Finite time horizon versus stationary control

Transcription

1 Finite time horizon versus stationary control Enrique Zuazua BCAM Basque Center for Applied Mathematics & Ikerbasque Bilbao, Basque Country, Spain WG, PDE-BCAM, Sept, 213 Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

2 Table of Contents Motivation 1 Motivation 2 Evolution versus steady state control 3 Long time numerical simulations 4 General conclusions Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

3 Motivation Motivation In various fields of Science, Engineering and Industry control and design issues play often a key role. And often times problems are formulated in long time intervals. This is for instance the case in celestial mechanics, climate, etc. And when that occurs several issues become of a key importance: Convergence of dynamic controls versus steady-state ones; Long-time numerical simulations. Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

4 Evolution versus steady state control Table of Contents 1 Motivation 2 Evolution versus steady state control 3 Long time numerical simulations 4 General conclusions Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

5 Motivation Evolution versus steady state control Engineering and physical processes do not obey a single modeling paradigm. Often times, both time-dependent and steady models are available and appropriate. This yields a number of different possibilities when facing optimal design, control and inverse problems. What model do we adopt? The time-dependent or the steady state one. Steady-state models are often understood as a simplification of the time evolution one, assuming (some times rigorously but most often without proof) that the time-dependent solution stabilizes around the steady-state as t. Main question Does the solution of the time-evolution control (or design or inverse problem) converge as t to the control of the steady state problem as well? Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

6 Evolution versus steady state control This issue is particularly important in aeronautical optimal design, 1 a mature but still rapidly evolving field where huge challenges arise and, in which, in particular, many problems related to design and control are still widely open. Often times people employ steady state models and the corresponding control ones while, from a mathematical point of view, the evolution problem is better understood. The reason for this is very simple: In the context of Nonlinear PDE steady state problems are hard to solve. In particular uniqueness is hard to prove. Accordingly sensitivity analysis is difficult as well. By the contrary, for evolution problems, under suitable assumptions on the nonlinearity, initial-boundary value problems are uniquely solvable, solutions depending smoothly on the data. 1 A. Jameson. Optimization Methods in Computational Fluid Dynamics (with Ou, K.), Encyclopedia of Aerospace Engineering, Edited by Richard Blockley and Wei Shyy, John Wiley Sons, Ltd., 21. Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

7 Evolution versus steady state control Turnpike theory Although the idea can be traced 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:... It is exactly like a turnpike paralleled by a network of minor roads. 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. Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

8 Evolution versus steady state control Mainly motivated by applications to economic models and game theory there is a literature concerned with this kind of stationary behavior in the transient time for long horizon control problems. In that context, such type of result goes under the name of turnpike property which was mostly investigated in the finite dimensional case. A. J. Zaslavski, Turnpike properties in the calculus of variations and optimal control. Nonconvex Optimization and its Applications, 8. Springer, New York, 26. Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

9 Evolution versus steady state control Time evolution control problem. Joint work with A. Porretta, SIAM J. Cont. Optim., to appear Consider the finite dimensional dynamics x t + Ax = Bu x() = x (1) where A M(N, N), B M(N, M), the control u L 2 (, T ; R M ), and x R N. Given a matrix C M(N, N), and some x R N, consider the optimal control problem min u J T (u) = 1 2 T ( u(t) 2 + C(x(t) x ) 2 )dt. There exists a unique optimal control u(t) inl 2 (, T ; R M ), characterized by the optimality condition u = B p t + A p = C C(x x ) p, (2) p(t )= Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

10 PDE Evolution versus steady state control Similar problems can be formulated in the context of PDE (heat equations, elasticity, fluids,...). In that case the equation evolves in Ω while the control is localized in ω Ω. Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

11 Evolution versus steady state control The steady state control problem The same problem can be formulated for the steady-state model Ax = Bu. Then there exists a unique minimum ū, and a unique optimal state x, of the stationary control problem min u J s (u) = 1 2 ( u 2 + C(x x ) 2 ), Ax = Bu, (3) which is nothing but a constrained minimization in R N ;andbyelementary calculus, the optimal control ū and state x satisfy A x = Bū, ū = B p, and A p = C C( x x ). Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

