Sensitivity of dynamic structures; case of a smart beam

B. Rousselet UNSA

juin 2000

1 Introduction

As we said in [?], the idea of well posed problems of mathematical physics was introduced and sudied by Hadamard in the 1920s [?]. The term ’well posed’ for initial and boundary-value problems in the modern literature has come to be defined to mean that (i) the problem has a solution, (ii) the solution is unique, (iii) the solution depends continuously on data defining the problem. The first two parts of this definition have a precise meaning but the third is indefinite as to what constitutes ’data’ . Continuity of the solution with respect to data is a classical theme; for evolution equation this is usually restricted to continuity with respect to initial data and right hand side; however we are here interested in continuty with respect to coefficients appearing in the bilinear forms (or in the coefficient of the operators involved in the partial differential equations) or in the shape of the domain Ω in which which the system is posed. Many papers have adressed the sensitivity of static structures: see for examples [?], [?]; a case of static structures with geometric non linearities and instabilities has been considered in [?], [?]; a case of a thermal problem in [?], [?], [?]. From a mathematical point of view, as noted previously in [?], the classical implicit function theorem is not applicable for equations of Petrowsky type; this is due to the following fact: with a right hand side in L^{2}(0, T, L^{2}(Ω)), the solution is not in gneral in L^{2}(0, T, H^{2m}(Ω)), see for example in paragraph 5.1.3 of [?]. We present here some abstract lipschitz continuity and differentiablity results which would lead to the same kind of properties for shape sensitivity (already considered in [?], [?]) or thickness sensitivity (see [?]) of dynamically loaded structures. A recent related result may be found in [?]. But the goal of this paper is really to start to adress the sensitivity of controlled systems. Shape sensitivity of piezoelectric wafers mounted on a plate has been considered in [?] for the approximation with a finite number of modes. Here we try to adress the sensitivity of the exactly controled continuous system. The term exactly controled means that the control actuator should bring the system to rest in a given finite time T such that y(T, x) = 0, y_{t}(T, x) = 0. This is somewhat different of stablizing the system and has been a field of active research for systems modeled by partial differentiable equations since the work of J.L. Lions ([?] and numerical approximations and implementations ([?, ?]); the use of eigenfunctons for exact control (see [?]) enables to compute exact control within a reasonable computing time and so the optimal design of such exactly controled systems becomes also reasonable. In this paper we restrict the approach to a model of smart beam used in [?] but the approach is very general and can be applied quite easily for shape sensitivity of exactly controled systems; such extensions are in preparation.

2 Second order evolution equation

We denote Ω, a bounded open subset of ℝ^{n}; it will be usually with a C^{∞} boundary Γ and situated locally on one side of Γ. We consider a Hilbert space V (Ω) with H_{0}^{m}(Ω) ⊆ V (Ω) ⊆ H^{m}(Ω); m is also the order of derivatives appearing in the bilinear form a(y, p); it is assumed to be symmetric, continuous, coercive on V. It depends on design variable h which may be either a parameter appearing in the equation such as thickness of a structure of variable thickness or the shape of the boundary Γ. For equations arising in solid mechanics, a denotes virtual work of internal forces, c, the virtual work of inertial forces and (f, p), the virtual work of external forces with p the virtual displacement. The folowing hypothesis will be assumed throughout the paper. Hypothesis 1. a is a symmetric bilinear form coercive on V Hypothesis 2. c is a symmetric bilinear form coercive on L^{2}(Ω) Hypothesis 3. for simplicity, we assume that the bilinear forms are not time dependent. We consider the abstract equation:

\frac{d}{dt}c(\frac{∂y}{∂t} , p) + a(y, p) = (f, p)  (2.1)

We recall the following theorems from [?]; in fact we are mainly recalling a priori inequalities which are in the proof of the quoted theorems. Theorem 2.1. Theorem 3 .8.1 of [?]

∥y∥_{V}^{2} + ∥\frac{∂y}{∂t} ∥_{L^{2}(Ω)}^{2} ≤ c (∥y_{0}∥_{V}^{2} + ∥y_{1}∥_{L^{2}(Ω)}^{2} + ∫_{0}^{T} ∥f(t)∥_{L^{2}(Ω)}^{2})  (2.2)

Theorem 2.2. (from proof of Theorem 3 .9.3 of [?])

