Pontryagin Maximum Principle for Distributed-Order Fractional Systems

We consider distributed-order non-local fractional optimal control problems with controls taking values on a closed set and prove a strong necessary optimality condition of Pontryagin type. The possibility that admissible controls are subject to pointwise constraints is new and requires more sophisticated techniques to include a maximality condition. We start by proving results on continuity of solutions due to needle-like control perturbations. Then, we derive a differentiability result on the state solutions with respect to the perturbed trajectories. We end by stating and proving the Pontryagin maximum principle for distributed-order fractional optimal control problems, illustrating its applicability with an example.

Disarrangements and instabilities in augmented one-dimensional hyperelasticity

In the present work, the overall nonlinear elastic behaviour of a one-dimensional multi-modular structure incorporating possible imperfections at the discrete (microscale) level is derived with respect to both tensile and compressive applied loads. The model is built up through the repetition of n units, each one comprising two rigid rods having equal lengths, linked by means of pointwise constraints capable of elastically limiting motions in terms of relative translations (sliders) and rotations (hinges). The mechanical response of the structure is analysed by varying the number n of the elemental moduli, as well as in the limit case of an infinite number of infinitesimal constituents, in light of the theory of (first-order) structured deformations (SDs), which interprets the deformation of any continuum body as the projection, at the macroscopic scale, of geometrical changes occurring at the level of its sub-macroscopic elements. In this way, a wide family of nonlinear elastic behaviours is generated by tuning internal microstructural parameters, the tensile buckling and the classical Euler's elastica under compressive loads resulting as special cases in the so-called continuum limit —say when n → ∞ . Finally, by plotting the results in terms of the first Piola–Kirchhoff stress versus macroscopic stretch, it is for the first time demonstrated that such SD-based one-dimensional models can be used to generalize some standard hyperelastic behaviours by additionally taking into account instability phenomena and concealed defects.

Total variation regularization of multi-material topology optimization

This work is concerned with the determination of the diffusion coefficient from distributed data of the state. This problem is related to homogenization theory on the one hand and to regularization theory on the other hand. An approach is proposed which involves total variation regularization combined with a suitably chosen cost functional that promotes the diffusion coefficient assuming prespecified values at each point of the domain. The main difficulty lies in the delicate functional-analytic structure of the resulting nondifferentiable optimization problem with pointwise constraints for functions of bounded variation, which makes the derivation of useful pointwise optimality conditions challenging. To cope with this difficulty, a novel reparametrization technique is introduced. Numerical examples using a regularized semismooth Newton method illustrate the structure of the obtained diffusion coefficient.