1/26 Bang bang control of elliptic and parabolic PDEs Michael Hinze (joint work with Nikolaus von Daniels & Klaus Deckelnick) Jackson, July 24, 2018
2/26 Model problem (P) min J(y, u) = 1 u U ad,y Y ad 2 y z 2 + α D 2 u 2 U subject to y = G(Bu). y = G(Bu) iff Ay = Bu in D, plus b.c. (plus i.c.) Examples Ay = d i,j=1 x j (a ij y xi ) + d i =1 b i y xi + cy uniformly elliptic operator, Y = H 1 (Ω) Ay = y t d i,j=1 x j (a ij y xi ) + d i =1 b i y xi + cy parabolic operator, Y = W (0, T ). Constraint sets: U ad = {a u b} and Y ad = {y c}, or { y δ}. Bang bang control: α = 0.
3/26 Existence and uniqueness of solutions, optimality conditions For every u we have a unique y(u) = G(Bu). So we may minimize the reduced functional J(u) = J(G(Bu), u) instead. Problem (P) admits a unique solution u with corresponding state y = G(Bu). J (u), v u U,U 0 for all v F ad = {v U ad, G(Bv) Y ad }.
4/26 Necessary optimality conditions (α > 0), Y ad = {y c} Constraint qualification (Slater condition) ũ U ad such that G(Bũ) < c in D Then there exist µ M( D) and some p such that with y and u there holds Ay = Bu A p = y z + µ u = P Uad ( 1 α RB p), µ 0, y c in D and D(c y)dµ = 0, Regularity elliptic case: p W 1,s (Ω) for all s < d/(d 1). parabolic case: p L s (W 1,σ ) for all s, σ [1, 2) with 2/s + d/σ > d + 1. Low regularity of adjoint states introduces difficulties in the numerical analysis
5/26 Only state constraints (α > 0), U ad U Optimality conditions: Ay = Bu A p = y z + µ u = 1 α RB p, µ 0, y c in D and y)dµ = 0, D(c The state y in general is smoother than the adjoint p. The optimal control and the adjoint state p are coupled via an algebraic relation a discretization of p induces a discretization of u. A discretization of y ideally should deliver feasible discrete states.
6/26 Only control constraints (α > 0), Y ad Y Optimality conditions: Ay = Bu A p = y z u = P Uad ( 1 α RB p), The adjoint p in general is smoother than the state y. The optimal control and the adjoint state p are coupled via an algebraic relation a discretization of p induces a discretization of u. Reduction to p, say delivers AA p BP Uad ( 1 α RB p) = Az, which in the parabolic case is a bvp for p in space-time.
7/26 Control constraints, α = 0 The function u U ad is a solution of the optimal control problem iff there exists an adjoint state p such that y = G(u), p = G(y z) and (αu + RB p, v u) 0 for all v U ad. Recall: u = P Uad ( 1 α RB p) for α > 0, i.e. u = a, αu + RB p > 0, 1 α RB p, αu + RB p = 0, b, αu + RB p < 0, If α = 0: u = a, RB p > 0, [a, b], RB p = 0, = b, RB p < 0. Control is determined through the sign of RB p. Control in the generic case is either at upper or lower bound. Use homothopy α 0 to compute bang bang control.
8/26 Tailored discretization Discrete counterpart to (P): (P h ) min J h (y h, u h ) s.t. PDE h (y h ) = B h (u h ) u h U h ad,y h Y h ad Questions: Appropriate choice of U h ad and Ansatz for u h? Appropriate choice of Y h ad and Ansatz for y h? Aim: Capture as much structure as possible of (P) on the discrete level.
9/26 Discretization a variational concept 1 Discrete optimal control problem: (P h ) min J h (u) = 1 u U ad 2 y h z 2 + α D 2 u 2 U subject to y h = G h (Bu) and y h I h c. Here, y h (u) = G h (Bu) denotes e.g. the This problem is still dimensional. p.l. and continuous fe approximation to y(u) (elliptic case), dg(0) in time and p.l. and continuous fe in space approximation to y(u) (parabolic case), i.e. a(y h, v h ) = Bu, v h for all v h X h. The control is not discretized explicitely! 1 M. Hinze: A variational discretization concept in control constrained optimization: the linear-quadratic case. J. Comput. Optim. Appl. 30, 45-63 (2005)
10/26 Discrete necessary optimality conditions (α > 0) Problem (P h ) admits a unique solution u h U ad. Furthermore, uniform convergence of discrete states implies discrete Slater condition G h (Bũ) < I h b in D for h small enough Then there exist µ h M( D) and some p h X h such that with y h and u h there holds a(y h, v h ) = Bu h, v h v h X h, a(w h, p h ) = D (y h z)w h + D w h dµ h w h Y h, u h = P Uad ( 1 α RB p h ) µ h 0, y h I h c, and D(I h c y h )dµ h = 0.