∥y∥_{L^{2}(Ω)}^{2} + ∥\frac{∂y}{∂t} ∥_{V′}^{2} ≤ c (∥y_{0}∥_{L^{2}(Ω)}^{2} + ∥y_{1}∥_{V′}^{2} + ∫_{0}^{T} ∥f(t)∥_{V′}^{2})  (2.3)

Theorem 2.3. (from proofs of theorem 5.2.1 and lemma 5.2.1 of [?])

∥\frac{∂y}{∂t} ∥_{H^{m}(Ω)}^{2} + ∥\frac{∂^{2}y}{∂t^{2}} ∥_{L^{2}(Ω)}^{2} ≤ c [∥y_{0}∥_{H^{2m}(Ω)}^{2} + ∥y_{1}∥_{H^{m}(Ω)}^{2} + ∫_{0}^{T} ∥\frac{∂f}{∂t} ∥_{L^{2}(Ω)}^{2}] (2.4)

∥y∥_{H^{2m}(Ω)}^{2} ≤ c [∥y_{0}∥_{H^{2m}(Ω)}^{2} + ∥y_{1}∥_{H^{m}(Ω)}^{2} + ∫_{0}^{T} ∥f(t)∥_{L^{2}(Ω)}^{2} + ∫_{0}^{T} ∥\frac{∂f}{∂t} ∥_{L^{2}(Ω)}^{2}] (2.5)

Corollary 2.1. With zero initial data:

∥\frac{∂y}{∂t} ∥_{H^{2m}(Ω)}^{2} ≤ c [∫_{0}^{T} ∥f(t)∥_{H^{m}(Ω)}^{2} + ∫_{0}^{T} ∥\frac{∂^{2}f}{∂t^{2}} ∥_{L^{2}(Ω)}^{2}]  (2.6)

The proof is based on theorem 5.7.1 of [?] and the use of theorem of intermediate derivatives (theorem 1.2.3 of [?]).

3 Lipschitz continuity of the solution with respect to design variables

This section is devoted to Lipschitz continuity of the solution with respect to design variables while the following is considering differentiability with respect to design variables. Two situations will be considered one with additional regularity on the data, one with basic regularity on the data. notations: We denote h, the design variable; a_{h}(z, p), the bilinear form for the value h of the design variable; δa = a_{h+δh} − a_{h} and similarly δy = y_{h+δh} − y_{h}, δf = f_{h+δh} − f_{h}

3.1 With basic regularity

Hypothesis 4 (Lipschitz 1).

f ∈ L^{2}(0, T, L^{2}(Ω)), y_{0} ∈ V, y_{1} ∈ L^{2}(Ω) (3.1)
∀z ∈ V ∀p ∈ V |δa(z, p)| ≤c∥δh∥_{X}∥z∥_{V} ∥p∥_{V}  (3.2)
∥δCz∥_{V′} ≤ c∥δh∥_{X}∥z∥_{V′} ∀p ∈ V |(δf, p)| ≤c∥δh∥_{X}∥p∥_{V}  (3.3)

Proposition 3.1. With hypothesis ?? we have:

∥δy∥_{L^{2}(Ω)}^{2} + ∥δẏ∥_{V′}^{2} ≤ c∥δh∥_{X}^{2} ∫_{0}^{T} ∥y∥_{V}^{2}  (3.4)

or h ⟼ y is lipschitzian from X to L^{∞}(0, T, L^{2}(Ω)) and h ⟼ ẏ is lipschitzian from X to L^{∞}(0, T, V′).

