This paper provides a derivation of Zipf-Pareto laws directly from the principle of least effort. A probabilistic functional of efficiency is introduced as the consequence of an extension of the ...nonadditivity of the efficiency of thermodynamic engine to a large number of living agents assimilated to engines, all randomly distributed over their output. Application of the maximum calculus to this efficiency yields the Zipf's and Pareto's laws.
In this paper, we solve an optimal control problem for a class of time-invariant switched stochastic systems with multi-switching times, where the objective is to minimise a cost functional with ...different costs defined on the states. In particular, we focus on problems in which a pre-specified sequence of active subsystems is given and the switching times are the only control variables. Based on the calculus of variation, we derive the gradient of the cost functional with respect to the switching times on an especially simple form, which can be directly used in gradient descent algorithms to locate the optimal switching instants. Finally, a numerical example is given, highlighting the validity of the proposed methodology.
The generalized uncertainty relation applicable to quantum and stochastic systems is derived within the stochastic variational method. This relation not only reproduces the well-known inequality in ...quantum mechanics but also is applicable to the Gross–Pitaevskii equation and the Navier–Stokes–Fourier equation, showing that the finite minimum uncertainty between the position and the momentum is not an inherent property of quantum mechanics but a common feature of stochastic systems. We further discuss the possible implication of the present study in discussing the application of the hydrodynamic picture to microscopic systems, like relativistic heavy-ion collisions.
•The variational method for stochastic variables is generalized to the Hamiltonian formulation.•We derived, for the first time, the generalized uncertainty relation applicable to quantum and stochastic systems.•The finite minimum uncertainty is not an inherent property of quantum physics but a common feature in stochastic dynamics.•The possible limitation of the hydrodynamic approach is discussed from the point of view of the finite minimum uncertainty.
Due to non-linear factors such as the rate capacity and the recovery effect, the shape of the battery discharge curve plays a significant role in the overall lifetime of the batteries. Accordingly, ...this paper proposes a simple heuristic battery-aware speed scheduling policy for periodic and non-periodic real-time tasks in Dynamic Voltage Scaling (DVS) systems with non-negligible leakage/static power. A set of comprehensive analysis has been conducted to compare the battery efficiency of the proposed policies with an optimal solution, which could be derived via the Calculus of Variations (CoV). These evaluations have taken into account both periodic and non-periodic tasks in DVS-based systems. Our experiments have shown a maximum of 7% difference between the optimal solution and the simple heuristic speed scheduling for realistic settings of the battery model. By considering the calculated optimal speed scheduling for different tasks (with different utilizations), a two-phase algorithm has been proposed, in which a speed approximation function is being calculated offline based on curve fitting, while the best execution speed is applied online. The results show a maximum of 17.7% and 11.3% battery charge saving for non-periodic and periodic tasks in comparison to the baseline critical frequency method, respectively.
The aim of the paper is to propose a paradigm shift for the variational approach to brittle fracture. Both dynamics and the limit case of statics are treated in a same framework. By contrast with the ...usual incremental approach, we use a space–time principle covering the whole loading and crack evolution. The emphasis is given on the modelling of the crack extension by the internal variable formalism and a dissipation potential as in plasticity, rather than Griffith’s original approach based on the surface area. The new formulation appears to be more fruitful for generalization than the standard theory.
Since the seminal work of Emmy Noether, it is well know that all conservations laws in physics, e.g., conservation of energy or conservation of momentum, are directly related to the invariance of the ...action under a family of transformations. However, the classical Noether’s theorem cannot yield information about constants of motion for non-conservative systems since it is not possible to formulate physically meaningful Lagrangians for this kind of systems in classical calculus of variation. On the other hand, in recent years the fractional calculus of variation within Lagrangians depending on fractional derivatives has emerged as an elegant alternative to study non-conservative systems. In the present work, we obtained a generalization of the Noether’s theorem for Lagrangians depending on mixed classical and Caputo derivatives that can be used to obtain constants of motion for dissipative systems. In addition, we also obtained Noether’s conditions for the fractional optimal control problem.
In the paper, we proposed an approach for studying strong and weak extremums in non-smooth vector problems of calculus of variation, namely, in classic variational problems with fixed ends and with a ...free right end, and also in a variational problem with higher derivatives. The essence of the proposed approach is to introduce a Weierstrass type variation characterized by a numerical parameter. Necessary conditions for minimum containing as corollaries the Weierstrass condition, its local modification and also the Legendre and transversality conditions are obtained. In the case when the Legendre condition degenerates, equality and inequality type necessary conditions are obtained for the weak local minimum. The examples showing the content-richness of the obtained main results are given.
We develop a discretization method for solving the minimal energy configuration of bilayer plates based on a mathematical model developed in Schmidt (2007), Bartels et al. (2017). Our discretization ...method employs C1-spline functions. A highlight of the method involves a trick to handle the nonlinear isometry constraint in such a way that not only numerical integration becomes unnecessary, but also that the final optimization problems are in the form of degree 4 polynomial optimization problems (POP). We develop two different versions of the method, one resulted in a constrained degree 4 POP involving a small tolerance ɛ, another resulted in an unconstrained degree 4 POP involving a large penalty parameter μ. We develop a mathematical analysis, based on the direct method and techniques in Γ-convergence, to show how ɛ and μ can be chosen according to the grid size so that the minimizers of the discrete problems converge to that of the continuum variational problem as the grid size goes to zero. We corroborate the theory through a series of computational experiments, and also report an unexpected finding related to the asymmetry of the discretized problems.
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