On the controllability of entropy solutions of scalar conservation laws at a junction via lyapunov methods

In this note, we prove a controllability result for entropy solutions of scalar conservation laws on a star-shaped graph. Using a Lyapunov-type approach, we show that, under a monotonicity assumption on the flux, if u and v are two entropy solutions corresponding to different initial data and same in-flux boundary data (at the exterior nodes of the star-shaped graph), then u ≡ v for a sufficiently large time. In order words, we can drive u to the target profile v in a sufficiently large control time by inputting the trace of v at the exterior nodes as in-flux boundary data for u. This result can also be shown to hold on tree-shaped networks by an inductive argument. We illustrate the result with some numerical simulationsThis work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement NO: 694126-DyCon), the Air Force Office of Scientific Research (AFOSR) under Award NO: FA9550-18-1-0242, the Grant MTM2017-92996-C2-1-R COSNET of MINECO (Spain), the Alexander von Humboldt-Professorship program, the European Unions Horizon 2020 research and innovation programme under the Marie SklodowskaCurie grant agreement No.765579-ConFlex, and the Transregio 154 Project “Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks” of the Deutsche Forschungsgemeinschaft. N. De Nitti is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilita e le loro ` Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). We gratefully acknowledge M. Musch for implementing the numerical simulations of Section 4. We also thank B. Andreianov, J.-A. Barcena-Petisco, G. M. Coclite, C. Donadello, and V. Perrollaz for helpful ´ conversations related to the topic of this work. Finally, we express our gratitude to the anonymous referees for their careful reports, which greatly improved the quality of the manuscrip

Second-order Shape Optimization for Geometric Inverse Problems in Vision

We develop a method for optimization in shape spaces, i.e., sets of surfaces modulo re-parametrization. Unlike previously proposed gradient flows, we achieve superlinear convergence rates through a subtle approximation of the shape Hessian, which is generally hard to compute and suffers from a series of degeneracies. Our analysis highlights the role of mean curvature motion in comparison with first-order schemes: instead of surface area, our approach penalizes deformation, either by its Dirichlet energy or total variation. Latter regularizer sparks the development of an alternating direction method of multipliers on triangular meshes. Therein, a conjugate-gradients solver enables us to bypass formation of the Gaussian normal equations appearing in the course of the overall optimization. We combine all of the aforementioned ideas in a versatile geometric variation-regularized Levenberg-Marquardt-type method applicable to a variety of shape functionals, depending on intrinsic properties of the surface such as normal field and curvature as well as its embedding into space. Promising experimental results are reported

On the global controllability of scalar conservation laws with boundary and source controls

We provide global and semi-global controllability results for hyperbolic conservation laws on a bounded domain, with a general (not necessarily convex)flux and a time-dependent source term acting as a control. The results are achieved for, possibly critical, both continuously differentiable states and BV states. The proofs are based on a combination of the return method and on the analysis of the Riccati equaiton for the space derivative of the solution.Comment: 22 pages, 5 figure

Calculation of chemical and phase equilibria

Bibliography: pages 167-169.The computation of chemical and phase equilibria is an essential aspect of chemical engineering design and development. Important applications range from flash calculations to distillation and pyrometallurgy. Despite the firm theoretical foundations on which the theory of chemical equilibrium is based there are two major difficulties that prevent the equilibrium state from being accurately determined. The first of these hindrances is the inaccuracy or total absence of pertinent thermodynamic data. The second is the complexity of the required calculation. It is the latter consideration which is the sole concern of this dissertation