3.2 With additional regularity

Hypothesis 5 (Lipschitz 2).

f ∈ L^{2}(0, T, L^{2}(Ω)), \frac{∂f}{∂t} ∈ L^{2}(0, T, L^{2}(Ω)) y_{0} ∈ H^{2m}(Ω), y_{1} ∈ H^{m}(Ω) (3.5)

∀z ∈ H^{2m}(Ω) ∥δAz∥_{L^{2}(Ω)} ≤c∥δh∥_{X}∥z∥_{H^{2m}(Ω)}  (3.6)
|(δf, p)| ≤c∥δh∥_{X}∥p∥_{L^{2}(Ω)}  (3.7)
∥δCz∥_{L^{2}(Ω)} ≤c∥δh∥_{X}∥z∥_{L^{2}(Ω)}  (3.8)

Proposition 3.2. With hypothesis ?? we have:

∥δy∥_{H^{m}(Ω)}^{2} + ∥δẏ∥_{L^{2}(Ω)}^{2} ≤ c∥δh∥_{X}^{2} ∫_{0}^{T} ∥y∥_{H^{2m}(Ω)}^{2}  (3.9)

or e ⟼ y is lipschitzian from X to L^{∞}(0, T, V) and e ⟼ ẏ is lipschitzian from X to L^{∞}(0, T, L^{2}(Ω))

With still some more regularity, we get the following proposition; it was proved in the context of shape sensitivity in [?], chapter 4 and presented in [?],[?]. We state it here in an abstract setting. Hypothesis 6 (Lipschitz 3).

f ∈ L^{2}(0, T, L^{2}(Ω)), \frac{∂f}{∂t} ∈L^{2}(0, T, L^{2}(Ω)), \frac{∂^{2}f}{∂t^{2}} ∈ L^{2}(0, T, L^{2}(Ω)) (3.10)

y_{0} ∈ H^{2m}(Ω), y_{1} ∈ H^{m}(Ω)  (3.11)
∀z ∈ H^{2m}(Ω) ∥δAz∥_{L^{2}(Ω)} ≤c∥δh∥_{X}∥z∥_{H^{2m}(Ω)}  (3.12)
|(δf, p)| ≤c∥δh∥_{X}∥p∥_{L^{2}(Ω)}  (3.13)
|(δ \frac{∂f}{∂t} , p)| ≤c∥δh∥_{X}∥p∥_{L^{2}(Ω)}  (3.14)
∥δCz∥_{L^{2}(Ω)} ≤c∥δh∥_{X}∥z∥_{L^{2}(Ω)}  (3.15)

Proposition 3.3. With hypothesis ?? we have:

∥δy∥_{H^{2m}(Ω)}^{2} + ∥δẏ∥_{H^{m}(Ω)^{2}} + ∥δÿ∥_{L^{2}(Ω)}^{2} ≤ c∥δh∥_{X}^{2} ∫_{0}^{T} (∥y∥_{H^{2m}(Ω)}^{2} + ∥ẏ∥_{H^{2m}(Ω)}^{2}) (3.16)

or e ⟼ y is lipschitzian from X to L^{∞}(0, T, V) and e ⟼ ẏ is lipschitzian from X to L^{∞}(0, T, L^{2}(Ω)) Proof of previous propositions. We denote with ˙ the time derivative. The proof is based on the simple remark that δy is solution of:

∀p ∈ V c(δÿ, p) + a(δy, p) = −δc(ÿ_{h+δh}, p) − δa(y_{h+δh}, p) + (δf, p) (3.17)

For proposition ??, we first use theorem ??; we obtain that y is in L^{2}(0, T, V) so that the right hand side of ?? is in L^{2}(0, T, V′). Now we apply estimation (??) to equation ??; the initial conditions of equation ?? are zeros so that:

∥δy∥_{L^{2}(Ω)}^{2} + ∥δẏ∥_{V′}^{2} ≤ c ∫_{0}^{T} ∥δAy∥_{V′}^{2} + ∥δf∥_{V′}  (3.18)

with hypothesis ?? (Lipschitz 1) we get (??). For proposition ??, we first use theorem ??; we obtain that y is in L^{2}(0, T, H^{2m}(Ω)) so that the right hand side of (??) is in L^{2}(0, T, L^{2}(Ω)); Now we apply estimation ?? to equation ??;the initial conditions of equation ?? are zeros so that:

∥δy∥_{H^{m}(Ω)}^{2} + ∥δẏ∥_{L^{2}(Ω)}^{2} ≤ c ∫_{0}^{T} ∥δAy∥_{L^{2}(Ω)}^{2} + ∥δf∥_{L^{2}(Ω)}  (3.19)

