2.4 First order analysis via formal Lagrange technique
2.4.2 Partial shape derivatives
Definition 7(Lagrangian):
The Lagrangian of the set optimal control problem minimize J(B;uJ,uB,yJ,yB):= 1 which is an equivalent reformulation15of the original set optimal control problem (2.30), is defined as
L:O ×L2(J)×L2(B)˚ ×H2(J)×H2(B)˚ ×L2(J,∆)×L2(B˚,∆)
As already mentioned, one advantage of the formal Lagrange technique is, that all variables are treated as independent. Hence, there is no need for applying the chain rule, and in this sense the formalism only relies on partial derivatives. Thus, the notion of a partial shape derivatives is required in the present context.
By way of illustration, let f be a shape functional (whose domain actually is a vector bundle and should not be written as Cartesian product)
f :O ×X→R, (B,x)7→ f(B,x),
which is assumed to be shape differentiable atA ∈ O. In addition, letx(B)∈Xbe uniquely determined by the choice of the setBand be shape differentiable atA. Then the chain rule16reveals, that the(total) shape derivativeconsists of two separate parts
• thepartial shape derivative, which describes the explicit dependency off on the shape variableBand
• the partial derivative with respect to the function variablexcomposed with thelocal shape derivative x0(A)[V], which describes the implicit dependency of f on the shape variable, which is caused by the shape dependent variablex
15Confer the first part of the proof ofTheorem 5. It is possible to analogously define a Lagrangian for original set optimal control problem, too. However, this approach yields the sightly different necessary conditions which are obtained inAppendix B.
16It should be noted that applicability of the chain rule requires suitable notions of derivatives; Hadamard differentiability is required in particular. This topic is discussed in [44, p. 170 and Chp. 9 Sec. 2.3] in the context of shape calculus.
In particular, the derivative of an integral domain shape functional f(B) := R whereas the derivative of an integral boundary shape functional f(B) := R
∂Bg(B)(whose integrand is the trace of a distributed functiong(B) =G(B)|∂B) yields (see [151, Eq. (2.174)])
In addition, one is often confronted with the situation, that the integrand of an integral boundary shape functional is the product of the trace of a distributed function and another function which cannot be seen as the trace of a distributed function, this is
f(B):= Z
∂BG(B)|∂Bh(B).
At this, the decomposition into partial and local shape derivative is given by17 d f(A;V) = The situation gets even more involved if a normal derivative is part of the integrand
f(B):= Z
∂B∂nG(B).
In order to derive its (partial) shape derivative one can make use of the oriented distance function to be introduced next. As a start define thedistance functionfrom a pointx∈R2to a setM⊂R2. x ∈ ∂M, cf. [44, Chp. 7 Thm. 8.5]. Furthermore, the gradient∇bMof the oriented distance function is an extension of the outer unit normal vector field locally inBρ(x), (ibidem). Consequently, the oriented distance functionbB exists for anyB ∈ O, is C1,1-regular in a tubular neighborhood ofβ = ∂Band its gradient is an extension of the unit normal vector fieldnB. Due to Rademacher’s theorem the second derivative D2bB exists almost everywhere in this neighborhood. The set index ofbBwill be omitted in the following, since its connection with the set will be obvious.
With this notion at hand one can resume the derivation of the shape derivative of f d f(A;V) = Thus, one has to analyze the second summand in more detail
∇G(A)· ∇b
17Due to [151, Eqs. (2.173), (2.163) and (2.169)] there holds
g0(A)[V] =G0(A)[V]|∂A+∂nG(A)V·n. (2.64)
2.4.3 New necessary conditions 55
= ∇G(A)0[V]· ∇b+∇G(A)·(∇b)0[V]
∂A+ ∇ ∇G(A)· ∇b
· ∇b
|∂AV·n
=∇ G0(A)[V]· ∇b
∂A+ ∇G(A)∂A·(∇b)0[V]∂A +D2G(A)∇b· ∇b+D2b∇G(A)· ∇b
| {z }
∇G(A)·D2b∇b=0, sinceD2b∇b=0, cf. [44, p. 372]
∂AV·n
=∂nG0(A)[V] + ∇G(A)∂A·(∇b)0[V]∂A+∂nnG(A). Altogether this results in
d f(A;V) = Z
∂A ∂nnG(A) +κ ∂nG(A)V·n+ Z
∂A∂nG0(A)[V] + ∇G(A)∂A·(∇b)0[V]∂A.18 Another frequent case is that test functions are contained in the integrand. Test functions do not depend on the shape explicitly and henceforth their local shape derivative vanishes. With respect to the partial shape derivative they behave like shape dependent functions (since the explicit shape dependency is neglected then), and consequently the most common situations are covered within the above considered cases.