11/26 Discrete necessary optimality conditions (α > 0) Problem (P h ) admits a unique solution u h U ad. Furthermore, uniform convergence of discrete states implies discrete Slater condition G h (Bũ) < I h b in D for h small enough Then there exist µ h M( D) and some p h X h such that with y h and u h there holds a(y h, v h ) = Bu h, v h v h X h, a(w h, p h ) = D (y h z)w h + D w h dµ h w h Y h, u h = P Uad ( 1 α RB p h ) µ h 0, y h I h c, and D(I h c y h )dµ h = 0.
12/26 Discrete necessary optimality conditions (α > 0) Problem (P h ) admits a unique solution u h U ad. Furthermore, uniform convergence of discrete states implies discrete Slater condition G h (Bũ) < I h b in D for h small enough Then there exist µ h M( D) and some p h X h such that with y h and u h there holds a(y h, v h ) = Bu h, v h v h X h, a(w h, p h ) = D (y h z)w h + D w h dµ h w h Y h, u h = P Uad ( 1 α RB p h ) µ h 0, y h I h c, and D(I h c y h )dµ h = 0. Post processing 2 exploits the projection formula with a discrete adjoint state associated to a fully discrete control. 2 C. Meyer, A. Rösch. Superconvergence properties of optimal control problems. SIAM J. Control Optim. 43:970-985 (2004)
13/26 α = 0: optimality conditions for discrete problem The variational-discrete optimal control problems admits a solution u h U ad, which is unique in the case α > 0. The state y h is unique (also in the case α = 0). Let u h U ad be a solution of the optimal control problem. Then there exists a unique adjoint state p h such that y h = G h (u h ), p h = G h (y h z) and (αu h + RB p h, v u h ) 0 for all v U ad. Recall u h = P Uad ( 1 α RB p h ) for α > 0, i.e. u h = a, αu h + RB p h > 0, 1 α RB p h, αu h + RB p h = 0, b, αu h + RB p h < 0. If α = 0: u h = a, RB p h > 0, [a, b] RB p h = 0, = b, RB p h < 0.
Sketch of concept in 1d Bang bang control problems 14/26
15/26 Variational versus conventional discretization (α > 0) 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 1 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1
16/26 Variational discretization in the bang bang case α = 0 u(x) = sign(p(x)), p(x) = 1 128 sin(8πx 1)sin(8πx 2 )
17/26 Parabolic bang bang with α homothopy p(t, x) = T 2πa 2πat sin ( T ) sin 2πx 1 sin 2πx 2, a = 2, a 1 = 0.2, b 1 = 0.4 bounds. where B p(t) = sin 2πx 1 sin 2πx 2 p(t, x)dx. Ω u(t) = { a 1, B p > 0, b 1, B p < 0, Here, α = h 2 = k 4/3 with h = 2 l. Furthermore, with this coupling 3 u u k,h,α L 1 ch 2. 3 N. von Daniels, M. Hinze: Variational discretization of a control-constrained parabolic bang-bang optimal control problem. J. Comput. Math., to appear
18/26 Algorithms for P α h Define G h (u) = u P Uad ( 1 α p h(y h (u))). The optimality condition reads G h (u) = 0 and motivates the fix point iteration u given, do until convergence u + = P Uad ( 1 α p h(y h (u))), u = u +. 1. Is this algorithm numerically implementable? Yes, whenever for given u it is possible to numerically evaluate the expression P Uad ( 1 α p h(y h (u))) in the i th iteration, with an numerical overhead which is independent of the iteration counter of the algorithm.
