Numerical results for two model problems

the equations (6.2.2), (6.2.3) and (6.2.4) can be written as a mixed integer linear equation system Ax = b in the variables x = (x1, . . . , x(N+1)M)

with sparse coefficient matrix A. The integral part of the cost function (6.0.1) can be approximated by

NM

X

i=1

(xi− zi)2= (x − z)⊤(x − z) = x⊤x −2z⊤x + z⊤z, (6.2.7)

where ziis a discretization of xd. Using that z⊤zis constant, these costs

can be written as x⊤Qx − c⊤xwith Q = 1 and c = 2z. Moreover, the costs cW q(t) δt for a positive constant c can be encoded in Q = (qi,j)

by setting κ = c/N, qi,i= κ, qi+1,i= −1₂κand qi,i+1= −1₂κfor i = 1 =

NM +1, . . . , (N + 1)M. We remark that for stability of this method, the above discretization scheme requires the N and M chosen such that the Courant-Friedrichs-Lewy condition holds.

The discretization scheme has been implemented in MATLAB [44] and is used in Section 6.3 for verification of results obtained with the gradient method proposed in Section 6.1.

6.3 Numerical results for two model problems

In this Section, we summarize numerical results for two model prob- lems. Both are of academic nature, but they do demonstrate the viabil- ity of characterizing optimal switching signals based upon the gradient formula in Theorem 3.2.1.

The first very simple example mainly serves as a verification of the indirect method using gradient information for computing approxima- tions of optimal switching signals as proposed in Section 6.1.

Example 6.3.1: (Bang-bang type approximation of a traveling wave) Con- sider a traveling wave xd given by

xd(t, s) =

1

2sin(5π(t − s)) + 1, 0 6 t 6 1, 0 6 s 6 1 (6.3.1) which oscillates between 0 and 1. For the problem (6.0.1) with λ(t, s) ≡ 1 and f ≡ 0, the control task in this example consists of approximating the wave xd by switching the boundary values u(t) between the two

extremal values of the wave u0= 0, u1= 1, where we take the relaxation

parameter c = 0.0075 to avoid chattering.

It should be clear that for this problem, we cannot expect exact con- trollability, but we are seeking for a binary control q∗(·) minimizing the L2-distance between x and xd over the finite time horizon [0, T ] with

T = 1. Also observe that the optimal control of the relaxed problem with ˆu ∈ [0, 1] is not of bang-bang type. Thus, alternative methods as considered in Section 6.1 and Section 6.2 are required for the computa- tion of optimal controls.

For the projected gradient method considered in Section 6.1, the pa- rameters in this example where chosen as follows: For Algorithm 6.1.1, we used α = 0.1, β = 0.5, ε = 0.05 and for initialization, we use ¯q(·) given by ¯τ1, . . . , ¯τK with K = 25 equidistantly placed in [0, T ] and

q(0) = 1 as depicted in part a) of Figure 6.2 on page 111. For Algo- rithm 6.1.3, we used ∆s = 0.01 and ∆h = 0.001, thus a total number of about 1100 particles for the mesh-free solver. The costs of the so- lution corresponding to the choice ¯q(·) are J( ¯q(·)) = 0.0398 and the performance of the projected gradient method is shown in Table 6.2 on page 111. The computational time was 16.72 seconds.

For the MIQP formulation considered in Section 6.2, we used in this example a mesh with M = 50 and N = 50. The mixed integer quadratic program was solved using ILOG CPLEX [31], which terminated with a standard integer optimal (tolerance) termination criterion after a com- putation time of 18.74 seconds.

The optimal switching control approximations ˜q∗_PG(·) and ˜q∗ MIQP(·)

are also shown in part a) of Figure 6.2. The example is chosen such that the final time plot of x(t, ·) with T = 1 shows the complete his- tory of the control action. The final time plots of the solution ˜xGP(T , ·)

and ˜xMIQP(T , ·) corresponding to ˜q∗PG(·) and ˜q∗MIQP(·), respectively, are

shown in part b) of Figure 6.2. 

We see from Figure 6.2 on page 111 that in Example 6.3.1 both optimal switching control approximations ˜q∗_PG(·) and ˜q∗_MIQP(·) obtained with different methods almost coincide.

The second example testifies the potential viability of the gradient based optimization approach in a more difficult setting that could be part of a network optimization problem.

6.3 Numerical results for two model problems

(a) Two mode inlet control.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.2 0 0.2 0.4 0.6 0.8 1 1.2

(b) Inflow data in mode 1 (solid) and mode 2 (dashed).

Figure 6.1: Illustration of the control task in Example 6.3.2.

