Numerical Optimal Control of the Wave Equation - Ingenieurmathematik
Proceedings 5th MATHMOD Vienna, February 2006
(I.Troch, F.Breitenecker, eds.)
Numerical Optimal Control of the Wave Equation: Optimal Boundary Control of a String to Rest in Finite Time M. Gerdts, Universit¨ at Hamburg, G. Greif, Technische Universit¨ at Darmstadt, H. J. Pesch, Universit¨ at Bayreuth, Germany Corresponding Author: H. J. Pesch Universit¨ at Bayreuth, Lehrstuhl f¨ ur Ingenieurmathematik Universit¨ atsstr. 30, 95440 Bayreuth, Germany e-mail: [email protected]
Abstract. In many real-life applications of optimal control problems with constraints in form of partial differential equations (PDEs), hyperbolic equations are involved which typically describe transport processes. Because of their nature being able to transport discontinuities of initial or boundary conditions into the domain on which the solution lives or even to develop discontinuities in the presence of smooth data, these problems constitute a severe challenge for both theory and numerics of PDE constrained optimization. In the present paper, optimal control problems for the well-known wave equation are investigated. The intention is to study the order of the numerical approximations for both the optimal state and the optimal control variables while analytical solutions are known here. The numerical method chosen here is a full discretization method based on appropriate finite differences by which the PDE constrained optimal control problem is transformed into a nonlinear programming problem (NLP). Hence we follow here the approach first discretize, then optimize, which allows us to make use not only of powerful methods for the solution of NLPs, but also to compute sensitivity differentials, a necessary tool for real-time control. Keywords. Optimal control of partial differential equations; optimal control of hyperbolic equations; optimal control of the wave equation; first discretize, then optimize. 1. Introduction Owing to its importance for engineering applications, the field of PDE constrained optimization has become increasingly popular. In the near future, mathematical optimization methods will be able to solve problems whose complexity has so far allowed only the application of simulation-based methods. Hence, there is a strong need for new efficient methods for PDE constrained optimization which are capable of tackling real-life engineering applications, including methods for real-time control. Those problems may be nonlinear and consist of coupled systems of high dimension and different type with complicated right hand sides and possibly non-standard initial and boundary conditions; see, e. g., the highly complex nonlinear PDE constrained optimal control problem describing the dynamical behaviour of certain fuel cell systems [5], [6], and [21]. Without doubt Lion’s book [20] is still the standard for optimal control problems with linear equations and convex functionals. Nonconvex problems with semilinear equations of elliptic and parabolic type are in the focus of Tr¨ oltzsch’s new book [23], particularly addressing questions of existence of solutions and optimal controls, the derivation of necessary conditions and adjoint equations as well as of second order sufficient conditions. While the theory of optimal control for elliptic and parabolic equations is well developed in the semilinear case, although the associated optimal control problems generally are nonconvex, hyperbolic equations are not as well understood despite the fact that many dynamical processes are, at least partly, of hyperbolic nature; see, e. g., the aforementioned references on fuel cell control. Linear hyperbolic equations are treated, besides in Lion’s book, in Ahmed and Teo [1]. An introduction into the control of vibrations can be found in Krabs [15]. Oscillating elastic networks are investigated by Lagnese, Leugering et al. [16]–[17], [19] and Gugat [9]. However, semilinear hyperbolic problems are still not well understood because of the weaker smoothness of the solution operators. The most recent progress is due to S. Ulbrich [24], [25]. In this paper we will investigate a well understood optimal control problem concerned with the classical wave equation. The present paper is particularly based on a series of papers by Gugat [8], [10]–[13].
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Proceedings 5th MATHMOD Vienna, February 2006
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Gugat has presented analytical solutions for various optimal boundary control problems associated with the wave equation as one of the constraints. In contrast, we are going to solve these problems numerically in order to show the order of approximation for their optimal solutions. For, we want to propagate a numerical approach for the numerical solution of PDE constrained optimal control problems if hyperbolic equations are involved. The method of choice proposed here is either a full discretization method, in case of small size problems, or the vertical method of lines, in case of medium size problems. For large size problems only model reduction methods may today give a chance for their solution such as proper orthogonal decomposition; see, e. g., Hinze, Volkwein [14]. Concerning small and medium size problems, appropriate difference methods for the spatial discretization, resp. semidiscretization must be applied in order to approximate the hyperbolic equation correctly. Clearly, both approaches belong to the class first discretize, then optimize. They possess the advantage that the full power of methods for NLPs and ODE constrained optimal control problems can be used including their possibilities for a numerical sensitivity analysis and therefore real-time control purposes; see, e. g., B¨ uskens [2]–[4]. The latter is a must if optimal solutions are to be applied in practise. No doubt, that the approach first optimize, then discretize generally may give more theoretical inside and safety, but as in ODE constrained optimization its advantage does not pay for its drawbacks when complicated real-life applications are to be treated. 2. Optimal Control Problems 2.1 The continuous optimal control problem Let L > 0, T > 0, and a > 0 be real numbers, and let u0 ∈ L1 [0; L], where L1 [0; L] denotesR the set of all x Lebesgue integrable functions on the interval [0; L]. Furthermore, let the mapping x → 0 u1 (s) ds be 1 also in L [0; L] for a function u1 . We define ||(v, w)||1, [0;T ] :=
Z
T
(|v(t)| + |w(t)|) dt . 0
Then we can state the following optimal control problem: ! ||(yL , yR )||1, [0;T ] = min
(1)
subject to the constraints yL , yR ∈ L1 [0; T ] ,
(2)
utt (x, t) = a2 uxx (x, t) , (x, t) ∈ (0; L) × (0; T ) ,
(3)
u(x, 0) = u0 (x) , ut (x, 0) = u1 (x) , x ∈ [0; L] ,
(4)
u(0, t) = yL (t) , u(L, t) = yR (t) , t ∈ [0; T ] ,
(5)
u(x, T ) = 0 , x ∈ [0; L] and ut (x, T ) = 0 , x ∈ (0; L) .
