇤

x = εuxx,

x ∈ R, t > 0. Then2 : If ε = 0, u(·, t) ∼ N(·, t) as t → ∞; If ε > 0, u(·, t) ∼ uM(·, t) as t → ∞, uM is the constant sign self-similar solution of the viscous Burgers equation (defined by the mass M of u0), while N is the so-called hyperbolic N-wave.

2Y. J. Kim & A. E. Tzavaras, Diffusive N-Waves and Metastability in the Burgers

Equation, SIAM J. Math. Anal. 33(3) (2001), 607–633.

E. Zuazua

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Long time control: Numerics and turnpike Long time numerics

Lack of commutativity

lim

t!1 lim ε!0 6= lim ε!0 lim t!1

E. Zuazua

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Long time control: Numerics and turnpike Long time numerics

Conservative schemes

Let us consider now numerical approximation schemes for the inviscid problem (ε = 0): 8 > < > : un+1

j

= uj

n − ∆t

∆x ⇣ gn

j+1/2 − gn j−1/2

⌘ , j ∈ Z, n > 0. u0

j = 1 ∆x

R xj+1/2

xj−1/2 u0(x)dx,

j ∈ Z. The approximated solution u∆ is given by u∆(t, x) = un

j ,

xj−1/2 < x < xj+1/2, tn ≤ t < tn+1, where tn = n∆t and xj+1/2 = (j + 1

2)∆x.

The large time dynamics of these discrete systems depends on how much numerical viscosity they add to ensure stability!

E. Zuazua

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Long time control: Numerics and turnpike Long time numerics

Example: N-waves for the hyperbolic Burgers equation

E. Zuazua

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Long time control: Numerics and turnpike Long time numerics

Why?

For the Lax-Friedrichs scheme diffusion is linear, it is invariant with respect to time and one reproduces the diffusive dynamics as t → ∞: −(∆x)2 2∆t wxx. But, for the Engquist-Osher and Godunov schemes the viscosity is non-linear of the order, roughly, of −(w2)xx ∼ −2wwxx. Taking into account that we have the uniform a priori bound |w(x, t)| ≤ Ct−1/2 they amount of viscosity → 0 as t → ∞. Nonlinear numerical viscosity is better. It self-adapts to solutions’ behaviour. Need of choosing the right numerical scheme to reproduce the right dynamics.

E. Zuazua

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Long time control: Numerics and turnpike Turnpike

Outline

1 Motivation 2 Long time control: Numerics and turnpike

Motivation Long time numerics Turnpike

3 Constraints

Motivation

4 1-d constrained heat equation 5 Parameter depending control problems

Motivation Averaged control Greedy algorithms

6 Conclusion and open problems

E. Zuazua

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Long time control: Numerics and turnpike Turnpike

Origins

Although the idea goes back to John von Neumann in 1945, Lionel W. McKenzie traces the term to Robert Dorfman, Paul Samuelson, and Robert Solow’s ”Linear Programming and Economics Analysis” in 1958, referring to an American English word for a Highway: ... There is a fastest route between any two points; and if the

rigin and destination are close together and far from the

turnpike, the best route may not touch the turnpike. But if the

rigin and destination are far enough apart, it will always pay to

get on to the turnpike and cover distance at the best rate of travel, even if this means adding a little mileage at either end.

E. Zuazua

Turnpike, constraints & greedy Marrakech, Avril 2018 17 / 50

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Long time control: Numerics and turnpike Turnpike

E. Zuazua

Turnpike, constraints & greedy Marrakech, Avril 2018 18 / 50

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Long time control: Numerics and turnpike Turnpike

Examples where controls seem to fail the turnpike property

0.5 1 1.5 2 2.5 3 3.5 4 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 Discrete control computed on the non uniform mesh t 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1

t

Typical dynamics of controls for wave and heat like equations, as solutions

f the corresponding adjoint systems.
E. Zuazua

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Long time control: Numerics and turnpike Turnpike

The control problem

Let n ≥ 1 and T > 0, Ω be a simply connected, bounded domain of Rn with smooth boundary Γ, Q = (0, T) × Ω and Σ = (0, T) × Γ: 8 < : yt − ∆y = f 1ω in Q y = 0

n

Σ y(x, 0) = y0(x) in Ω. (1) 1ω = the characteristic function of ω of Ω where the control is active. We assume that y0 ∈ L2(Ω) and f ∈ L2(Q) so that (1) admits a unique solution y ∈ C

[0, T] ; L2(Ω)
∩ L2

0, T; H1

0(Ω)

.

y = y(x, t) = solution = state, f = f (x, t) = control

E. Zuazua

Turnpike, constraints & greedy Marrakech, Avril 2018 20 / 50

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Long time control: Numerics and turnpike Turnpike

We want to minimise the cost: J(f ) = 1 2 Z T Z

ω

f 2dxdt + 1 2 Z

Ω

|y(x, T) − yd|2dx (2) making a compromise between reaching the target ud and energy consumption f . The classical optimality system (Pontryaguin’s principle) guarantees that the control is of the form f = ϕ where ϕ is the solution of the adjoint equation: 8 < : −ϕt − ∆ϕ = 0 in Q ϕ = 0

n Σ

ϕ(T) = y(T) − yd in Ω. (3) Lack of turnpike?

