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An extended beam theory for smart materials applications. II: Static formation problems. (English) Zbl 0911.35039
[cf. part I Appl. Math. Optimization 34, No. 3, 279-298 (1996; Zbl 0856.73026)].
The formability problem for generalised Timoshenko beams is considered using surface and embedded actuators. First the complete potential energy $$V$$ is written down in terms of the vertical and horizontal displacements, Young’s modulus, the shear modulus and the actuator positions. Using Hamilton’s principle, the partial differential equations of the equilibrium position are found, together with the appropriate boundary conditions. Then an abstract approach to existence and uniqueness is given, being based on the Hilbert space $${\mathcal H}=H^2(0,L)\oplus H^1([0,L]\times[0,h])$$. Subspaces of $${\mathcal H}$$ are determined by the appropriate boundary conditions and using a combination of coercivity, the Fredholm alternative and the Lax-Milgram theorem, existence of a unique weak solution is proved.
Approximate formability is shown by using a standard duality approach, although the technical details are quite involved. The periodic case is studied in some detail as it is important in the case of smart materials. Finally, a joint optimisation of the controls and actuator densities in this case is given.

##### MSC:
 35J25 Boundary value problems for second-order elliptic equations 49L99 Hamilton-Jacobi theories 74K10 Rods (beams, columns, shafts, arches, rings, etc.)
##### Keywords:
Timoshenko beams; actuators; unique weak solution
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