With these deliberations at hand it is possible to compute the derivative of the Lagrangian. Since all vari-ables of the Lagrangian are independent – in particular, all functions space varivari-ables are independent of the choice of the setB ∈ O19– the (total) shape derivative ofLcoincides with its partial shape derivative
dL(B; . . . ;V) =∂BL(B; . . . ;V).
2.4.3 New necessary conditions
In order to derive first order necessary conditions for the set optimal control problem (2.61), one needs to compute all partial derivatives of the LagrangianLat the optimum (cf. [52, Prop. 1.6 on p. 170])
¯
o:= (A; ¯uI, ¯uA, ¯yI, ¯yA; ¯pI, ¯pA, ¯qA, ¯σI, ¯σA).
The experiences ofSection 2.3say, that the strict inequality constraint (2.61g) has no influence on local op-timality and consequently the cone of admissible directions of variation is not restricted. The Lagrangian is a convex-concave functional with respect to the function space variables. To the best of the author’s knowledge, the qualitative dependency on the set variable cannot be classified but nonlinear, which is due to the underlying manifold structure of the shape space, seeSection 2.6. Nevertheless,Lhas a critical point a the optimum ¯o. Hence, each partial derivative ofLevaluated at ¯ohas a null. These necessary conditions for optimality will be derived in the following.
As a start, regard the derivatives with respect to the control variables.
0= ∂uJL(o¯)(h) = Z
Iλ(u¯I −ud)h+hp¯I, ∀h∈L2(J), 0= ∂uBL(o¯)(h) =
Z
A˚λ(u¯A−ud)h+hp¯A+hq¯A, ∀h∈L2(B)˚ . Consequently, the fundamental lemma of the calculus of variations yields
0=λ(u¯I −ud) +p¯I a. e. inI, (2.68a) 0=λ(u¯A−ud) +p¯A+q¯A a. e. in ˚A. (2.68b)
18The local shape derivative of the gradient of the oriented distance function is given by [44, Chp. 9 Eq. (4.38)]
(∇b)0[V] = (DV)∇b· ∇b
∇b−(DV)>∇b−D2b V. (2.67)
19However,Definition 7reveals on closer examination, that the function spaces of the variables do dependent on the setB. This implicit dependency can be overcome by regarding the variables as the restrictions of some else, which are defined on the holdallΩ. This method offunction space embeddingis used in [44, p. 565ff.]. Alternatively one can regard the Lagrangian as a functional on a vector bundle on a shape related manifold, seeParagraph 2.6.3. It is important to notice however, that the space dependency does not imply a predetermination of function space variables if the set variable is fixed.
The partial derivatives with respect to the state variables yield the adjoint equations with the help of Green’s formula (cf.RemarktoLemma 3)
0= ∂yJL(o¯)(h) = Whereas the derivatives with respect to the multipliers yield the original constraints, as usual, the partial shape derivative of the Lagrangian can be simplified, with the help of different equations as indicated
0=∂B L(o¯);V
Hence, one obtains weak continuity of the optimal control across the interfaceγ
¯
uI|γ=u¯A|γ a. e. onγ. (2.70)
20Use the same reasoning as in the first part of the proof ofLemma 8which yielded (2.49).
57 In summary, the vanishing partial shape derivative of the LagrangianLis compatible with the inequality constraint (2.61f), since weak continuity of the optimal control actually is a necessary condition for the considered optimal control problem, cf. (2.5).
Remark:
Weak continuity of the optimal control across the interface between active and inactive set is the analog to the continuity of the Hamiltonian across junction points of boundary arcs for autonomous problems in OC-ODE, see [122, p. 22 (iii)], [75, Eq. (5.15)].
ThisSectionends with a corollary, in which some principle results of sections2.3and2.4are collected.
Corollary 4:
Let A ∈ Oand (u¯I, ¯uA, ¯yI, ¯yA) ∈ H2(I)×H2(A)˚ ×H2(I)×H2(A)˚ be the unique solution of the set optimal control problems (2.30), and respectively (2.61).
Then there holds:
1. The necessary conditions ofCorollary 3, which were obtained via the reduction technique of Sec-tion 2.3coincide with the saddle point characterizing equations of the Lagrangian, i. e. the equality constraints of (2.61) and (2.68)–(2.70).
2. The necessary conditions are compatible with the strict inequality constraintymin<y¯I <ymaxinI, i. e. (2.30g).
Proof. 1) The first assertion is obvious, except that equation (2.69f) has no analog within the conditions of Corollary 3. However, this equation is unnecessary, since it only determines the additional multiplier ¯σB to be the Neumann trace of ¯pA.
2) The optimal state obviously respects the strict inequality constraint, since the inactive setI is defined such that this condition is fulfilled, cf.Definition 3.