European Congress on Computational Methods in Applied Sciences and Engineering

ECCOMAS 2000

Barcelona, 11-14 September 2000 \begin{matrix}c \\ ◯ \end{matrix}ECCOMAS

DESIGN OF EXACTLY CONTROLED STRUCTURES

B. Rousselet^{⋆}

⋆ Laboratoire J. A. Dieudonné, U.M.R. C.N.R.S. n^{0}6621, Université de Nice-Sophia Antipolis, F-06108 Nice Cedex 2, email: br@math.unice.fr \underline{http://www-math.unice.fr}

Key words: Exact control, smart sructures, beam, optimal design, shape optimization

Abstract. We consider dynamic response of mechanical structures; in a first stage we recall basic ideas for exactly controling such a structure: ie how to bring it to rest in a finite time: y(T) = 0 , ẏ(T) = 0. and how to perform a numerical implementation based on ideas of Bourquin and Destuynder. We consider various cases such as beams, elasticity, plates. In a second stage, we adress the optimal design of such a structure; we consider as functional: the energy used to bring the system to rest; the design variable may be for example, the distributed thickness of a beam or of a plane elastic slab; it may be also the shape of the boundary of the stucture. Using ideas of Rousselet (1982) [?, ?], we can prove the design sensitivity of such systems and we obtain a very simple expression of this sensitivity; we present some numerical illustrations.

B. Rousselet

1 INTRODUCTION

The present contribution brings together 3 fields: structural mechanics, optimal design and control. Although 2 of these 3 fields have traditional connections, the interaction of these 3 fields together is quite new. We refer to ([?],[?]) for a presentation of a smart beam model as well as references on the use of piezolelctric devices for reducing structural vibrations. As in this paper we consider exactly controled structures: which means we want to bring a structure to rest in a given time T. A first problem to solve is then controlability. There is a classical algebraic condition for controlability of finite dimensional systems . For distributed systems, this is a difficult mathematical problem which is in progress recently by using the so-called HUM method introduced by J.L. Lions ([?]) see also ([?, ?]) . The general result ot be expected is that a distributed system is exactly controlable for T large enough. In ([?]) this is proved for a smart beam model. For the moment let us describe the control problem for a very simple example: a one degree of freedom sprin mass system. The state equation is:

m\frac{∂^{2}y}{∂t^{2}} + ky = u  (1)

with initial conditions:

y(0) = y_{0} dsurdyt = y_{1}  (2)

We adress the minimization of the functional:

J(u) = \frac{1}{2} ∫_{0}^{T} u^{2}dt  (3)

with the final conditions to be exactly fullfiled:

y(T) = 0 \frac{∂y}{∂T} = 0  (4)

Controlablity means existence of a control u that bring the state from the initial to the final condition. This is assumed troughout this paper. In order to derive necessary optimality conditions for this control problem, 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. To get a tractable necesary condition, it is convenient to consider the state equation with final conditions and enforce the initial conditions with Lagrange multipliers p1, p0.

L(y, u, p, p_{0}, p_{1}) = J(u) − ∫_{0}^{T} p(mÿ + ky − u) − mp1(y(0) − y_{0}) + mp_{0}(ẏ(0) − y_{1}) (5)

Then:

\frac{∂L}{∂u} = 0 ⇒ u + p = 0 the optimality condition  (6)

\frac{∂L}{∂p} = 0 gives back the state equation and \frac{∂L}{∂p_{i}} = 0 gives back the initial conditions.

\frac{∂L}{∂y}z = 0 provides after integration by parts:  (7)
mp̈ + kp = 0 the adjoint state  (8)
p(0) = p_{0} ṗ(0) = p_{1}  (9)

This adjoint state equation is to be solved with u + p = 0 and the state equation with final conditions and initial conditions. The unknown are the initial conditions p_{0}, p_{1} of the adjoint state; we should determine their value so that the initial conditions of the state be satisfied. We introduce the linear maping:

Λ : \binom{p_{0}}{p_{1}} ⟼ \binom{ẏ(0)}{−y(0)}

The given problem amounts to solve the 2 × 2 linear system:

Λ\binom{p_{0}}{p_{1}} = \binom{y_{1}}{−y_{0}}  (10)

It turns out that Λ is symetric and coercitive which implies the controlability.

The optimal design problem is now to minimize the energy provided to the structure by the actuator (??); the design variables to be considered are the mass m and the spring rigidity k. It can be proved:

\frac{∂J}{∂m} = − ∫_{0}^{T} \begin{matrix}... \\ y \end{matrix}p \frac{∂J}{∂k} = − ∫_{0}^{T} yp  (11)

It should be emphasized that the adjoint state p has been introduced for control not for design sensitivity. The situation is somewhat analogous to the design sensitivity of the compliance of a static system which can be derived without any adjoint state.

2 A SMART BEAM MODEL:

2.1 Euler Bernoulli beam model

Without transverse shear, the strain work is:

a_{e}(y, ŷ) = ∫_{0}^{l} EI(e)y_{xx}ŷ_{xx}  (12)

where y denotes the normal deflection and ŷ the virtual normal deplacement. Work of

quantity of accelerations ( neglecting rotational inertia):

A(\frac{∂^{2}y}{∂t^{2}} , ŷ) = ∫_{0}^{l} ρe\frac{∂^{2}y}{∂t^{2}} ŷ  (13)

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

L(ŷ) = Au_{1}ŷ(a) + Bu_{2}ŷ′(a) + ∫_{a}^{b} Cu_{3}ŷ  (14)

Principle of virtual work

A(\frac{∂^{2}y}{∂t^{2}} , ŷ) + a_{e}(y, ŷ) = L(ŷ)  (15)

with initial conditions:

y(0, x) = y_{0}(x) \frac{∂y}{∂t} (0, x) = y_{1}(x)  (16)