E. Zuazua

Turnpike, constraints & greedy Marrakech, Avril 2018 21 / 50

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Long time control: Numerics and turnpike Turnpike

Better balanced controls

Let us now consider the control f minimising a compromise between the norm of the state and the control among the class of admissible controls: min 1 2 hZ T Z

Ω

|y|2dxdt + Z T Z

ω

|f |2dxdt + 1 2 Z

Ω

|y(x, T) − yd|2dx i . Then the Optimality System reads yt − ∆y = −ϕ1ω in Q y = 0 on Σ y(x, 0) = y0(x) in Ω −ϕt − ∆ϕ =y in Q ϕ = 0 on Σ. ϕ(T) = y(T) − yd in Ω. We now observe a coupling between ϕ and y on the adjoint state equation!

E. Zuazua

Turnpike, constraints & greedy Marrakech, Avril 2018 22 / 50

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Long time control: Numerics and turnpike Turnpike

New Optimality System Dynamics

What is the dynamic behaviour of solutions of the new fully coupled OS? For the sake of simplicity, assume ω = Ω. The dynamical system now reads yt − ∆y = −ϕ ϕt + ∆ϕ = −y This is a forward-backward parabolic system. A spectral decomposition exhibits the characteristic values µ±

j = ±

q 1 + λ2

j

where (λj)j≥1 are the (positive) eigenvalues of −∆. Thus, the system is the superposition of growing + diminishing real exponentials.

E. Zuazua

Turnpike, constraints & greedy Marrakech, Avril 2018 23 / 50

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Long time control: Numerics and turnpike Turnpike

The turnpike property for the heat equation

This new dynamic behaviour, combining exponentially stable and unstable branches, is compatible with the turnpike behavior. Controls and trajectories exhibit the expected dynamics:

E. Zuazua

Turnpike, constraints & greedy Marrakech, Avril 2018 24 / 50

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Long time control: Numerics and turnpike Turnpike

In this particular example turnpike means that optimal pairs (f (t), y(t)) are exponentially close to the steady-state optimal pair characterised as the minima for the functional Js(g) = 1 2 Z

ω

g2dxdt + 1 2 Z

Ω

|z(x) − zd|2dx (4) where z = z(x) solves ⇢ −∆z = g1ω in Ω z = 0

n

∂Ω, (5) Namely ||y(t) − z|| + ||f (t) − g|| ≤ C[exp(−µt) + exp(−µ(T − t))] if T >> 1.

E. Zuazua

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Long time control: Numerics and turnpike Turnpike

The turnpike path

E. Zuazua

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Long time control: Numerics and turnpike Turnpike

Conclusions

1 The turnpike property holds for a broad class of linear PDE models,

both conservative (waves) and dissipative (heat) provided:

1

The cost functional penalises both state and control;

2

The system under consideration is stabilizable.

2 Steady-state optimal pairs can be used to initialise the iterative

approximation of the time-evolution Optimality System.

3 Extensions to nonlinear PDEs are in its infancy. 4 Other more general notions can be introduced: periodic-turnpike.

E. Zuazua

Turnpike, constraints & greedy Marrakech, Avril 2018 27 / 50

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Long time control: Numerics and turnpike Turnpike

Simulations on the semilinear heat equation by S. Volkwein and S. Roog (Konstanz-Germany) So far analytical results allow us proving the turnpike property fort small targets/deformations but numerical simulations are much more robust and seem to indicate that the property also holds for large ones.

E. Zuazua

Turnpike, constraints & greedy Marrakech, Avril 2018 28 / 50

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Long time control: Numerics and turnpike Turnpike

Our references

A. Porretta & E. Z. , Long time versus steady state optimal control, SIAM
J. Control Optim., 51 (6) (2013), 4242-4273.
E. Tr´

elat & E. Z., The turnpike property in finite-dimensional nonlinear

ptimal control, JDE, 218 (2015) , 81-114.
M. Gugat, E. Tr´

elat, E. Zuazua, Optimal Neumann control for the 1D wave equation: Finite horizon, infinite horizon, boundary tracking terms and the turnpike property, Systems and Control Letters, 90 (2016), 61-70.