(6)
We require that all consistency conditions are fulfilled, i. e., u0 (0) = yL (0), u0 (L) = yR (0), u(0, T ) = yL (T ) = 0, and u(L, T ) = yR (T ) = 0. Hence we have to control a vibrating string with the state variable u(x, t) satisfying the wave equation (3) from an arbitrary initial state (4) to rest in final time (6) by means of the boundary conditions (5) via the control functions (2) such that the functional (1) is minimized. In Gugat [11], Theorem 2, an analytical solution for this optimal control problem is given making this problem particularly valuable for numerical tests. However, due to the lack of smoothness of the functional (1), which would constitute an artificial difficulty, we change to a smooth version of problem (1)–(6) by minimizing Z T ! (7) ||(yL , yR )||22, [0;T ] := |yL (t)|2 + |yR (t)|2 dt = min . 0
Note that the change of the functional does not change the optimal solution with respect to the state and control variables. Solely the optimal value of the functional is different. Henceforth, we consider the optimal control problem (7) subject to (2)–(6).
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Proceedings 5th MATHMOD Vienna, February 2006
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2.2 The discrete optimal control problem Introducing the time step size k > 0 and the spatial step size h > 0 so that q := T /k and m := L/h are natural numbers, the discrete version of (7) subject to (2)–(6) is given as follows q X j=1
! |yL (tj )|2 + |yR (tj )|2 = min
(8)
subject to the constraints Qh,k uji (YL;k , YR;k ) = 0 ,
(9)
u(xi , 0) = u0 (xi ) , ut (xi , 0) = u1 (xi ) , i = 0, . . . , m ,
(10)
j j u(x0 , tj ) = yL , u(xm , tj ) = yR , j = 0, . . . , q ,
(11)
u(xi , tq ) = 0 , i = 0, . . . , m ,
and
u(xi , tq ) − u(xi , tq−1 ) = 0 , i = 1, . . . , m − 1 , k
(12)
j j q > 0 where xi := i h, tj := j k, uji = u(xi , tj ), yL := yL (tj ), yR := yR (tj ), YL;k := yL , . . . , yL , and q > 0 YR;k := yR , . . . , yR . The difference operator Qh,k represents an appropriate discretization scheme for the wave equation specified later.
3. Numerical Results The following numerical tests are performed for four examples: Example 1: Let be L = 1, T = 3.25, a = 1, u0 (x) = 0, u1 (x) = sin(π x). Example 2: Let be L = π, T = 2 π, a = 1, u0 (x) = sin x, u1 (x) = cos x. Example 3: Let be L = π, T = 2 π, a = 2, and 4 − x for 0 ≤ x ≤ 41 π , π 1 3 4 u0 (x) = π x − 2 for 4 π ≤x ≤ 4 π, 4 − π x + 4 for 43 π ≤ x ≤ 43 π ,
Example 4: Let be L = 2, T = 6, a = 1, and ( x for 0 ≤ x ≤ 1 , u0 (x) = x − 2 for 1 ≤ x ≤ 2 ,
and u1 (x) = 0 ,
and u1 (x) = 0 ,
x ∈ [0; π] .
x ∈ [0; 2] .
All computations were carried through by means of Matlab. For the optimization the Matlabroutine fmincon, an SQP method has been used. When applying numerical difference schemes to the wave equation we have two possibilities. The first class of difference schemes is based on an equivalent formulation of the Cauchy problem associated with the wave equation, i. e., utt (x, t) = a2 uxx (x, t) , (x, t) ∈ (0; L) × R+ ,
(13)
u(x, 0) = u0 (x) , ut (x, 0) = u1 (x) , x ∈ [0; L] ,
(14)
ut (x, t) + a vx (x, t) = 0 ,
(15)
vt (x, t) + a ux (x, t) = 0 , (x, t) ∈ (0; L) × R+ ,
(16)
u(x, 0) = u0 (x) , Z 1 x v(x, 0) = − u1 (s) ds + C , x ∈ [0; L] . a 0
(17)
as follows,
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The constant C is arbitrary and can be set to zero since the component v is determined up to a constant only. The second class tackles the wave equation (13) directly. The following well-known schemes are candidates for an application to the system (15), (16), if certain restrictions are obeyed, see, e. g., Strikwerda [22]: The FTFS scheme (’forward-time forward-space’), also known as upwind scheme: j
j
u − ui uj+1 − uji i + a i+1 = 0; k h
(19)
The FTBS scheme (’forward-time backward-space’), also known as upwind scheme: uj − uji−1 uj+1 − uji i +a i = 0; k h
(20)
The Lax-Friedrich scheme: uj+1 − i
1 2
(uji+1 + uji−1 ) uj − uji−1 + a i+1 = 0; k 2h
(21)
The Lax-Wendroff scheme: uj+1 = uji − i
a2 k 2 ak j j j j ui+1 − uji−1 + u − 2 u + u . i+1 i i−1 2h 2 h2
(22)
In contrast, the following scheme directly applies to the wave equation (13). Approximating both second derivatives by central difference quotients of second order one obtains j j j j−1 2 2 uj+1 = 2 (1 − α ) u + α u + u for j ≥ 1 , (23) i i i+1 i−1 − ui u1i = (1 − α2 ) u0i +
1 2 0 α ui+1 + u0i−1 + k u1 (xi ) 2
(24)
with α := a hk . Equation (24) serves as the second starting line on t = k for the multistep method (23) besides the initial conditions (14) on t = 0. All of the above schemes can be rewritten so that uj+1 is a linear combination of values of u at time i instances j and j − 1, for example the formula uj+1 = (1 + a p) uji − a p uji+1 =: Ph,k uji i
with
p :=
k h