For proposition ?? , we first use theorem ?? ; we obtain that y is in L^{2}(0, T, H^{2m}(Ω)); with ??, we get also that ẏ is in L^{2}(0, T, H^{2m}(Ω)); now we get that the right hand side of (??) is in L^{2}(0, T, L^{2}(Ω)) and also its time derivative; now we use again theorem ?? to obtain:

∥δy∥_{H^{2m}(Ω)}^{2} + ∥δẏ∥_{H^{m}(Ω)}^{2} + ∥δÿ∥_{L^{2}(Ω)}^{2} ≤  (3.20)
c∥δh∥_{X}^{2} ∫_{0}^{T} {∥δAy∥_{L^{2}(Ω)}^{2} + ∥δCÿ∥_{L^{2}(Ω)}^{2} + ∥δf∥_{L^{2}(Ω)}^{2} +  (3.21)
∥δAẏ∥_{L^{2}(Ω)}^{2} + ∥δCÿ′∥_{L^{2}(Ω)}^{2} + ∥δḟ∥_{L^{2}(Ω)}^{2}}  (3.22)

and we conclude with the estimations assumed in hypothesis ?? of the proposition.

4 Differentiability with respect to design variables

Let us first consider equation ?? formally differentiated with respect to an abstract design variable h ∈ X. We denote da(z, p) = \frac{∂a(z,p)}{∂h}dh where the derivative is considered with z and p being fixed; simalarly dy = \frac{∂y}{∂h}dh denotes a differential considered with x being fixed; it is the derivative of h ⟼ y_{h}(x).

∀p ∈ V c(dÿ, p) + a(dy, p) = −dc(ÿ, p) − da(y, p) + (df, p)  (4.1)

Let us first consider this equation with hypothesis ??; with this assumtion theorem ?? states that y ∈ V and ÿ ∈ V′; the right hand side of ?? satisfies hypothesis suitable for theorem ?? and we get that this equation has a solution dy ∈ L^{2}(Ω) and dẏ ∈ V′. Unfortunately, the proof, with the hypothesis ?? that dy is indeed the differential of h ⟼ y_{h}(x) would rely on estimates which are not classical. It will be possible to prove it under hypothesis ??. Hypothesis 7. we assume hypothesis ?? and moreover

∀z ∈ H^{2m}(Ω) ∥dAz∥_{L^{2}(Ω)} ≤c∥dh∥_{X}∥z∥_{H^{2m}(Ω)}  (4.2)
∥dCz∥_{L^{2}(Ω)} ≤c∥dh∥_{X}∥z∥_{L^{2}(Ω)}  (4.3)
∥δ^{2}Az∥_{L^{2}(Ω)} ≤c∥dh∥_{X}^{2}∥z∥_{H^{2m}(Ω)}  (4.4)
∥δ^{2}Cz∥_{L^{2}(Ω)} ≤c∥dh∥_{X}^{2}∥z∥_{L^{2}(Ω)}  (4.5)

(4.6)

Theorem 4.1. Under hypothesis ??, the solution of ?? is the differential of

h ⟼ y h ⟼ ẏ  (4.7)
X ⟶ V X ⟶ V′  (4.8)

Proof starts by expanding all quantities up to second order: for example, the bilinear form: a_{h+dh} = a_{h} + da + δ^{2}a; we consider equation ?? for the value h + dh of the design variable and expand it up to second order; we simplify the expansion with equation ?? for the value h of the design variable and with the differentiated equation ??; we obtain after some manipulations:

∀p ∈ V c_{h}(δ^{2}ÿ, p) + a_{h}(δ^{2}y, p) =  (4.9)
−dc(δ̈y, p) − da(δy, p) + (δ^{2}f, p) − δ^{2}c(ÿ_{h+dh}, p) − δ^{2}a(y_{h+dh}, p) (4.10)

If we look at the right hand side, we notice that hypothesis ?? provides δy ∈ L^{2}(Ω) so that dAδy ∈ dAL^{2}(Ω); the latter space is larger than V′; however it seems that the solution of this equation would be well defined by using transposition theory (see [?]); however estimates of the solution does not seem standard so that we restrict the situation to hypothesis ??; with this hypothesis, we know with proposition ?? that y, δy ∈ H^{2m}(Ω) so that the right hand side is L^{2}(Ω) and using theorem ?? :