A. Porretta and E. Z., Remarks on long time versus steady state optimal

control, Springer INdAM Series “Mathematical Paradigms of Climate Science”, F. Ancona, P. Cannarsa, Ch. Jones, A. Portaluri, 15, 2016, 67-89.

E. Zuazua

Turnpike, constraints & greedy Marrakech, Avril 2018 29 / 50

SLIDE 30

Constraints

Outline

1 Motivation 2 Long time control: Numerics and turnpike

Motivation Long time numerics Turnpike

3 Constraints

Motivation 1-d constrained heat equation

4 Parameter depending control problems

Motivation Averaged control Greedy algorithms

5 Conclusion and open problems

E. Zuazua

Turnpike, constraints & greedy Marrakech, Avril 2018 31 / 49

SLIDE 32

Constraints Motivation

R. Escobedo, A. Iba˜

nez and E.Zuazua, Optimal strategies for driving a mobile agent in a “guidance by repulsion” model, Communications in Nonlinear Science and Numerical Simulation, 39 (2016), 58-72.

E. Zuazua

Turnpike, constraints & greedy Marrakech, Avril 2018 32 / 50

SLIDE 33

1-d heat equation

The constrained Dirichlet control problem

Consider the 1-d heat equation yt(t, x) = ∂2

xy(t, x)

t > 0, x ∈ (0, 1) y(t, 0) = u0(t), y(t, 1) = u1(t) (t > 0), with initial condition y0. The aim is to control this system to a constant steady state y1 > 0 y(T, x) = y1 (x ∈ [0, 1]) , with the control constraint u0(t) ≥ 0 , u1(t) ≥ 0 (t > 0).

E. Zuazua

Turnpike, constraints & greedy Marrakech, Avril 2018 36 / 50

SLIDE 34

Constraints 1-d heat equation

Outline

1 Motivation 2 Long time control: Numerics and turnpike

Motivation Long time numerics Turnpike

3 Constraints

Motivation 1-d constrained heat equation

4 Parameter depending control problems

Motivation Averaged control Greedy algorithms

5 Conclusion and open problems

E. Zuazua

Turnpike, constraints & greedy Marrakech, Avril 2018 34 / 49

SLIDE 35

Constraints Motivation

Optimal L2 controls do not understand natural constraints: Numerical simulations, back to Glowinski and Lions in the 80’s and 90’s

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1

t

0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 0.05 0.1 0.15 0.2 0.25

t

This is so since controls are just restrictions to ω of solutions of the adjoint system −ϕt − ∆ϕ = 0.

E. Zuazua

Turnpike, constraints & greedy Marrakech, Avril 2018 34 / 50

SLIDE 36

1-d heat equation

Existence of controls for long time horizons

Theorem If the target y1 is a positive steady state, there exists a time T large enough and positive controls u0, u1 ∈ H1(0, T) such that y(T, ·) = y1. Idea of the proof. The staircase principle. Move from the initial datum to the final target slowly, by small amplitude controls, avoiding large oscillations of solutions, to avoid that they violate the constraints. 3

3J. Loh´

eac, E. Tr´ elat and E. Zuazua, Minimal controllability time for the heat equation under unilateral state or control constraints, M3AS, 27 (2017), no. 9, 1587-1644.

E. Zuazua

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SLIDE 37

1-d heat equation

The staircase principle

E. Zuazua

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t

u

1 2 3 4 5 u1 u0

given path of controls control determined neighborhood of the path of controls of width 2σ

SLIDE 38

1-d heat equation

Minimal control time

The constrained Dirichlet control problem

Let T be the minimal to control the system from y0 to y1, under non-negativity constraints. Theorem Let y0 2 L2(0, 1), y1 2 R∗

+ with y0 6= y1. Then,

1 T := T

y0, y1

> 0,

2 there exist non-negative controls u0, u1 2 M(0, T) such that the

solution y with controls u0, u1 satisfies y(T, ·) = y1. The solution y, of the Dirichlet control problem with controls in the set of Radon measures, is defined by transposition. T

y0, y1

> 0 even if y0 < y1.

E. Zuazua

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SLIDE 39

1-d heat equation

Numerical experiments I

The constrained Dirichlet control problem

E. Zuazua

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SLIDE 40

1-d heat equation

Numerical experiments II

The constrained Dirichlet control problem

E. Zuazua

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SLIDE 41

Parameter depending control problems

Outline

1 Motivation 2 Long time control: Numerics and turnpike

Motivation Long time numerics Turnpike

3 Constraints

Motivation

4 1-d constrained heat equation 5 Parameter depending control problems

Motivation Averaged control Greedy algorithms

6 Conclusion and open problems

E. Zuazua

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SLIDE 43

Parameter depending control problems Motivation 1 Control of PDE theory lacks of unity. Often times rather different

analytical tools are required to tackle different models/problems.