12 We assume that Evolution versus steady state control The pair (A, B) is controllable, (4) or, equivalently, that the matrices A, B satisfy the Kalman rank condition Rank BABA 2 B... A N 1 B = N. (5) Then there exists a linear stabilizing feedback law L M(M, N) andc, µ>suchthat x t + Ax = B(Lx) x() = x = x(t) ce µt x t >. (6) Concerning the cost functional, we assume that the matrix C is such that The pair (A, C) is observable (7) which means that the following algebraic condition holds: Rank CCACA 2... CA N 1 = N. (8) Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

13 Evolution versus steady state control Under the above controllability and observability assumptions, we have the following result. Theorem Assume that (5) and (8) hold true. Then we have 1 T min J T u L 2 (,T ) T min u R N J s and 1 T u(t) ū 2 + C(x(t) x) 2 dt T where ū is the optimal control of J s and x the corresponding optimal state. In particular, we have 1 bt x(t) dt x, (b a)t at 1 bt u(t) dt x (b a)t at for every a, b [, 1]. Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

14 Proof. Evolution versus steady state control We use the optimality conditions defining the adjoint states p and p, which give (x x) t + A(x x) =B(u ū) u ū = B (p p) (p p) t + A (p p) =C C(x x) From the optimality system we get [(x x)(p p)] t = B(u ū)(p p) C(x x) 2 which implies T u ū 2 + C(x x) 2 dt =[(x x)(p() p)] + [(x(t ) x) p]. Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

15 Evolution versus steady state control If (A, B) is controllable, then there exists c, independent of T,suchthat, for every f L 2 (, T ; R N ), q T R N, the solution of q t + A q = f q(t )=q T (9) satisfies, with µ is as above, T q() 2 c B q 2 dt + T f 2 dt + e 2µT q T 2. (1) Indeed, multiplying the adjoint equation by x of (6), we get q() x = q T x(t ) T q(x t + Ax) dt + T f xdt = q T x(t ) T B q Lx dt + T f xdt, and using the exponential decay of x(t) we obtain T T 1 q() x C x B q 2 dt + f 2 2 dt + C x e µ T q T which suffices with x = q(). Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

16 Evolution versus steady state control Using the observability inequality (1) we have T 1 (p() p) c C(x x) 2 2 T 1 dt + B (p p) 2 2 dt + p. (11) Similarly, in the equation of x x we use the observability inequality for (A, C) which is ensured by (8): T x(t ) x c T 1 u ū 2 dt + C(x(t) x) 2 dt + x x 2 2. (12) Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

17 Evolution versus steady state control Hence T u ū 2 + C(x x) 2 dt c (13) by the previous estimates (11) and (12). We conclude 1 T u ū 2 + C(x x) 2 dt C T T. This of course also implies the convergence of the averaged minimum level to the stationary minimum. Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

18 Evolution versus steady state control Rate of convergence Following [CLLP] 2,ifB and C are coercive we also have u ū 2 + C(x x) 2 = B (p p) 2 + C(x x) 2 γ p p 2 + x x 2 hence we deduce from the optimality system [(x x)(p p)] t = B (p p) 2 C(x x) 2 γ (x x)(p p), for some γ>. Since (x x)(p p) is bounded at t =andt = T due to (11), (12) and (13), we obtain e γ(t t) K [(x x)(p p)](t) Ke γt for some K >. Integrating we get bt u 2 x 2 ū + x ds K e γat + e γ(1 b)t at which implies an exponential rate of convergence. 2 P. Cardaliaguet, J.-M. Lasry, P.-L. Lions, A. Porretta, Long time average of Mean Field Games, NetworkHeterogeneousMedia,7(2),212 Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

19 Comments Evolution versus steady state control The same results hold for linear PDE and in particular heat and wave equations. Note that the problem in the time interval [, T ]ast can be rescaled into the fixed time interval [, 1] by the change of variables t = Ts: εx s + Ax = Bu, s [, 1]. In the limit as ε the steady-state equation emerges: Ax = Bu. This becomes a classical singular perturbation control problem. Note however that, in this setting, the role that the controllability and observability properties of the system play is much less clear than when dealing with it as T. The problem in widely open for nonlinear models. Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