∥δ^{2}y∥_{V}^{2} + ∥δ^{2}ẏ∥_{L^{2}(Ω)}^{2} ≤  (4.11)

c ∫_{0}^{T} (∥dCδÿ∥_{L^{2}(Ω)}^{2} + ∥Aδy∥_{L^{2}(Ω)}^{2} + ∥δ^{2}Cÿ_{y+dh}∥_{L^{2}(Ω)}^{2} + ∥δ^{2}Ay_{h+dh}∥_{L^{2}(Ω)}^{2})

and we conclude with the estimates of the hypothesis ??

5 A smart beam model:

5.1 Euler Bernoulli beam model

Without transverse shear, the strain work is:

a_{e}(y, ŷ) = ∫_{0}^{l} EI(e)y_{xx}ŷ_{xx}  (5.1)

where y denotes the normal deflection and ŷ the virtual normal deplacement. Work of quantity of accelerations ( neglecting rotational inertia):

c_{e}(\frac{∂^{2}y}{∂t^{2}} , ŷ) = ∫_{0}^{l} ρe\frac{∂^{2}y}{∂t^{2}} ŷ  (5.2)

Work of external control forces: u_{1}, u_{2}, u_{3}:

L(ŷ) = Au_{1}ŷ(t, a) + Bu_{2} \frac{∂ŷ}{∂t}(t, a) + ∫_{a}^{b} Cu_{3}ŷ  (5.3)

This is a possible mathematical model of the action of a piezoelectric wafer on a beam as proposed in [?]. Let H_{0}^{2}(Ω) ⊂ V ⊂ H^{2}(Ω). Principle of virtual work

∀ŷ ∈ V c(\frac{∂^{2}y}{∂t^{2}} , ŷ) + a_{e}(y, ŷ) = L(ŷ)  (5.4)

with initial conditions:

y(0, x) = y_{0}(x), \frac{∂y}{∂t} (0, x) = y_{1}(x)  (5.5)

Lipschitz continuty and differentiability results of the previous section may be applied easily to this model.

5.2 Control

If u_{1}, u_{2} ∈ L^{2}(0, T) u_{3} ∈ L^{2}(0, T, L^{2}(Ω)), y_{0} ∈ L^{2}(Ω), y_{1} ∈ V′ as a consequence of theorem ??, the solution of (??) is in V^{0} where

V^{0} = {ϕ ∈ L^{2}(0, T, L^{2}(Ω)), ϕ̇ ∈ L^{2}(0, T, V′)}  (5.6)

Moreover if A = 0 B = 0 y_{0} ∈ V, y_{1} ∈ L^{2}(Ω), as a consequence of theorem ?? the solution of ?? is in V^{1} where

V^{1} = {ϕ ∈, L^{2}(0, T, V), ϕ̇ ∈ L^{2}(0, T, L^{2}(Ω))}  (5.7)

For control purposes, it is convenient to write the equations in weak form with a test function that depends on time and space; state equation with initial conditions:

y(0, x) = y_{0}(x) \frac{∂y}{∂t} (0, x) = y_{1}(x)  (5.8)

∫_{0}^{T} ∫_{0}^{l} ρe\frac{∂^{2}y}{∂t^{2}} ϕ + ∫_{0}^{T} ∫_{0}^{l} EI(e)y_{xx}ϕ_{xx} =  (5.9)
∫_{0}^{T} Au_{1}ϕ(t, a) + Bu_{2}ϕ_{x}(t, a) + ∫_{0}^{T} ∫_{a}^{b} Cu_{3}ϕ

Fonctional to minimize:

J(v) = \frac{1}{2} ∫_{0}^{T} (u_{1}^{2} + u_{2}^{2}) + \frac{1}{2} ∫_{0}^{T} ∫_{a}^{b} u_{3}^{2}  (5.10)

We minimize J with the final conditions.

y(T, x) = 0 \frac{∂y}{∂t} (T, x) = 0  (5.11)