2.2 Control

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)  (17)

∫_{0}^{T} ∫_{0}^{l} ρe\frac{∂^{2}y}{∂t^{2}} ϕ + ∫_{0}^{T} ∫_{0}^{l} EI(e)y_{xx}ϕ_{xx} =  (18)
∫_{0}^{T} Au_{1}ϕ(a) + Bu_{2}ϕ′(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} Cu_{3}^{2}  (19)

We minimize J with the final conditions.

y(T, x) = 0 \frac{∂y}{∂t} (T, x) = 0  (20)

Controlability has been adressed by Ph. Destuynder ([?]). 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 inforce initial conditions. The lagrangian is written:

L(y, u; p, p_{0}, p_{1}) = J(u) −  (21)

∫_{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(a) + Bu_{2}p′(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))

then, \frac{∂L}{∂u_{i}} = 0 yields:

u_{1} = −Ap(a) u_{2} = −Bp′(a) u_{3} = −Cp, x ∈ [a, b]  (22)

\frac{∂L}{∂p} = 0 gives back the state equation \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  (23)

with initial conditions: p(0, x) − p_{0}(x) = 0 \frac{∂p}{∂t}(0, x) − p_{1}(x) = 0 and boundary conditions: p(t, x) = 0 \frac{∂p}{∂ν} = 0 x ∈ [0, T] × [0, l]

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)}  (24)

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). Lemme 1

(p̃_{0}, p̃_{1})Λ\binom{p_{0}}{p_{1}} =  (25)

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

∫_{0}^{T} Au_{1}p(a) + Bu_{2}p′(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}} =  (26)

\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.

Sensitivity of the smart beam It is natural to minimize the energy provided by the actuators to the structure; it is given by (??). the design variable is the thickness e which depends on x. Proposition 0.1

\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 differrentiated state equation in which we set ϕ = p and integration by parts. Justification is analogous to the case without control see ([?, ?, ?]).

3 CONTROL AND SHAPE OPTIMIZATION FOR WAVE EQUATION

This is just an introductory example; the state equation.

\frac{∂y}{∂t} − Δy = uχ_{ω}  (27)
y(0, x) = y_{0}(x) \frac{∂y}{∂t} (0, x) = y_{1}(x)  (28)
y = 0 (t, x) ∈ [0, T] × δΩ  (29)

the control problem is to minimize:

J(u) = \frac{1}{2} ∫_{0}^{l} ∫_{ω} u^{2}  (30)

We assume that T is large enough for the system to be controlable. We set the problem in a space of functions which satisfy the final conditions; we enforce initial conditions with Lagrange multipliers.

L(y, u; p, p_{0}, p_{1}) = \frac{1}{2} ∫_{0}^{l} ∫_{ω} u^{2}  (31)

∫_{0}^{l} ∫_{Ω} p\frac{∂^{2}y}{∂t^{2}} + ∫_{0}^{T} ∫_{Ω} ∇p.∇y + puχ_{ω} +

∫_{Ω} p_{1}(y(0, x) − y_{0}(x)) − p_{0}(\frac{∂y}{∂t} (0, x) − y_{1}(x))

then , \frac{∂L}{∂u} = 0 yields

u + p = 0 (t, x) ∈ [0, T] × ∂ω  (32)

\frac{∂L}{∂p} = 0 gives back the state equation and

\frac{∂L}{∂y}z = 0 yields the adjoint state equation.

Lemme 2

(p̃_{0}, p̃_{1})Λ\binom{p_{0}}{p_{1}} =  (33)

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

∫_{0}^{T} ∫_{Ω} p̃pχ_{ω}

3.0.1 Shape optimization

We minimize the energy (??) provided to the structure by the actuator; the design variable is the shape of the boundary; the sensitivity of this functional is given by: Proposition 0.2

δJ = ∫_{0}^{T} ∫_{Ω} pδpχ_{ω} + ∫_{0}^{T} ∫_{Ω} p^{2}χ_{ω}divF =
∫_{0}^{T} ∫_{Ω}((DF + DF^{t})∇y, ∇p) +
∫_{0}^{T} ∫_{Ω}(∇y, ∇p)divF +
∫_{0}^{T} ∫_{Ω} \frac{∂^{2}y}{∂t^{2}} divF  (34)

The proof uses the differentiated state equation in which we set ϕ = p and integration by parts. Justification is analogous to the case without control see ([?, ?, ?]).

4 CONCLUSION

We have presented a natural functional to minimize for control and design of an exactly controled structure. The design variable may be a coefficient describing properties of the structure, its variable thickness or the shape of the boundary; the topology sensitivity is in preparation. We emphasize for this functional the design sensitivity may be expressed without the use of any adjoint state additional to the one used for control of the structure. The situation is similar to the design of static structures where the compliance of the structure is considered: its design sensitivity may be expressed without any adjoint state. Numerical experiments are in progress following for the control, the approach of ([?, ?]).

RÉFÉRENCES

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[3] Ph. Destuynder, A. Saidi, smart materials and flexible structures, Control and cybernetyics, 26, 2, 161-205 (1997).

[4] Ph. Destuynder, A mathematical analysis of a smart-beam which is equipped with piezoelectric actuators.. Control & cybernetics, 28, 503-530, (1999).

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[6] B. Rousselet Quelques résultats en optimisation de domaines, thèse d’état, Université de Nice, France, 1982.

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[8] B. Rousselet, Static and dynamic loads, pointwise constraintin structural optimization, pp 225-238, Optimization, theory and optimization, Hiriart-Urruty & Oettli & Stoer eds, Lect. notes in pure & applied math., Marcel Dekker, (1983).