20 Long time numerical simulations Table of Contents 1 Motivation 2 Evolution versus steady state control 3 Long time numerical simulations 4 General conclusions Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

21 Long time numerical simulations Geometric integration Numerical integration of the pendulum Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

22 Long time numerical simulations Climate modelling Climate modeling is a grand challenge computational problem, a research topic at the frontier of computational science. Simplified models for geophysical flows have been developed aim to: capture the important geophysical structures, while keeping the computational cost at a minimum. Although successful in numerical weather prediction, these models have a prohibitively high computational cost in climate modeling. Xu Wang, wangzhu/ Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

23 Sonic boom Long time numerical simulations Goal: the development of supersonic aircraft that are sufficiently quiet so that they can be allowed to fly supersonically over land. The pressure signature created by the aircraft must be such that, when it reaches the ground, (a) it can barely be perceived by the human ear, and (b) it results in disturbances to man-made structures that do not exceed the threshold of annoyance for a significant percentage of the population. Juan J. Alonso and Michael R. Colonno, Multidisciplinary Optimization with Applications to Sonic-Boom Minimization, Annu. Rev. Fluid Mech. 212, 44: Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

24 Long time numerical simulations Joint work with L. Ignat & A. Pozo Consider the 1-D conservation law with or without viscosity: Then: u t + u 2 x = εu xx, x R, t >. If ε =, u(, t) N(, t) ast ; If ε>, u(, t) u M (, t) ast, u M is the constant sign self-similar solution of the viscous Burgers equation (defined by the mass M of u ), while N is the so-called hyperbolic N-wave. In both cases: u(x, t) t 1/2 F (x/ t), t. Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

25 Long time numerical simulations Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

26 Long time numerical simulations Conservative schemes for the inviscid equation Let us consider now numerical approximation schemes u n+1 j = un j t gj+1/2 n x g j 1/2 n, j Z, n >. uj = 1 xj+1/2 x x u j 1/2 (x)dx, j Z, The approximated solution u is given by u (t, x) =u n j, x j 1/2 < x < x j+1/2, t n t < t n+1, where t n = n t and x j+1/2 =(j ) x. Is the large tine dynamics of these discrete systems, a discrete version of the continuous one? Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

27 Long time numerical simulations 3-point conservative schemes 1 Lax-Friedrichs g LF (u, v) = u2 + v 2 4 x t v u 2, 2 Engquist-Osher 3 Godunov g EO (u, v) = u(u + u ) 4 min g G (u, v) = max w [u,v] w [v,u] + v(v v ), 4 w 2 2, if u v, w 2 2, if v u. Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

28 Long time numerical simulations Numerical viscosity We can rewrite three-point monotone schemes in the form u n+1 j u n j t + (un j+1 )2 (u n j 1 )2 4 x = R(u n j, u n j+1) R(u n j 1, u n j ) where the numerical viscosity R can be defined in a unique manner as For instance: R(u, v) = Q(u, v)(v u) 2 = λ 2 R LF (u, v) = v u, 2 R EO (u, v) = λ 4 (v v u u ), u v 2 2 2g(u, v). λ R G 4 sign( u v )(v 2 u 2 ), v u, (u, v) = λ(v v u u ), elsewhere. 4 Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

29 Properties Long time numerical simulations These three schemes are well-known to satisfy the following properties: They converge to the entropy solution They are monotonic They preserve the total mass of solutions They are OSLC consistent: u n j 1 un j+1 2 x L 1 L decay with a rate O(t 1/2 ) 2 n t Similarly they verify uniform BV loc estimates But do they capture correctly the asymptotic behavior of solutions as t? Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

30 Long time numerical simulations Main result: Viscous effective behavior Theorem (Lax-Friedrichs scheme) Consider u L 1 (R) and x and t such that λu n, 1, λ = t/ x. Then,foranyp [1, ), the numerical solution u given by the Lax-Friedrichs scheme satisfies lim t 1 2 (1 1 ) p u (t) w(t) =, t L p (R) where the profile w = w M is the unique solution of w t + w 2 2 x = ( x)2 2 w xx, x R, t >, w() = M δ, with M = R u. Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