19/26 Semi smooth Newton algorithm for α > 0 2. Does the fix point algorithm converge? Yes, if α > RB S h S hb L(U), since P Uad is non expansive. Condition too restrictive for our purpose semi smooth Newton method applied to G h (u) = 0: u given, solve until convergence G h (u)u+ = G h (u) + G h (u)u, u = u+. 1. This algorithm is implementable whenever the fix point iteration is, since G h (u) + G h (u)u = P U ad ( 1 α p h(u)) 1 α P U ad ( 1 α p h(u)) S h S hu. 2. For every α > 0 this algorithm is locally fast convergent 4. 4 M. Hinze, M. Vierling: Variational discretization and semi-smooth Newton methods; implementation, convergence and globalization in pde constrained optimzation with control constraints. Optim. Meth. Software 27:933-950 (2012)
20/26 Error analysis, sketch for elliptic case 5 It is well known that y y h + α u u h y y h (u) + p p h (y(u)) So one expects estimates for y y h also in the case α = 0. Estimates for u u h? Estimate for the states (S = {x Ω p(x) 0} Ω) y y h C (h 2 + (b a) p R h p L 1 (Ω S) + p R h p L u u h L 1 (S) ), p p h L C y y h + p R h p L, follow from 0 (p p h, u h u) = (R h p p h, u h u) + (p R h p, u h u) I + II. I 1 2 y y h 2 + 1 2 y R hy 2 II = Ω S (p R h p)(u h u) + S (p R h p)(u h u). 5 K. Deckelnick, M. Hinze: A note on the approximation of elliptic control problems with bang-bang controls. Comput. Optim. Appl. 51:931-939 (2012)
21/26 Error estimates If the adjoint state p in the solution with some β (0, 1] satisfies the structural assumption C > 0 ɛ > 0 L({x Ω; p(x) ɛ}) Cɛ β, one for h small enough gets a unique variational discrete control u h, which together with the discrete state y h and the discrete adjoint p h satisfies y y h + p p h L C 1 h2 2 β + p R h p L, u u h L 1 C β h2β 2 β + p R h p L.
22/26 Sketch of proof for β = 1 u u h L 1, y y h, p p h L C {h 2 + p R h p L } Sketch of proof: 0 (p p h, u h u) = (R h p p h, u h u) + (p R h p, u h u) I + II. I 1 2 y y h 2 + 1 2 y R hy 2 II = S (p R h p)(u h u). Combine now u u h L 1 (b a)l({p > 0, p h 0} {p < 0, p h 0}) {p > 0, p h 0} {p < 0, p h 0} { p(x) p p h } L({ p(x) p p h }) C p p h u u h L 1 C p p h p p h p R h p + R h p p h R h p p h C y y h to estimate II.
23/26 Numerical example with 2 switching points, fix-point iteration u(x) = sign( sin(mπx)), y(x) = sin(mπx), and p(x) = sin(mπx), m=2. 1 0.8 exact discrete FE grid 0.2 exact discrete FE grid 0.15 0.6 0.1 0.4 0.05 0.2 0 0 0.05 0.2 0.1 0.4 0.15 0.6 0.2 0.8 0.25 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.25 0.3 0.35 0.4 0.45 Experimental order of convergence: u u h L 1 : 3.00077834 Function values 1.99966106 p p h L : 1.99979367 y y h L : 1.9997965 p p h L 2 : 1.99945711
24/26 Special cases 1. If the desired state z is reachable with a feasible control, then y y h + p p h L Ch 2. 2. If p C 1 ( Ω) satisfies min p(x) > 0, where K = {x Ω p(x) = 0}. x K Then, the structural assumption is satisfied with β = 1. 3. If p W 2, (Ω) and satisfies the structural assumption, then y y h + p p h L + u u h L 1 Ch 2 log h γ(d).
25/26 Further reading M. Hinze: A variational discretization concept in control constrained optimization: the linear-quadratic case. J. Comput. Optim. Appl. 30:45-63 (2005). C. Meyer, A. Rösch. Superconvergence properties of optimal control problems. SIAM J. Control Optim. 43:970-985 (2004). K. Deckelnick, M. Hinze: A note on the approximation of elliptic control problems with bang-bang controls. Comput. Optim. Appl. 51:931-939 (2012). E. Casas, D. Wachsmuth, and G. Wachsmuth. Sufficient Second-Order Conditions for Bang-Bang Control Problems. SIAM J. Control Optim. 55:3066-3090 (2017). D. Wachsmuth and G. Wachsmuth. Regularization error estimates and discrepancy principle for optimal control problems with inequality constraints. Control and Cybernetics 40:1125-1158 (2011). N. von Daniels, M. Hinze: Variational discretization of a control-constrained parabolic bang-bang optimal control problem. J. Comput. Math., accepted for publication. N. von Daniels. Tikhonov regularization of control-constrained optimal control problems. Comput. Optim. Appl. 70:295-320 (2018). U. Felgenhauer. On Stability of Bang-Bang Type Controls. SIAM J. Control Optim. 41:1843-1867 (2003). M. Seydenschwanz. Convergence results for the discrete regularization of linear-quadratic control problems with bang-bang solutions. Comput. Optim. Appl. 61:731-760 (2015). D. Wachsmuth. Adaptive regularization and discretization of bang-bang optimal control problems. ETNA 40:249-267 (2013). D. Wachsmuth. Robust error estimates for regularization and discretization of bang-bang control problems. Comp. Opt. Appl. 62:271-289 (2014).
26/26 End Thank you very much for your attention!