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This paper investigates the sensitivity analysis of a scalar mechanical contact problem described by a boundary value problem involving the Tresca's friction law. The sensitivity analysis is performed with respect to right-hand source and boundary terms perturbations. In particular, the friction threshold involved in the Tresca's friction law is perturbed, which constitutes the main novelty of the present work with respect to the existing literature. Hence, we introduce a parameterized Tresca friction problem and its solution is characterized by using the proximal operator associated with the corresponding perturbed nonsmooth convex Tresca friction functional. Then, by invoking the extended notion of twice epi-differentiability depending on a parameter, we prove the differentiability of the solution to the parameterized Tresca friction problem, characterizing its derivative as the solution to a boundary value problem involving Signorini unilateral conditions. Finally, numerical simulations are provided in order to illustrate our main result. [ABSTRACT FROM AUTHOR]

The present work was primarily motivated by our findings in the literature of some flaws within the proof of the second-order Legendre necessary optimality condition for fractional calculus of variations problems. Therefore, we were eager to elaborate a correct proof and it turns out that this goal is highly nontrivial, especially when considering final constraints. This paper is the result of our reflections on this subject. Precisely, we consider here a constrained minimization problem of a general Bolza functional that depends on a Caputo fractional derivative of order 0 < α ≤ 1 and on a Riemann–Liouville fractional integral of order β > 0 , the constraint set describing general mixed initial/final constraints. The main contribution of our work is to derive corresponding first- and second-order necessary optimality conditions, namely the Euler–Lagrange equation, the transversality conditions and, of course, the Legendre condition. A detailed discussion is provided on the obstructions encountered with the classical strategy, while the new proof that we propose here is based on the Ekeland variational principle. Furthermore, we underline that some subsidiary contributions are provided all along the paper. In particular, we prove an independent and intrinsic result of fractional calculus stating that it does not exist a nontrivial function which is, together with its Caputo fractional derivative of order 0 < α < 1 , compactly supported. Moreover, we also discuss some evidences claiming that Riemann–Liouville fractional integrals should be considered in the formulation of fractional calculus of variations problems in order to preserve the existence of solutions. [ABSTRACT FROM AUTHOR]

In biomechanics, recent mathematical models allow one to predict the muscular force response to functional electrical stimulations. The main concern of the present paper is to deal with the computation of optimized electrical pulses trains (for example in view of maximizing the final force response). Using the fact that functional electrical stimulations are modeled as Dirac pulses, our problem is rewritten as an optimal sampled-data control problem, where the control parameters are the pulses amplitudes and the pulses times. We establish the corresponding Pontryagin first-order necessary optimality conditions and we show how they can be used in view of numerical simulations. [ABSTRACT FROM AUTHOR]