represents the FTFS scheme. Hereby Ph,k denotes the difference operator for the FTFS scheme, resp. Qh,k uji := uj+1 − Ph,k uji = 0. Analogously we can rewrite all schemes except (23), (24) as one step i methods, i. e., uj+1 = Ph,k uji . i In order to be convergent, these schemes must be consistent and stable according to the well-known Lax Equivalence Theorem. More precisely, a consistent finite difference scheme for a PDE, for which the Cauchy problem is wellposed, is convergent (e. g. w. r. t. the L2 -norm) if and only if it is stable; see for example [22]. In order to prove that, it is sufficient to show convergence solely for the simple scalar transport equation ut (x, t) + a ux (x, t) = 0 , since the system (15), (16) can be decoupled by a similarity transformation yielding a hyperbolic system with eigenvalues λ1,2 = ±a; see, e. g., Larsson, Thome [18] It is an easy exercise to show then that the upwind schemes are both of consistency order O(k + h) =: (1, 1). However, the FTFS scheme is stable only if a < 0 and −1 ≤ a p ≤ 0, whereas the FTBS scheme is stable only if a > 0 and 0 ≤ a p ≤ 1. This explains the names: The spatial difference quotients have to be chosen against the wind, i. e., against the direction of the information flow.
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The situation for the Lax-Friedrich scheme is more complicated. It is only consistent of order (1, 1) if k −1 h2 → 0, which can be fulfilled if k = Λ(h) with a sufficiently smooth function Λ with Λ(0) = 0. The scheme is stable if |a p| ≤ 1, and hence convergent then. The Lax-Wendroff scheme is consistent of order O(k 2 + h2 ) =: (2, 2) and stable if also |a p| ≤ 1, hence convergent. In case of general hyperbolic systems the CFL condition |a p| ≤ 1 has to be replaced by |λ p| ≤ 1 for all eigenvalues λ.
3.1 Results for Example 1 The following two figures show the results for Example 1 obtained by means of the multistep scheme (23), (24). Exact and approximated optimal boundary controls coincide perfectly; see Fig. 1. The associated state is given in Fig. 2.
exact exact Figure 1: Example 1: Exact and approximate optimal boundary controls yL (t) (left) and yR (t) app app (right) resp. yL (t) (left) and yR (t) (right) using the multistep scheme (23), (24) with h = k = 0.00625
Figure 2: Example 1: Approximate optimal state variable u(x, t) on the time interval [0; 3.25] using the multistep scheme (23), (24) with h = k = 0.0125
3.2 Results for Example 2 Figures 3 and 4 show the results for Example 2 obtained by means of the Lax-Wendroff scheme (22). Again the coincidence between numerical approximation and exact optimal solution is convincing. Solely at t = π, where the first derivative of the optimal boundary control is discontinuous, a slight oszillation arises.
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exact exact Figure 3: Example 2: Exact and approximate optimal boundary controls yL (t) (left) and yR (t) app app (right) resp. yL (t) (left) and yR (t) (right) using the Lax-Wendroff scheme (22) with h = k = π/140
Figure 4: Example 2: Approximate optimal state variable u(x, t) on the time interval [0; 2 π] using the Lax-Wendroff scheme (22) with h = k = π/140 3.3 Results for Example 3 Figures 5 and 6 show the results for Example 3 obtained by means of the Lax-Friedrich scheme (21). Again the coincidence between numerical approximation and exact optimal solution is perfect. However, note that these results can only obtained if a p = 1 which is the maximum value of the stability interval. The same result holds when using the multistep scheme (23), (24). For a p < 1 oscillations occur for both methods.
exact exact Figure 5: Example 3: Exact and approximate optimal boundary controls yL (t) (left) and yR (t) app app (right) resp. yL (t) (left) and yR (t) (right) using the Lax-Friedrich scheme (21) with h = π/101 and k = π/202
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Figure 6: Example 3: Approximate optimal state variable u(x, t) on the time interval [0; π] using the Lax-Friedrich scheme (21) with h = π/101 and k = π/202
3.4 Results for Example 4 For Example 4, Figs. 7 and 8 show the superiority of the second order multistep scheme (23), (24) compared to the simple first order upwind scheme (20) in the resolution of the discontinuities despite the larger step size of the multistep method. The associated approximate state variables are given in Fig. 9.
exact exact Figure 7: Example 4: Exact and approximate optimal boundary controls yL (t) (left) and yR (t) app app (right) resp. yL (t) (left) and yR (t) (right) using the upwind scheme (20) with h = k = 0.01
exact exact Figure 8: Example 4: Exact and approximate optimal boundary controls yL (t) (left) and yR (t) app app (right) resp. yL (t) (left) and yR (t) (right) using the multistep scheme (23), (24) with h = k = 0.05
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Figure 9: Example 4: Approximate optimal state variable u(x, t) on the time interval [0; 6] using the upwind scheme (20) (left) resp. the multistep scheme (23), (24) (right) both with h = k = 0.05
3.5 Convergence Analysis for Example 2 For the weighted step size ratio ap = 1 we were able to produce very precise results for all optimal control examples investigated. The following Tables 1 and 2 list the discrete L 2 - norm errors for the approximated optimal control variables and the numerically achieved convergence orders for all difference schemes investigated for Example 2. Similar results were obtained for Example 1 and 4. Note that the results for the first order methods (20), (21), and (22) coincide up to the first four digits so that they together are given in Table 2. The results for Example 3 are given in Tables 3 and 4 below. As we can see, the convergence order drops down to 0.5 for both the first and the second order methods which is due to the nonsmooth control variables.
