2.2 Optimal Control of PDEs
In this section we start with the definition of the required function spaces. A detailed introduction can be found in [32]. Especially, the precise definition of a Lipschitz domain and weak derivatives can be found there. Furthermore, we assume that the definition of the Lebesgue spaces Lp(Ω) with the corresponding Lp-norm
kukLp(Ω) = Z
Ω|u(x)|p dx 1/p
, p∈[1,∞), kukL∞(Ω) = ess sup
x∈Ωku(x)k
is known. Functions which only differ on a set of measure zero belong to the same equivalence class. Next, we introduce the so called Sobolev spaces
Definition 2.6
Let Ω ∈Rn with boundary Γ := ∂Ω be open. For k ∈ N0, p ∈ [1,∞], we define the Sobolev space Wk,p(Ω) by
Wk,p(Ω) :={u∈Lp(Ω) :uhas weak derivativesDαu∈Lp(Ω)for all|α| ≤k} with the associated norm
kukWk,p(Ω):=
X
|α|≤k
kDαukpLp(Ω)
1/p
, p∈[1,∞) kukWk,∞(Ω):= X
|α|≤k
kDαukL∞(Ω).
• For the important case p= 2 we define Hk(Ω) :=Wk,2(Ω).
• The closure of C0∞(Ω) in Wk,p(Ω) is denoted by W0k,p(Ω).
• The dual space of the Hilbert space H01(Ω) is denoted by H−1(Ω).
In the context of parabolic PDEs it is reasonable to interpret the solution y(x, t) as an abstract function (also called vector-valued function) y(t) = y(·, t)∈ X with values in the Banach space X.
Definition 2.7
For 1≤p <∞ we denote by Lp(a, b;X) the Banach space of abstract functions y: [a, b]→X with the property
Z b
a ky(t)kpX dt <∞. (2.16) The corresponding norm is given by
kykLp(a,b;X):=
Z b
a ky(t)kpX dt p
. (2.17)
2 Optimal Control of PDEs
Analogously, we define the Banach space L∞(a, b;X)by kykL∞(a,b;X) := ess sup
t∈[a,b]ky(t)kX <∞. (2.18) For a Hilbert space V we introduce the space
W(0, T;V) :={y:y∈L2(0, T;V), yt∈L2(0, T;V∗)} (2.19) with the associated norm
kykW(0,T;V):=
Z T 0
(ky(t)k2V +kyt(t)k2V∗)dt
. (2.20)
For the important case V =H1(Ω) we write
W(0, T) := {y:y∈L2(0, T;H1), yt∈L2(0, T;H−1)}.
Next, we present the Friedrichs-Poincar´e inequality (also known an Friedrichs in-equality). The proof can be found in [24].
Theorem 2.8
Let Ω⊂ Rn denote a bounded Lipschitz domain and let Γ0 ⊂Γ be a measurable set such that |Γ0|>0. Then there exists a constant c > 0 such that
Z
Ω|y|2dx≤c Z
Ω|∇y|2 dx (2.21)
for all y ∈H1(Ω) that satisfy y= 0 on Γ0.
This useful inequality will play an important role in our theoretical investigations.
Therefore, in Chapter 3 we will use a version of this inequality where the best Friedrichs constant is explicitly given.
Now, we have the necessary tools to analyse the semilinear heat equation with a distributed control function u(x, t) and Neumann control v(x, t) on the boundary
yt(x, t)−∆y(x, t) +f(y(x, t)) = u(x, t) inQ:= Ω×(0, T) (2.22a)
∂νy(x, t) = v(x, t) on Σ := Γ×(0, T) (2.22b)
y(0, x) = y0(x) in Ω (2.22c)
In most practical applications only one of these control functions is present in the system.
Remark 2.9
It should be mentioned that most theorems presented in this section are (under suit-able conditions) valid for more general PDEs :
2.2 Optimal Control of PDEs
• The negative Laplacian can be replaced by an elliptic differential operator
Ay(x) = − Xn i,j=1
Di(aij(x)Djy(x)), x∈Ω satisfying Pn
i,j=1aij(x)ξiξj ≥γ|ξ|2, γ >0, ∀ξ ∈Rn.
• The nonlinear function f may depend on x and t, i.e., f(x, t, y).
• A nonlinear function b(x, t, y) can be incorporated on the boundary.
• A homogeneous Dirichlet condition can be imposed on the boundary.
In general, one cannot expect to find classical solutions of (2.22) and, thus, we are interested in so calledweak solutions. For this purpose we multiply equation (2.22a) with φ∈H1(Ω) and integrate over Ω. Using integration by parts we obtain
Z
Ω
yt(x, t)φ(x)dx+ Z
Ω∇y(x, t)∇φ(x)dx+ Z
Ω
f(y(x, t))φ(x)dx
= Z
Ω
u(x, t)φ(x)dx+ Z
Γ
v(x, t)φ(x)dS. (2.23)
This representation also forms the basis of the Galerkin method presented in Section 2.3.1.
Definition 2.10
A function y ∈W(0, T) is called weak solution of (2.22) if
• (2.23) holds for each φ ∈H1(Ω) and a.e. time 0≤t≤T
• and the initial condition y(0) =y0 is satisfied.
The following assumptions are required to prove the existence of a weak solution:
Assumption 2.11
Let Ω⊂ Rn, n ≥ 1, be a bounded Lipschitz domain (for n = 1 a bounded interval).