Controlability of plates and beam have been adressed by Ph. Destuynder ([?]); here we assume that there exists u such that (??) is satisfied. In order to derive necessary optimality conditions, we use a Lagrangian; this is in contrast with Ph. Destuynder ([?]) and the introduction of exact controlabity by J.L. Lions ([?]) where penalty arguments are used. We consider the state equation with final conditions and use lagrange multipliers to enforce initial conditions; to simplify we assume that V = H_{0}^{2}(Ω): mechanically it means that the beam is clamped at both ends. The Lagrangian is written:

L(y, u; p, p_{0}, p_{1}) = J(u) −  (5.12)

∫_{0}^{T} ∫_{0}^{l} ρe\frac{∂^{2}y}{∂t^{2}} p − ∫_{0}^{T} ∫_{0}^{l} EI(e)y_{xx}p_{xx} +

∫_{0}^{T} {Au_{1}p(t, a) + Bu_{2}p_{x}(t, a)} + ∫_{0}^{T} ∫_{a}^{b} Cu_{3}p +

∫_{0}^{l} \{p_{1}[y(0, x) − y_{0}(x)] − p_{0}[\frac{∂y}{∂t} (0, x) − y_{1}(x)]\}  (5.13)

with

y ∈ V, y(T, x) = 0 = ẏ(T, x)  (5.14)
u = (u_{1}, u_{2}, u_{3}) ∈ L^{2}(0, T) × L^{2}(0, T) × L^{2}(0, T, L^{2}(Ω)), p ∈ V (5.15)

then, \frac{∂L}{∂u_{i}} = 0 yields:

u_{1}(t) = −Ap(t, a), u_{2}(t) = −B \frac{∂p}{∂x}(t, a), and for x ∈ [b, c], u_{3} = −Cp (5.16)

\frac{∂L}{∂p} = 0 gives back the state equation and \frac{∂L}{∂y}z = 0 yields the adjoint state equation:

∫_{0}^{T} ∫_{0}^{l} ρe\frac{∂^{2}p}{∂t^{2}}z + ∫_{0}^{T} ∫_{0}^{l} EI(e)y_{xx}p_{xx} = 0  (5.17)

with initial conditions: ρep(0, x) − p_{0}(x) = 0 ρe\frac{∂p}{∂t}(0, x) − p_{1}(x) = 0 and boundary conditions: p(t, x) = 0 \frac{∂p}{∂x} = 0 x ∈ [0, T] × \{0, l\} as we have assumed for simplicity that the beam is clamped at both ends. To study the necesary conditions, following J.L. Lions ([?]), we introduce:

Λ : \binom{p_{0}}{p_{1}} ⟼ \binom{\frac{∂y}{∂t} (0, x)}{−y(0, x)}  (5.18)

The choice of the order of the target insures symetry and coercivity of Λ; and this implies controlability (ie existence of a control which brings the state from given initial conditions to final conditions). The exact control is given by (??) with p_{0}, p_{1} being solution of:

Λ\binom{p_{0}}{p_{1}} = \binom{y_{1}}{−y_{0}}  (5.19)

Lemme 5.1.

(p̃_{0}, p̃_{1})Λ\binom{p_{0}}{p_{1}} =  (5.20)

∫_{0}^{l} −p̃_{1}y_{0}(x) + p̃_{0} \frac{∂y}{∂t} (0, x) =

∫_{0}^{T} Aũ_{1}p(t, a) + Bũ_{2}p_{x}(t, a) + ∫_{0}^{T} ∫_{0}^{l} Cu_{3}p

and

J(u_{1}, u_{2}, u_{3}) = \frac{1}{2}(p_{0}, p_{1})Λ\binom{p_{0}}{p_{1}} =  (5.21)

\frac{1}{2} ∫_{0}^{T} A^{2}p(a)^{2} + B^{2}p′(a)^{2} + ∫_{0}^{T} ∫_{0}^{l} C^{2}p^{2}

This identity is derived from the adjoint state equation multiplied by y and integrated by parts.

5.3 Sensitivity of a smart beam

It is natural to minimize the energy provided by the actuators to the structure; it is given by (??) and more explicitely by (??). In other words, we try to minimize with respect to design, the functional already minimized with respect to control. The design variable is the thickness e which depends on x. In the following proposition, it should be emphasized that the design sensitivity involves the adjoint state used for control but no adjoint state for design sensitivity; in this respect the situation is analogous to the one encountered when we consider the sensitivity of the compliance of a static structure. Proposition 5.1.