h 0.3142 0.1571 0.0785 0.0393 0.0196 0.0098
k 0.3142 0.1571 0.0785 0.0393 0.0196 0.0098
N 11 21 41 81 161 321
error k.k2 0.1500 0.1051 0.0526 0.0372 0.0264 0.0264
order k.k2 0.5136 0.4999 0.4983 0.4989 0.4983
Table 1: Numerical convergence and error analysis of the multistep method (23), (24) for Example 2
h 0.3142 0.1571 0.0785 0.0393 0.0196 0.0098
k 0.3142 0.1571 0.0785 0.0393 0.0196 0.0098
N 11 21 41 81 161 321
error k.k2 0.2217 0.1386 0.0986 0.0711 0.0510 0.0363
order k.k2 0.6776 0.4911 0.4715 0.4815 0.4900
Table 2: Numerical convergence and error analysis of the upwind scheme (20), the Lax-Friedrich scheme (21), and the Lax-Wendroff scheme (22) for Example 2 For Example 3 the optimal control variables are continuous and for both the second and the first order methods the numerically achieved convergence order seems to be 1.5, which seems to be surprising at the first glance particularly concerning the first order methods. However, note that the theoretical convergence orders obtained for smooth data are only a lower bound.
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h 0.3142 0.1571 0.0785 0.0393 0.0196 0.0098
k 0.1571 0.0785 0.0393 0.0196 0.0098 0.0049
N 11 21 41 81 161 321
(I.Troch, F.Breitenecker, eds.)
error k.k2 0.1524 0.0577 0.0212 0.0077 0.0027 9.75 × 10−4
order k.k2 1.4001 1.4434 1.4698 1.4841 1.4690
Table 3: Numerical convergence and error analysis of the multistep method (23), (24) for Example 3
h 0.3142 0.1571 0.0785 0.0393 0.0196 0.0098
k 0.1571 0.0785 0.0393 0.0196 0.0098 0.0049
N 11 21 41 81 161 321
error k.k2 0.1795 0.0901 0.0368 0.0145 0.0056 0.0021
order k.k2 0.9950 1.2897 1.3442 1.3674 1.4150
Table 4: Numerical convergence and error analysis of the upwind scheme (20), the Lax-Friedrich scheme (21), and the Lax-Wendroff scheme (22) for Example 3 For more numerical results, e. g., for the optimal boundary control of a string to an arbitrary terminal state, see [7]. Conclusions Optimal control problems for hyperbolic equations such as the classical wave equation are a particular challenge both for theory and numerics because of the existence of nonsmooth solutions. Numerical computations for the optimal boundary control of a string to rest in finite time, for which analytical solutions are known, have shown that the approach first discretize, then optimize is able to produce reliable results if the step size ratio is chosen close to the stability limit. The numerically achieved convergence order drops down to 0.5 for the first as well as the second order methods investigated in the paper, if the optimal boundary controls are discontinuous, whereas in the continuous case the order is 1.5 for both first and second order methods. These results obviously give rise to further questions, for example what is the best possible convergence order in the case of nonsmooth optimal boundary controls.
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[6] Chudej, K., Pesch, H. J., Sternberg, K.: Numerical Simulation of Molten Carbonate Fuel Cells. In: E. P. Hofer, E. Reithmeier (Eds.), Modeling and Control of Autonomous Decision Support Based Systems. — Aachen: Shaker Verlag, 227–234, 2005. [7] Greif, G.: Numerische Simulation und optimale Steuerung der Wellengleichung. Diploma Thesis, Chair of Engineering Sciences, Department of Mathematics, University of Bayreuth, Germany, 2005. [8] Gugat, M.: Analytic solutions of L∞ -optimal control problems for the wave equation. Journal of Optimization Theory and Applications 114, No. 2 (2002) 397–421. [9] Gugat, M.: Optimal Control of Networked Hyperbolic Systems: Evaluation of Derivatives. Advanced Modeling and Optimization 7 (2005) 9–37. [10] Gugat, M.: Optimal Boundary Control of a String to Rest in Finite Time. In: A. Seeger (Ed.), Recent Advances in Optimization, Lectures Notes in Economics and Mathematical Systems 563. — Springer: New York, 149–162, 2006. [11] Gugat, M.: L1 -Optimal Boundary Control of a String to Rest in Finite Time. In: F. Jarre, C. Lemarechal, J. Zowe (Eds.), Optimization and Applications. — Oberwolfach Reports, 2006. [12] Gugat, M., Leugering, G.: Solutions of Lp -norm minimal control problems for the wave equation. Computational and Applied Mathematics 21, No. 1 (2002) 227–244. [13] Gugat, M., Leugering, G., Sklyvar, G.: Lp -Optimal Boundary Control for the Wave Equation. SIAM Journal on Control and Optimization 44(2005) 49–74. [14] Hinze, M., Volkwein, S.: Proper Orthogonal Decomposition Surrogate Models for Nonlinear Dynamical Systems: Error Estimates and Suboptimal Control . — In: P. Benner, V. Mehrmann, D. Sorensen (Eds): Dimension Reduction of Large-Scale Systems. Lecture Notes in Computational and Applied Mathematics, Berlin: Springer, 2005, 261–306. [15] Krabs, W.: Optimal Control of Undamped Linear Vibrations. Lemgo: Heldermann, 1995. [16] Lagnese, J. E., Leugering, G.: Dynamic domain decomposition in approximate and exact boundary control in problems of transmission for wave equations. SIAM J. Control and Optimization 38 (2000), 503–537. [17] Lagnese, J. E., Leugering, G., Schmidt, E. J. P. G.: Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures. Boston: Birkh¨ auser, 1984. [18] Larsson, S., Thome, V.: Partial Differential Equations with Numerical Methods. Texts in Applied Mathematics 45. New York: Springer, 2003. [19] Leugering, G.: A Domain Decomposition of Optimal Control