The function f is assumed to be locally Lipschitz, continuous and monotonically increasing.
Theorem 2.12
Suppose that Assumption 2.11 hold. Then the semilinear parabolic initial value problem (2.22) has a unique weak solution y ∈ W(0, T)∩C( ¯Q) for any triple u∈Lr(Q), v ∈Ls(Σ) and y0 ∈C( ¯Ω)with r > n/2 + 1, and s > n+ 1.
The proof of an even more general result can be found in [81]. The uniqueness is quite important, because it guarantees the existence of a unique control to state mapping S :Lr(Q)×Ls(Σ)→W(0, T)∩C( ¯Q),(u, v)7→y.
2 Optimal Control of PDEs
After these results concerning the state equation we introduce the general cost func-tional, cf. [91],
minJ(y, u, v) :=
Z
Ω
φ(y(x, T))dx+ Z Z
Q
ϕ(y(x, t), u(x, t))dxdt Z Z
Σ
ψ(y(x, t), v(x, t))dSdt. (2.24) Assumption 2.13
• Ω⊂Rn is a bounded Lipschitz domain.
• The functions φ, ϕ and ψ are twice differentiable with respect to y, u and v.
• Let ϕ and ψ be convex with respect to u and v, respectively.
• We have fy(y)≥0 and y0 ∈C( ¯Ω).
• The sets of admissible control values are given by
U={u∈L∞(Q) :ua(x, t)≤u(x, t)≤ub(x, t) for a.e.(x, t)∈Q} V={v ∈L∞(Σ) :va(x, t)≤v(x, t)≤vb(x, t) for a.e.(x, t)∈Σ} with ua, ub ∈L∞(Q), ua(x, t)≤ub(x, t) and va, vb ∈L∞(Σ), va(x, t)≤vb(x, t).
The existence of an optimal control can be shown with these assumptions, cf. [91].
Theorem 2.14
Suppose that Assumption 2.13 holds. Then the optimal control problem with cost functional (2.24) and PDE (2.22) has at least one optimal pair (¯u,¯v)∈U×V with associated state y¯=y(¯u,v).¯
We close the investigation of this optimal control problem by presenting the first order optimality conditions, cf. [91].
Theorem 2.15
Suppose that Assumption 2.13 is satisfied and let (¯u,¯v) an optimal control pair for the optimal control problem with cost functional (2.24) and state equation (2.22).
Let the adjoint state p ∈ W(0, T) ∩L∞(Q) be the solution of the corresponding adjoint equation
−pt(x, t)−∆p(x, t) +fy(¯y(x, t))p(x, t) =ϕy(¯y(x, t),u(x, t))¯ in Q (2.25a)
∂νp(x, t) =ψy(¯y(x, t),v(x, t))¯ onΣ (2.25b) p(x, T) =φy(¯y(x, T)) inΩ. (2.25c) Then the variational inequalities
Z Z
Q
(p(x, t) +ϕu(¯y(x, t),u(x, t)))(u(x, t)¯ −u(x, t))¯ dxdt≥0 ∀u∈U (2.26a) Z Z
Σ
(p(x, t) +ψv(¯y(x, t),v(x, t)))(v¯ (x, t)−v(x, t))¯ dxdt≥0 ∀v ∈V (2.26b) are satisfied.
2.2 Optimal Control of PDEs Obviously, we have exactly the same structure as in Theorem 2.4 where we investi-gated the abstract optimal control system: The necessary conditions consist of the state equation (2.22), the adjoint equation (2.25) and the variational inequalities (2.26).
Example 2.16
In this example we consider the semilinear heat equation with distributed control and a quadratic cost functional. To shorten the notation we neglect the function arguments. We look at the optimal control problem
minJ(y, u) = 1
2ky−ydkL2(Q)+λ
2kukL2(Q) (2.27) subject to
yt−∆y+f(y) = u inQ
∂νy= 0 onΣ
y(0) =y0 inΩ
where λ >0 denotes a regularization parameter and yd the desired state. Then the adjoint equation is given by
−pt−∆p+fy(¯y)p= ¯y−yd inQ
∂νp= 0 onΣ
p(T) = 0 inΩ.
For the variational inequality we obtain Z Z
Q
(p+λu)(u¯ −u)¯ ≥0 ∀u∈U.
By using the projection operator P[ua,ub](v) := max{ua,min{ub, v}}, the varia-tional inequality can be written in the following useful form, cf. [91],
¯
u(x, t) =P[ua,ub]
−1 λp(x, t)
.
Remark 2.17
In Section 3.1 we analyse a larger class of semilinear PDEs which is not fully covered by the theorems in this section. This especially concerns the non monotonic Schl¨ogl equation presented in Section 5.1. However, in this case the transformationy(x, t) :=
eµtv(x, t) (for a particular choice of µ ∈ R) leads to a monotonically increasing nonlinearity, which can be addressed with the methods presented in this section, cf.
[20].
2 Optimal Control of PDEs
Remark 2.18
The treatment of Dirichlet boundary control is more complicated than the corre-sponding Neumann control because the standard variational formulation does not work in this case. However, the linear heat equation with Dirichlet boundary con-trol considered in this thesis was already studied in the early seventies by [68] using the so called transposition method. The existence and regularity results concerning the boundary controlled linear wave equation considered in Section 3.7.2 can also be found in [68]. A nice approach to deal with general linear quadratic optimal control problems is given by the theory of strongly continuous semigroups, see e.g. [15] and [67].