31 Why? Long time numerical simulations u n+1 j u n j t + (un j+1 )2 (u n j 1 )2 4 x = R LF (u n j, u n j+1) R LF (u n j 1, u n j ) with Thus R LF (u, v) = v u. 2 u n+1 j u n j t + (un j+1 )2 (u n j 1 )2 4 x 1 u n 2 j+1 + uj 1 n 2uj n ( x) 2 u xx. 2 Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

32 Long time numerical simulations Main result: Inviscid effective behavior Theorem (Engquist-Osher and Godunov schemes) Consider u L 1 (R) and x and t such that λu n, 1, λ = t/ x. Then,foranyp [1, ), the numerical solutions u given by Engquist-Osher and Godunov schemes satisfy the same asymptotic behavior but for the hyperbolic N wave w = w p,q unique solution of w t + w 2 2 =, x R, t >, x x, x <, w() = M δ, lim w(t, z)dz = p, x =, t q p, x >, with M = R u and x p = min x R u (z)dz and q =max x R x u (z)dz. Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

33 Proof Long time numerical simulations Scaling transformation: u λ (x, t) =λu(λx,λ 2 t) The asymptotic behavior of u(x, t) ast is reduced to the analysis of the behavior of the rescaled family u λ as λ but in the finite time horizon < t < 1. Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

34 Example Long time numerical simulations Let us consider the inviscid Burgers equation with initial data.5, x [ 1, ], u (x) =.15, x [, 2],, elsewhere. The parameters that describe the asymptotic N-wave profile are: M =.25, p =.5 and q =.3. We take x =.1 asthemeshsizefortheinterval[ 35, 8] and t =.5. Solution to the Burgers equation at t = 1 5 : Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

35 Long time numerical simulations EngquistïOsher (t=) Godunov (t=) Nïwave Solution.8 LaxïFriedrichs (t=) Nïwave Solution ï.2 ï.2 ï.2 ï1 1 Evolution of masses ï1 1 Evolution of masses Total mass + mass ï mass.2.1 ï1 1 Evolution of masses.3 Total mass + mass ï mass Enrique Zuazua (BCAM) Total mass + mass ï mass Nïwave Solution Finite time horizon versus stationary control WG, Sept / 4

36 Long time numerical simulations Similarity variables Let us consider the change of variables given by: s =ln(t + 1), ξ = x/ t +1, w(ξ,s) = t +1u(x, t), which turns the continuous Burgers equation into 1 w s + 2 w ξw =, ξ R, s >. The asymptotic profile of the N-wave becomes a steady-state solution: ξ, 2p <ξ< 2q, N p,q (ξ) =, elsewhere, ξ Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

37 Examples Long time numerical simulations Convergence of the numerical solution using Engquist-Osher scheme (circle dots) to the asymptotic N-wave (solid line). We take ξ =.1 and s =.5. Snapshots at s =, s =2.15, s =3.91, s =6.55, s = 2 and s = 1. Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

38 Examples Long time numerical simulations Numerical solution using the Lax-Friedrichs scheme (circle dots), taking ξ =.1 and s =.5. The N-wave (solid line) is not reached, as it converges to the diffusion wave. Snapshots at s =, s =2.15, s =3.91, s =6.55, s = 2 and s = 1. Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

39 Conclusions Long time numerical simulations Within the class of convergent numerical schemes we have shown the need of discriminating those that are asymptotically correct. We have shown the significant reduction on the computational cost when using the intrinsic similarity variables. Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

40 Table of Contents General conclusions 1 Motivation 2 Evolution versus steady state control 3 Long time numerical simulations 4 General conclusions Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4

41 Lots to be done on: General conclusions The analysis of how time-evolution control and steady-state one are related for nonlinear problems. Similar issues in the context of shape design and inverse problems. Development of numerical algorithms preserving large time asymptotics for nonlinear PDEs (other works of our team on dispersive equations, dissipative wave equations,...) The two issues overlap when trying to effectively compare numerical approximations Enrique Zuazua (BCAM) Finite time horizon versus stationary control WG, Sept / 4