Dynamic control Greedy, turnpike, constraints and some applications Enrique Zuazua 1 DeustoTech-Bilbao & UAM-Madrid, Spain enrique.zuazua@deusto.es http://cmc.deusto.es Marrakech, Avril 2018 1 ERC Advanced Grant DYCON-Dynamic Control ( http://cmc.deusto.es/dycon/ ) and ANR Project ICON E. Zuazua Turnpike, constraints & greedy Marrakech, Avril 2018 1 / 50
Motivation Table of Contents 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 2 / 50
Motivation 1 In past decades control theory for ODE and especially for PDE has evolved significantly. 2 In parallel, numerical analysis and scientific computing have experienced a significant development. 3 Important applications have been successfully addressed. 4 We have learned that, often times the state of the art in Control + Numerics does not su ffi ce. 5 This is particularly the case when facing the real challenges and applications that often involve added di ffi culties related to * Constraints * Long time horizons * Parameter dependence 6 All of them require further significant analysis. 7 In this lecture we shall present some of the contributions of our team, pointing towards some challenging open problems. E. Zuazua Turnpike, constraints & greedy Marrakech, Avril 2018 3 / 50
Motivation We shall briefly discuss three topics: 1 Long time control: Numerics and turnpike 2 Constraints 3 Parameter-depending problems E. Zuazua Turnpike, constraints & greedy Marrakech, Avril 2018 4 / 50
Long time control: Numerics and turnpike Table of Contents 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 5 / 50
Long time control: Numerics and turnpike 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
Long time control: Numerics and turnpike Motivation Sonic boom Goal: the development of supersonic aircrafts, su ffi ciently 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
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
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
Long time control: Numerics and turnpike Long time numerics Geometric/Symplectic integration Numerical integration of the pendulum (A. Marica) a a HAIRER, E., LUBICH, Ch., WANNER, G.. Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Di ff erential Equations. 2nd ed. Berlin : Springer, 2006, 644 p. E. Zuazua Turnpike, constraints & greedy Marrakech, Avril 2018 10 / 50
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: u 2 ⇤ ⇥ u t + x = ε u xx , x ∈ R , t > 0 . Then 2 : If ε = 0, u ( · , t ) ∼ N ( · , t ) as t → ∞ ; If ε > 0, u ( · , t ) ∼ u M ( · , t ) as t → ∞ , u M is the constant sign self-similar solution of the viscous Burgers equation (defined by the mass M of u 0 ), while N is the so-called hyperbolic N-wave. 2 Y. J. Kim & A. E. Tzavaras, Di ff usive N-Waves and Metastability in the Burgers Equation , SIAM J. Math. Anal. 33 (3) (2001), 607–633. E. Zuazua Turnpike, constraints & greedy Marrakech, Avril 2018 11 / 50
Long time control: Numerics and turnpike Long time numerics Lack of commutativity t !1 lim lim ε ! 0 6 = lim ε ! 0 lim t !1 E. Zuazua Turnpike, constraints & greedy Marrakech, Avril 2018 12 / 50
Long time control: Numerics and turnpike Long time numerics Conservative schemes Let us consider now numerical approximation schemes for the inviscid problem ( ε = 0): n − ∆ t 8 ⇣ ⌘ u n +1 = u j g n j +1 / 2 − g n j ∈ Z , n > 0 . , > j − 1 / 2 j < ∆ x R x j +1 / 2 1 u 0 j = x j − 1 / 2 u 0 ( x ) dx , j ∈ Z . > : ∆ x The approximated solution u ∆ is given by u ∆ ( t , x ) = u n x j − 1 / 2 < x < x j +1 / 2 , t n ≤ t < t n +1 , j , where t n = n ∆ t and x j +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 Turnpike, constraints & greedy Marrakech, Avril 2018 13 / 50
Long time control: Numerics and turnpike Long time numerics Example: N-waves for the hyperbolic Burgers equation E. Zuazua Turnpike, constraints & greedy Marrakech, Avril 2018 14 / 50
Long time control: Numerics and turnpike Long time numerics Why? For the Lax-Friedrichs scheme di ff usion is linear, it is invariant with respect to time and one reproduces the di ff usive dynamics as t → ∞ : − ( ∆ x ) 2 2 ∆ t w xx . But, for the Engquist-Osher and Godunov schemes the viscosity is non-linear of the order, roughly, of − ( w 2 ) xx ∼ − 2 ww xx . 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 Turnpike, constraints & greedy Marrakech, Avril 2018 15 / 50
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 Turnpike, constraints & greedy Marrakech, Avril 2018 16 / 50
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 origin and destination are close together and far from the turnpike, the best route may not touch the turnpike. But if the origin 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
Long time control: Numerics and turnpike Turnpike E. Zuazua Turnpike, constraints & greedy Marrakech, Avril 2018 18 / 50
Long time control: Numerics and turnpike Turnpike Examples where controls seem to fail the turnpike property Discrete control computed on the non uniform mesh 1 0.3 0.2 0.8 0.1 0.6 0 − 0.1 0.4 − 0.2 0.2 − 0.3 − 0.4 0 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t t Typical dynamics of controls for wave and heat like equations, as solutions of the corresponding adjoint systems. E. Zuazua Turnpike, constraints & greedy Marrakech, Avril 2018 19 / 50
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