Problems for Dynamic Networks of Elastic Strings. Computational Optimization and Applications 16 (2000) 5–29. [20] Lions, J. L.: Optimal Control of Systems Governed by Partial Differential Equations. Berlin: Springer, 1971. [21] Sternberg, K.: Simulation, Optimale Steuerung und numerische Sensitivit¨ atsanalyse einer Schmelzkarbonat-Brennstoffzelle. To be submitted as dissertation, Faculty for Mathematics and Physics, University of Bayreuth, Germany, 2006. [22] Strikwerda, J. C.: Finite Difference Schemes and Partial Differential Equations. Norwell: Kluwer Academic Publishers (formerly Chapman and Hall), 1997. [23] Tr¨ oltzsch, F.: Optimalsteuerung bei partiellen Differentialgleichungen. Wiesbaden: Vieweg, 2005. [24] Ulbrich, S.: A Sensitivity and Adjoint Calculus for Discontinuous Solutions of Hyperbolic Conservation Laws with Source Terms. SIAM J. on Control and Optimization 41, No. 3 (2002) 740–797. [25] Ulbrich, S.: Adjoint-Based Derivative Computations for the Optimal Control of Discontinuous Solutions of Hyperbolic Conservation Laws. Systems & Control Letters Vol. 48, No. 3-4 (2003), 309–324
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Numerical Optimal Control of the Wave Equation: Optimal Boundary Control of a String to Rest in Finite Time M. Gerdts, Universit¨ at Hamburg, G. Greif, Technische Universit¨ at Darmstadt, H. J. Pesch, Universit¨ at Bayreuth, Germany Corresponding Author: H. J. Pesch Universit¨ at Bayreuth, Lehrstuhl f¨ ur Ingenieurmathematik Universit¨ atsstr. 30, 95440 Bayreuth, Germany e-mail: [email protected]
Abstract. In many real-life applications of optimal control problems with constraints in form of partial differential equations (PDEs), hyperbolic equations are involved which typically describe transport processes. Because of their nature being able to transport discontinuities of initial or boundary conditions into the domain on which the solution lives or even to develop discontinuities in the presence of smooth data, these problems constitute a severe challenge for both theory and numerics of PDE constrained optimization. In the present paper, optimal control problems for the well-known wave equation are investigated. The intention is to study the order of the numerical approximations for both the optimal state and the optimal control variables while analytical solutions are known here. The numerical method chosen here is a full discretization method based on appropriate finite differences by which the PDE constrained optimal control problem is transformed into a nonlinear programming problem (NLP). Hence we follow here the approach first discretize, then optimize, which allows us to make use not only of powerful methods for the solution of NLPs, but also to compute sensitivity differentials, a necessary tool for real-time control. Keywords. Optimal control of partial differential equations; optimal control of hyperbolic equations; optimal control of the wave equation; first discretize, then optimize. 1. Introduction Owing to its importance for engineering applications, the field of PDE constrained optimization has become increasingly popular. In the near future, mathematical optimization methods will be able to solve problems whose complexity has so far allowed only the application of simulation-based methods. Hence, there is a strong need for new efficient methods for PDE constrained optimization which are capable of tackling real-life engineering applications, including methods for real-time control. Those problems may be nonlinear and consist of coupled systems of high dimension and different type with complicated right hand sides and possibly non-standard initial and boundary conditions; see, e. g., the highly complex nonlinear PDE constrained optimal control problem describing the dynamical behaviour of certain fuel cell systems [5], [6], and [21]. Without doubt Lion’s book [20] is still the standard for optimal control problems with linear equations and convex functionals. Nonconvex problems with semilinear equations of elliptic and parabolic type are in the focus of Tr¨ oltzsch’s new book [23], particularly addressing questions of existence of solutions and optimal controls, the derivation of necessary conditions and adjoint equations as well as of second order sufficient conditions. While the theory of optimal control for elliptic and parabolic equations is well developed in the semilinear case, although the associated optimal control problems generally are nonconvex, hyperbolic equations are not as well understood despite the fact that many dynamical processes are, at least partly, of hyperbolic nature; see, e. g., the aforementioned references on fuel cell control. Linear hyperbolic equations are treated, besides in Lion’s book, in Ahmed and Teo [1]. An introduction into the control of vibrations can be found in Krabs [15]. Oscillating elastic networks are investigated by Lagnese, Leugering et al. [16]–[17], [19] and Gugat [9]. However, semilinear hyperbolic problems are still not well understood because of the weaker smoothness of the solution operators. The most recent progress is due to S. Ulbrich [24], [25]. In this paper we will investigate a well understood optimal control problem concerned with the classical wave equation. The present paper is particularly based on a series of papers by Gugat [8], [10]–[13].