\frac{∂J}{∂e} ẽ = − ∫_{0}^{T} ∫_{0}^{l} ẽρ\frac{∂^{2}y}{∂t^{2}} p − ∫_{0}^{T} ∫_{0}^{l} E \frac{∂I(e)}{∂e}ẽy_{xx}p_{xx}

The proof uses the differentiated state equation in which we set ϕ = p and integration by parts. Justification relies on the case without control presented in section ?? ; shape and stuctural sensitivity of dynamic structures was presented first in ([?, ?, ?]). For design sensitivity, the equations to be considered are the state equation (??) of the control problem in which we replace the control by its expression (??) , the adjoint state equation (??), and equation (??).

∫_{0}^{T} ∫_{0}^{l} ρe\frac{∂^{2}y}{∂t^{2}} ϕ − ∫_{0}^{T} ∫_{0}^{l} EI(e)y_{xx}ϕ_{xx} +

∫_{0}^{T} {A^{2}p(t, a)ϕ(t, a) + B^{2}p_{x}(t, a)ϕ_{x}(t, a)} + ∫_{0}^{T} ∫_{a}^{b} Cpϕ = 0 (5.22)
y(T, x) = 0 = ẏ(T, x)  (5.23)
∫_{0}^{T} ∫_{0}^{l} ρe\frac{∂^{2}p}{∂t^{2}}z + ∫_{0}^{T} ∫_{0}^{l} EI(e)y_{xx}p_{xx} = 0  (5.24)
ρep(0, x) − p_{0}(x) = 0, ρe\frac{∂p}{∂t}(0, x) − p_{1}(x) = 0,  (5.25)
p(t, x) = 0, \frac{∂p}{∂x} = 0 for (t, x) ∈ [0, T] × \{0, l\}  (5.26)

Λ\binom{p_{0}}{p_{1}} = \binom{y_{1}}{−y_{0}}equivalent to y(0, x) = y_{0}(x) ẏ(0, x) = y_{1}(x)

(5.27)

We aknowledge that the sensitivity of the state equation is justified only in the case A = 0 = B by using theorem ??; then the righ hand side is in L^{2}(0, T, L^{2}(Ω)). Nethertheless the computation is formally the same. The proof of the general case is in preparation. The differentiated state equation is given by:

∫_{0}^{T} ∫_{0}^{l} ρe\frac{∂^{2}dy}{∂t^{2}} ϕ − ∫_{0}^{T} ∫_{0}^{l} EI(e)dy_{xx}ϕ_{xx} =

− ∫_{0}^{T} {A^{2}p(t, a)dp(t, a) + B^{2}p_{x}(t, a)dp_{x}(t, a)} − ∫_{0}^{T} ∫_{a}^{b} Cpϕ −(5.28)

∫_{0}^{T} ∫_{0}^{l} ρde\frac{∂^{2}y}{∂t^{2}} ϕ − ∫_{0}^{T} ∫_{0}^{l} EdI(e)y_{xx}ϕ_{xx}  (5.29)

(5.30)

In this equation we set ϕ = p, integrate by parts in time, use the adjoint state equation in which we set z = dy; we notice that

dy(T, x) = 0, dẏ(T, x), dy(0, x) = 0, dẏ(0, x) = 0  (5.31)

we get:

∫_{0}^{T} [A^{2}pdp + B^{2}p_{x}dp_{x} + ∫_{0}^{T} ∫_{b}^{c} pdp = − ∫_{0}^{T} ∫_{0}^{l}(ρde ẏp + EdI \frac{∂^{2}y}{∂x^{2}} \frac{∂^{2}p}{∂x^{2}} ) (5.32)

the left hand side is the derivative of the functional and the right hand side is the desired expression entirely explicit with respect to the design variable.

6 Conclusion

We have presented a general abstract approoach for design sensitivity of systems governed by variational second order evolution equations; it encompasses some previous results of the author and is widely applicable. Its use for design sensitivity of a model of a smart beam has been presented for illustrations. Other cases of application are in preparation.

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