Example 6.3.2: (Two-mode plug flow regulation) Consider a pipe of a network that can be controlled at the inlet s = 0 by choosing the in- flow of material concentration either from u1(t) or from u2(t), compare

Figure 6.1 (a). The plug flow in the pipe is assumed to satisfy the con- servation law

∂

∂tx(t, s) + ∂

∂s[ρ(s)x(t, s)] = 0 (6.3.2) with ρ(s) = 4₃(s−1)2+1₂. Suppose that a desired material distribution in the pipe is given by xd(t, s) = 1₂(s + 1)2. As in Example 6.3.1, we cannot

expect exact controllability, but we are again seeking for a binary control q∗(·) minimizing the L2_{-distance between x and x}

dover the finite time

horizon [0, T ] with T = 1 and include a regularization parameter c = 0.0075 to avoid chattering. So the control problem is of the form (6.0.1) with

λ(t, s) = ρ(s) and f(t, s) = − d

dsρ(s) = −3s + 3. (6.3.3) For the projected gradient method considered in Section 6.1, the pa- rameters α, β, ε, ∆s and ∆h are chosen as in Example 6.3.1. The search is started with ¯q(·) placing 35 switching times τk equidistantly in [0, T ]

with q(0) = 1 and a corresponding cost of J( ¯q(·)) = 0.454, cf. part a) of Figure 6.3 on page 112. The performance of the projected gradient method is shown in Table 6.4 on page 112. The computational time was 8.52 seconds.

this example a mesh with M = 80 and N = 160. The mixed integer quadratic program was solved using ILOG CPLEX [31], which termi- nated with a standard integer optimal (tolerance) termination criterion after a computation time of 50.24 seconds.

The optimal switching control approximations ˜q∗_PG(·) and ˜q∗ MIQP(·)

are also shown in part a) of Figure 6.3. As in Example 6.3.1, the final time plot of x(t, ·) with T = 1 shows the complete history of the control action. The final time plots of the solution xGP(T , ·) and xMIQP(T , ·) cor-

responding to ˜q∗_PG(·) and ˜q∗

MIQP(·), respectively, are shown in part b) of

Figure 6.3. 

We see from Figure 6.3 on page 112 that the optimal switching con- trol approximations ˜q∗_PG(·) and ˜q∗

MIQP(·) in Example 6.3.2 obtained with

different methods are quite similar. At this point, however, it shall be remarked that the choice ¯q(·) for the initialization of the search is of course crucial for gradient based optimization method since it searches for locally optimal solutions only. On the other hand, the mixed integer quadratic program based on a total discretization of the PDE searches for globally optimal solutions, but at exponential cost. A direct com- parison of the methods in terms of runtime, quality of the solution etc. is therefore not feasible. Nevertheless, we conclude from Example 6.3.1 and Example 6.3.2 that the gradient based optimization method is a promising alternative to mixed integer programming.

6.3 Numerical results for two model problems

Iter (i) J(qi_{(·)) k ¯h}i_{k ζ}i

1 3.980305e-01 3.602155e+00 2.016207e-02 2 1.795638e-01 2.894234e+00 8.180764e-04 3 1.750549e-01 2.557251e+00 9.178310e-04 4 1.556110e-01 2.208352e+00 1.376385e-03 5 1.506769e-01 1.849382e+00 2.739581e-03 6 1.292383e-01 1.510550e+00 3.346466e-03 7 1.251216e-01 1.185407e+00 5.192029e-03 8 1.073144e-01 8.756839e-01 8.849768e-03 9 9.043279e-02 5.858933e-01 1.874199e-02 10 8.962414e-02 3.146402e-01 6.195228e-02 11 6.754035e-02 2.685274e-02 –

Table 6.2: Performance of the projected gradient method for Exam- ple 6.3.1. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 (a) ¯q(·), ˜q∗ PG(·) and ˜q∗MIQP(·). 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 (b) ˜xGP(T , ·) and ˜xMIQP(T , ·).

Figure 6.2: Numerical results for Example 6.3.1. Figure a) shows the switching signal ¯q(·) used for initialization and the approximations of optimal switching signals ˜q∗_PG(·) and ˜q∗_MIQP(·) (from top to bot- tom). Figure b) shows the desired wave xd(T , ·) (dashed) and —

almost coinciding — the integer optimal solution approximations ˜xGP(T , ·) (solid) and ˜xMIQP(T , ·) (dashed-dotted) at T = 1.

Iter (i) J(qi_{(·)) k ¯h}i_{k ζ}i

1 4.542712e-01 2.731121e+00 1.450710e-02 2 3.685034e-01 1.459007e+00 9.248055e-03 3 3.387464e-01 1.128496e+00 1.095173e-02 4 2.986284e-01 8.649842e-01 1.059605e-02 5 2.778872e-01 5.753346e-01 3.308403e-02 6 2.566346e-01 3.304015e-01 1.149051e-02 7 2.559575e-01 2.334151e-01 5.730142e-02 8 2.247413e-01 1.232074e-01 9.954417e-02 9 2.100891e-01 9.796896e-02 5.285141e-02 10 2.097940e-01 7.331591e-02 5.407091e-02 11 2.096671e-01 3.089278e-02 –

Table 6.4: Performance of the projected gradient method for Exam- ple 6.3.2. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 (a) ¯q(·), ˜q∗ PG(·) and ˜q∗MIQP(·). 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 3.5 (b) ˜xGP(T , ·) and ˜xMIQP(T , ·).

Figure 6.3: Numerical results for Example 6.3.2. Figure a) shows the switching signal ¯q(·) used for initialization and the approxima- tions of optimal switching signals ˜q∗_PG(·) and ˜q∗_MIQP(·) (from top to bottom). Figure b) shows the desired material distribution xd(·)

(dashed) and the optimal solution approximations ˜xGP(T , ·) (solid)