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Gugat has presented analytical solutions for various optimal boundary control problems associated with the wave equation as one of the constraints. In contrast, we are going to solve these problems numerically in order to show the order of approximation for their optimal solutions. For, we want to propagate a numerical approach for the numerical solution of PDE constrained optimal control problems if hyperbolic equations are involved. The method of choice proposed here is either a full discretization method, in case of small size problems, or the vertical method of lines, in case of medium size problems. For large size problems only model reduction methods may today give a chance for their solution such as proper orthogonal decomposition; see, e. g., Hinze, Volkwein [14]. Concerning small and medium size problems, appropriate difference methods for the spatial discretization, resp. semidiscretization must be applied in order to approximate the hyperbolic equation correctly. Clearly, both approaches belong to the class first discretize, then optimize. They possess the advantage that the full power of methods for NLPs and ODE constrained optimal control problems can be used including their possibilities for a numerical sensitivity analysis and therefore real-time control purposes; see, e. g., B¨ uskens [2]–[4]. The latter is a must if optimal solutions are to be applied in practise. No doubt, that the approach first optimize, then discretize generally may give more theoretical inside and safety, but as in ODE constrained optimization its advantage does not pay for its drawbacks when complicated real-life applications are to be treated. 2. Optimal Control Problems 2.1 The continuous optimal control problem Let L > 0, T > 0, and a > 0 be real numbers, and let u0 ∈ L1 [0; L], where L1 [0; L] denotesR the set of all x Lebesgue integrable functions on the interval [0; L]. Furthermore, let the mapping x → 0 u1 (s) ds be 1 also in L [0; L] for a function u1 . We define ||(v, w)||1, [0;T ] :=
Z
T
(|v(t)| + |w(t)|) dt . 0
Then we can state the following optimal control problem: ! ||(yL , yR )||1, [0;T ] = min
(1)
subject to the constraints yL , yR ∈ L1 [0; T ] ,
(2)
utt (x, t) = a2 uxx (x, t) , (x, t) ∈ (0; L) × (0; T ) ,
(3)
u(x, 0) = u0 (x) , ut (x, 0) = u1 (x) , x ∈ [0; L] ,
(4)
u(0, t) = yL (t) , u(L, t) = yR (t) , t ∈ [0; T ] ,
(5)
u(x, T ) = 0 , x ∈ [0; L] and ut (x, T ) = 0 , x ∈ (0; L) .
(6)
We require that all consistency conditions are fulfilled, i. e., u0 (0) = yL (0), u0 (L) = yR (0), u(0, T ) = yL (T ) = 0, and u(L, T ) = yR (T ) = 0. Hence we have to control a vibrating string with the state variable u(x, t) satisfying the wave equation (3) from an arbitrary initial state (4) to rest in final time (6) by means of the boundary conditions (5) via the control functions (2) such that the functional (1) is minimized. In Gugat [11], Theorem 2, an analytical solution for this optimal control problem is given making this problem particularly valuable for numerical tests. However, due to the lack of smoothness of the functional (1), which would constitute an artificial difficulty, we change to a smooth version of problem (1)–(6) by minimizing Z T ! (7) ||(yL , yR )||22, [0;T ] := |yL (t)|2 + |yR (t)|2 dt = min . 0
Note that the change of the functional does not change the optimal solution with respect to the state and control variables. Solely the optimal value of the functional is different. Henceforth, we consider the optimal control problem (7) subject to (2)–(6).
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2.2 The discrete optimal control problem Introducing the time step size k > 0 and the spatial step size h > 0 so that q := T /k and m := L/h are natural numbers, the discrete version of (7) subject to (2)–(6) is given as follows q X j=1
! |yL (tj )|2 + |yR (tj )|2 = min
(8)
subject to the constraints Qh,k uji (YL;k , YR;k ) = 0 ,
(9)
u(xi , 0) = u0 (xi ) , ut (xi , 0) = u1 (xi ) , i = 0, . . . , m ,
(10)
j j u(x0 , tj ) = yL , u(xm , tj ) = yR , j = 0, . . . , q ,
(11)
u(xi , tq ) = 0 , i = 0, . . . , m ,
and
u(xi , tq ) − u(xi , tq−1 ) = 0 , i = 1, . . . , m − 1 , k
(12)
j j q > 0 where xi := i h, tj := j k, uji = u(xi , tj ), yL := yL (tj ), yR := yR (tj ), YL;k := yL , . . . , yL , and q > 0 YR;k := yR , . . . , yR . The difference operator Qh,k represents an appropriate discretization scheme for the wave equation specified later.
3. Numerical Results The following numerical tests are performed for four examples: Example 1: Let be L = 1, T = 3.25, a = 1, u0 (x) = 0, u1 (x) = sin(π x). Example 2: Let be L = π, T = 2 π, a = 1, u0 (x) = sin x, u1 (x) = cos x. Example 3: Let be L = π, T = 2 π, a = 2, and 4 − x for 0 ≤ x ≤ 41 π , π 1 3 4 u0 (x) = π x − 2 for 4 π ≤x ≤ 4 π, 4 − π x + 4 for 43 π ≤ x ≤ 43 π ,
Example 4: Let be L = 2, T = 6, a = 1, and ( x for 0 ≤ x ≤ 1 , u0 (x) = x − 2 for 1 ≤ x ≤ 2 ,
and u1 (x) = 0 ,
and u1 (x) = 0 ,
x ∈ [0; π] .
x ∈ [0; 2] .
