L1-Optimal boundary control of a string to rest in finite time Martin Gugat
Hyperbolic partial differential equations often appear as models in engineering, for example as systems of conservation laws that model fluid flow. The control of such systems is usually only possible with boundary controls, which in the mathematical model corresponds to control via the boundary conditions. To get some insight into the nature of optimal controls for such systems, we consider the following problem of optimal Dirichlet boundary control for the wave equation, where real numbersT >0,L >0 andc >0 are given such thatT ≥L/c.
P : minimize Z T
0
|u1(t)|+|u2(t)|dtsubject tou1, u2∈L1(0, T) and
y(x,0) =y0(x), yt(x,0) =y1(x), x∈(0, L) y(0, t) =u1(t), y(L, t) =u2(t), t∈(0, T) ytt(x, t) =c2yxx(x, t), (x, t)∈(0, L)×(0, T)
y(x, T) = 0, yt(x, T) = 0, x∈(0, L).
In general this problem does not have a unique solution. An explicit represen- tation of all solutions is given in the following theorem.
Theorem 1Assume that T ≥t0 =L/c, thaty0 ∈L1(0, L)and that Y1(x) = Rx
0 y1(s)ds∈L1(0, L). Let
α0(t) = y0(ct) + (1/c) Rct
0 y1(s)ds, t∈(0, t0), β0(t) = y0(L−ct)−(1/c) RL−ct
0 y1(s)ds, t∈(0, t0).
Choose a real numberrthat minimizes
I(r) =1 2
Z t0
0
|α0(t)−r|+|β0(t) +r|dt.
Let k= max{j∈N:j t0≤T}, ∆ =T−k t0.
Forj ∈ {0, ..., k},t∈(0,∆) letλj(t)≥0,νj(t)≥0 almost everywhere such that λj(α0−r)∈L1(0,∆),νj(β0+r)∈L1(0,∆) and
Xk j=0
λj(t) = 1 = Xk j=0
νj(t) almost everywhere on(0,∆).
Forj∈ {0, ..., k−1}, t∈(∆, t0) letµj(t)≥0, ωj(t)≥0 almost everywhere such thatµj(α0−r)∈L1(∆, t0),ωj(β0+r)∈L1(∆, t0)and
k−1X
j=0
µj(t) = 1 =
k−1X
j=0
ωj(t) almost everywhere on(∆, t0).
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Then a solution of ProblemP is given by u1, u2 defined in (1)–(8) and every solution of ProblemP is given by controlsu1,u2 of this form.
u1(t+jt0) = λj(t) [α0(t)−r]/2 ifj is even andt∈(0,∆), (1)
u1(t+jt0) = µj(t) [α0(t)−r]/2 ifj is even andt∈(∆, t0), (2)
u1(t+jt0) = −νj(t) [β0(t) +r]/2 ifj is odd andt∈(0,∆), (3)
u1(t+jt0) = −ωj(t) [β0(t) +r]/2 ifj is odd andt∈(∆, t0), (4)
u2(t+jt0) = νj(t) [β0(t) +r]/2 ifjis even andt∈(0,∆), (5)
u2(t+jt0) = ωj(t) [β0(t) +r]/2 ifjis even andt∈(∆, t0), (6)
u2(t+jt0) = −λj(t) [α0(t)−r]/2 ifjis odd andt∈(0,∆), (7)
u2(t+jt0) = −µj(t) [α0(t)−r]/2 ifjis odd andt∈(∆, t0).
(8)
The minimal value of Problem P is given by the integral I(r) with an optimal choice ofr. The solution of ProblemP is unique if and only if the minimal value of ProblemP is zero.
A proof of this result is given in [1]. This proof is based upon the travelling waves solution of the wave equation. It shows how the structure of the optimal controls is related to the characteristic curves. The solutions of the corresponding problems forLp-norms withp∈[2,∞) are given in [2], where a proof based upon Fourier series and moment problems is presented. For these problems, where
Z T
0
|u1(t)|p+|u2(t)|pdt
is minimized, the optimal controls are uniquely determined and have the same structure as in (1)–(8) but only with
λj(t) =νj(t) = 1/(k+ 1), µj(t) =ωj(t) = 1/k.
This is also true for p ∈ (1,2). For the corresponding L∞ problem where the objective function is an essential supremum, controls u1, u2 of the form as for p∈(1,∞) give the element of the solution set with minimal L2norm (see [2]).
Theorem 1 shows that if the problem dataT, L or c are changed, changes in the structure of the solution can occur ifT =kt0. Note that also for C∞ initial data, the optimal state may have jumps that are generated by the discontinuities of the optimal controls. ProblemP is related to control problems for nonlinear hyperbolic systems, see [3]. Hopefully Theorem 1 helps to exploit the structure of optimal controls for quasilinear hyperbolic systems.
References
[1] M. Gugat,L1-Optimal boundary control of a string to rest in finite time, Recent Advances in Optimization (Proceedings of the 12th French-German-Spanish Conference on Optimiza- tion), Lecture Notes in Economics and Mathematical Systems, Springer (2005), to appear.
[2] M. Gugat, G. Leugering, G. SklyarLp optimal boundary control for the wave equation, SIAM Journal on Control and Optimization (2005), to appear.
[3] M. GugatBoundary Controllability between Sub– and Supercritical Flow, SIAM Journal on Control and Optimization 42 (2003) 1056–1070.
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