2 Practical applications need of robust control theoretical results and

fast numerical solvers.

3 One of the key issues to be addressed in that direction is the control

f PDE models depending on parameters, that represent uncertain or

unknown quantities.

E. Zuazua

Turnpike, constraints & greedy Marrakech, Avril 2018 44 / 50

SLIDE 44

Parameter depending control problems Averaged control

Outline

1 Motivation 2 Long time control: Numerics and turnpike

Motivation Long time numerics Turnpike

3 Constraints

Motivation

4 1-d constrained heat equation 5 Parameter depending control problems

Motivation Averaged control Greedy algorithms

6 Conclusion and open problems

E. Zuazua

Turnpike, constraints & greedy Marrakech, Avril 2018 45 / 50

SLIDE 45

Parameter depending control problems Averaged control

Consider the transport equation with unknown velocity v, ft + vfx = 0, and take averages with respect to v. Then g(x, t) = Z f (x, t; v)ρ(v)dv then, for the Gaussian density ρ: ρ(v) = (4π)−1/2 exp(−v2/4) g(x, t) = h(x, t2); ht − hxx = 0. Dramatic change of type in the PDE under consideration. One can then employ parabolic techniques to control averages.

4 5

4E. Z., Automatica, 2014.
5Q. L¨

u, E. Z., Averaged controllability for Random Evolution Partial Differential Equations, J. Math. Pures Appl. 105 (2016) 367-414.

E. Zuazua

Turnpike, constraints & greedy Marrakech, Avril 2018 46 / 50

SLIDE 46

Parameter depending control problems Greedy algorithms

Outline

1 Motivation 2 Long time control: Numerics and turnpike

Motivation Long time numerics Turnpike

3 Constraints

Motivation

4 1-d constrained heat equation 5 Parameter depending control problems

Motivation Averaged control Greedy algorithms

6 Conclusion and open problems

E. Zuazua

Turnpike, constraints & greedy Marrakech, Avril 2018 47 / 50

SLIDE 47

Parameter depending control problems Greedy algorithms

But averaged control gives very little information on how efficiently each specific realisation of the system is actually controlled. Greedy algorithms have been developed in

M. Lazar & E. Z., Greedy controllability of finite dimensional linear

systems, Automatica, 74 (2016) 327-340. ensuring that the most relevant snapshots of the parameters are found to approximate optimally the manifolds (parameter-dependent one) of controls in an optimal way in the sense of Kolmogorov width. The subject is still mainly to be developed. So far the greedy choice depends on the initial data to be controlled

6

6M. Choulli E. Z., Lipschitz dependence of the coefficients on the resolvent and

greedy approximation for scalar elliptic problems, C. R. Acad. Sci. Paris, Ser I 354 (2016) 1174-1187.

E. Zuazua

Turnpike, constraints & greedy Marrakech, Avril 2018 48 / 50

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Conclusion

Conclusions and open problems

1 There are plenty of interesting analytical questions that are still open. 1

Are minimal time constrained controls of sparse nature?

2

Can one prove turnpike for nonlinear problems without smallness assumptions

3

Can greedy control be developed from a Control Systems perspective?

2 This emerges rather obviously and more importantly when facing

specific applications (gas transportation networks).

THANK YOU FOR YOUR ATTENTION!

E. Zuazua

Turnpike, constraints & greedy Marrakech, Avril 2018 50 / 50

Dynamic control Greedy, turnpike, constraints and some applications

Table of Contents

Table of Contents

Outline

Sonic boom

Outline

Geometric/Symplectic integration Numerical integration of the pendulum (A. Marica)a

Joint work with L. Ignat & A. Pozo, Math of Computation, 2014

Lack of commutativity

lim

t!1 lim ε!0 6= lim ε!0 lim t!1

Conservative schemes

Example: N-waves for the hyperbolic Burgers equation

Why?

Outline

Origins

Examples where controls seem to fail the turnpike property

The control problem

Better balanced controls

New Optimality System Dynamics

The turnpike property for the heat equation

The turnpike path

Conclusions

Our references

Table of Contents

Outline

nez and E.Zuazua, Optimal strategies for driving a mobile agent in a “guidance by repulsion” model, Communications in Nonlinear Science and Numerical Simulation, 39 (2016), 58-72.

The constrained Dirichlet control problem

Outline

Optimal L2 controls do not understand natural constraints: Numerical simulations, back to Glowinski and Lions in the 80’s and 90’s

Existence of controls for long time horizons

The staircase principle

Minimal control time

Numerical experiments I

Numerical experiments II

Table of Contents

Outline

Outline

Outline

Table of Contents

Conclusions and open problems