All computations were carried through by means of Matlab. For the optimization the Matlabroutine fmincon, an SQP method has been used. When applying numerical difference schemes to the wave equation we have two possibilities. The first class of difference schemes is based on an equivalent formulation of the Cauchy problem associated with the wave equation, i. e., utt (x, t) = a2 uxx (x, t) , (x, t) ∈ (0; L) × R+ ,
(13)
u(x, 0) = u0 (x) , ut (x, 0) = u1 (x) , x ∈ [0; L] ,
(14)
ut (x, t) + a vx (x, t) = 0 ,
(15)
vt (x, t) + a ux (x, t) = 0 , (x, t) ∈ (0; L) × R+ ,
(16)
u(x, 0) = u0 (x) , Z 1 x v(x, 0) = − u1 (s) ds + C , x ∈ [0; L] . a 0
(17)
as follows,
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The constant C is arbitrary and can be set to zero since the component v is determined up to a constant only. The second class tackles the wave equation (13) directly. The following well-known schemes are candidates for an application to the system (15), (16), if certain restrictions are obeyed, see, e. g., Strikwerda [22]: The FTFS scheme (’forward-time forward-space’), also known as upwind scheme: j
j
u − ui uj+1 − uji i + a i+1 = 0; k h
(19)
The FTBS scheme (’forward-time backward-space’), also known as upwind scheme: uj − uji−1 uj+1 − uji i +a i = 0; k h
(20)
The Lax-Friedrich scheme: uj+1 − i
1 2
(uji+1 + uji−1 ) uj − uji−1 + a i+1 = 0; k 2h
(21)
The Lax-Wendroff scheme: uj+1 = uji − i
a2 k 2 ak j j j j ui+1 − uji−1 + u − 2 u + u . i+1 i i−1 2h 2 h2
(22)
In contrast, the following scheme directly applies to the wave equation (13). Approximating both second derivatives by central difference quotients of second order one obtains j j j j−1 2 2 uj+1 = 2 (1 − α ) u + α u + u for j ≥ 1 , (23) i i i+1 i−1 − ui u1i = (1 − α2 ) u0i +
1 2 0 α ui+1 + u0i−1 + k u1 (xi ) 2
(24)
with α := a hk . Equation (24) serves as the second starting line on t = k for the multistep method (23) besides the initial conditions (14) on t = 0. All of the above schemes can be rewritten so that uj+1 is a linear combination of values of u at time i instances j and j − 1, for example the formula uj+1 = (1 + a p) uji − a p uji+1 =: Ph,k uji i
with
p :=
k h
represents the FTFS scheme. Hereby Ph,k denotes the difference operator for the FTFS scheme, resp. Qh,k uji := uj+1 − Ph,k uji = 0. Analogously we can rewrite all schemes except (23), (24) as one step i methods, i. e., uj+1 = Ph,k uji . i In order to be convergent, these schemes must be consistent and stable according to the well-known Lax Equivalence Theorem. More precisely, a consistent finite difference scheme for a PDE, for which the Cauchy problem is wellposed, is convergent (e. g. w. r. t. the L2 -norm) if and only if it is stable; see for example [22]. In order to prove that, it is sufficient to show convergence solely for the simple scalar transport equation ut (x, t) + a ux (x, t) = 0 , since the system (15), (16) can be decoupled by a similarity transformation yielding a hyperbolic system with eigenvalues λ1,2 = ±a; see, e. g., Larsson, Thome [18] It is an easy exercise to show then that the upwind schemes are both of consistency order O(k + h) =: (1, 1). However, the FTFS scheme is stable only if a < 0 and −1 ≤ a p ≤ 0, whereas the FTBS scheme is stable only if a > 0 and 0 ≤ a p ≤ 1. This explains the names: The spatial difference quotients have to be chosen against the wind, i. e., against the direction of the information flow.
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The situation for the Lax-Friedrich scheme is more complicated. It is only consistent of order (1, 1) if k −1 h2 → 0, which can be fulfilled if k = Λ(h) with a sufficiently smooth function Λ with Λ(0) = 0. The scheme is stable if |a p| ≤ 1, and hence convergent then. The Lax-Wendroff scheme is consistent of order O(k 2 + h2 ) =: (2, 2) and stable if also |a p| ≤ 1, hence convergent. In case of general hyperbolic systems the CFL condition |a p| ≤ 1 has to be replaced by |λ p| ≤ 1 for all eigenvalues λ.
3.1 Results for Example 1 The following two figures show the results for Example 1 obtained by means of the multistep scheme (23), (24). Exact and approximated optimal boundary controls coincide perfectly; see Fig. 1. The associated state is given in Fig. 2.
exact exact Figure 1: Example 1: Exact and approximate optimal boundary controls yL (t) (left) and yR (t) app app (right) resp. yL (t) (left) and yR (t) (right) using the multistep scheme (23), (24) with h = k = 0.00625
Figure 2: Example 1: Approximate optimal state variable u(x, t) on the time interval [0; 3.25] using the multistep scheme (23), (24) with h = k = 0.0125
3.2 Results for Example 2 Figures 3 and 4 show the results for Example 2 obtained by means of the Lax-Wendroff scheme (22). Again the coincidence between numerical approximation and exact optimal solution is convincing. Solely at t = π, where the first derivative of the optimal boundary control is discontinuous, a slight oszillation arises.
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exact exact Figure 3: Example 2: Exact and approximate optimal boundary controls yL (t) (left) and yR (t) app app (right) resp. yL (t) (left) and yR (t) (right) using the Lax-Wendroff scheme (22) with h = k = π/140
Figure 4: Example 2: Approximate optimal state variable u(x, t) on the time interval [0; 2 π] using the Lax-Wendroff scheme (22) with h = k = π/140 3.3 Results for Example 3 Figures 5 and 6 show the results for Example 3 obtained by means of the Lax-Friedrich scheme (21). Again the coincidence between numerical approximation and exact optimal solution is perfect. However, note that these results can only obtained if a p = 1 which is the maximum value of the stability interval. The same result holds when using the multistep scheme (23), (24). For a p < 1 oscillations occur for both methods.
exact exact Figure 5: Example 3: Exact and approximate optimal boundary controls yL (t) (left) and yR (t) app app (right) resp. yL (t) (left) and yR (t) (right) using the Lax-Friedrich scheme (21) with h = π/101 and k = π/202
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Figure 6: Example 3: Approximate optimal state variable u(x, t) on the time interval [0; π] using the Lax-Friedrich scheme (21) with h = π/101 and k = π/202
3.4 Results for Example 4 For Example 4, Figs. 7 and 8 show the superiority of the second order multistep scheme (23), (24) compared to the simple first order upwind scheme (20) in the resolution of the discontinuities despite the larger step size of the multistep method. The associated approximate state variables are given in Fig. 9.
exact exact Figure 7: Example 4: Exact and approximate optimal boundary controls yL (t) (left) and yR (t) app app (right) resp. yL (t) (left) and yR (t) (right) using the upwind scheme (20) with h = k = 0.01
exact exact Figure 8: Example 4: Exact and approximate optimal boundary controls yL (t) (left) and yR (t) app app (right) resp. yL (t) (left) and yR (t) (right) using the multistep scheme (23), (24) with h = k = 0.05
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Figure 9: Example 4: Approximate optimal state variable u(x, t) on the time interval [0; 6] using the upwind scheme (20) (left) resp. the multistep scheme (23), (24) (right) both with h = k = 0.05
3.5 Convergence Analysis for Example 2 For the weighted step size ratio ap = 1 we were able to produce very precise results for all optimal control examples investigated. The following Tables 1 and 2 list the discrete L 2 - norm errors for the approximated optimal control variables and the numerically achieved convergence orders for all difference schemes investigated for Example 2. Similar results were obtained for Example 1 and 4. Note that the results for the first order methods (20), (21), and (22) coincide up to the first four digits so that they together are given in Table 2. The results for Example 3 are given in Tables 3 and 4 below. As we can see, the convergence order drops down to 0.5 for both the first and the second order methods which is due to the nonsmooth control variables.
h 0.3142 0.1571 0.0785 0.0393 0.0196 0.0098
k 0.3142 0.1571 0.0785 0.0393 0.0196 0.0098
N 11 21 41 81 161 321
error k.k2 0.1500 0.1051 0.0526 0.0372 0.0264 0.0264
order k.k2 0.5136 0.4999 0.4983 0.4989 0.4983
Table 1: Numerical convergence and error analysis of the multistep method (23), (24) for Example 2
h 0.3142 0.1571 0.0785 0.0393 0.0196 0.0098
k 0.3142 0.1571 0.0785 0.0393 0.0196 0.0098
N 11 21 41 81 161 321
error k.k2 0.2217 0.1386 0.0986 0.0711 0.0510 0.0363
order k.k2 0.6776 0.4911 0.4715 0.4815 0.4900
Table 2: Numerical convergence and error analysis of the upwind scheme (20), the Lax-Friedrich scheme (21), and the Lax-Wendroff scheme (22) for Example 2 For Example 3 the optimal control variables are continuous and for both the second and the first order methods the numerically achieved convergence order seems to be 1.5, which seems to be surprising at the first glance particularly concerning the first order methods. However, note that the theoretical convergence orders obtained for smooth data are only a lower bound.
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h 0.3142 0.1571 0.0785 0.0393 0.0196 0.0098
k 0.1571 0.0785 0.0393 0.0196 0.0098 0.0049
N 11 21 41 81 161 321
(I.Troch, F.Breitenecker, eds.)
error k.k2 0.1524 0.0577 0.0212 0.0077 0.0027 9.75 × 10−4
order k.k2 1.4001 1.4434 1.4698 1.4841 1.4690
Table 3: Numerical convergence and error analysis of the multistep method (23), (24) for Example 3
h 0.3142 0.1571 0.0785 0.0393 0.0196 0.0098
k 0.1571 0.0785 0.0393 0.0196 0.0098 0.0049
N 11 21 41 81 161 321
error k.k2 0.1795 0.0901 0.0368 0.0145 0.0056 0.0021
order k.k2 0.9950 1.2897 1.3442 1.3674 1.4150
Table 4: Numerical convergence and error analysis of the upwind scheme (20), the Lax-Friedrich scheme (21), and the Lax-Wendroff scheme (22) for Example 3 For more numerical results, e. g., for the optimal boundary control of a string to an arbitrary terminal state, see [7]. Conclusions Optimal control problems for hyperbolic equations such as the classical wave equation are a particular challenge both for theory and numerics because of the existence of nonsmooth solutions. Numerical computations for the optimal boundary control of a string to rest in finite time, for which analytical solutions are known, have shown that the approach first discretize, then optimize is able to produce reliable results if the step size ratio is chosen close to the stability limit. The numerically achieved convergence order drops down to 0.5 for the first as well as the second order methods investigated in the paper, if the optimal boundary controls are discontinuous, whereas in the continuous case the order is 1.5 for both first and second order methods. These results obviously give rise to further questions, for example what is the best possible convergence order in the case of nonsmooth optimal boundary controls.
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