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

A singular controllability problem with vanishing viscosity

The aim of this paper is to answer the question: Do the controls of a vanishing viscosity approximation of the one dimensional linear wave equation converge to a control of the conservative limit equation? Our viscous term contains the fractional power of the Dirichlet Laplace operator and it is multiplied by a small parameter devoted to tend to zero. Our analysis, based on moment problems and biorthogonal sequences, enables us to evaluate the magnitude of the controls needed for each eigenmode and to show their uniform boundedness with respect to the vanishing parameter

Analysis and control of a scalar conservation law modeling a highly re-entrant manufacturing system

In this paper, we study a scalar conservation law that models a highly re-entrant manufacturing system as encountered in semi-conductor production. As a generalization of \cite{CKWang}, the velocity function possesses both the local and nonlocal character. We prove the existence and uniqueness of the weak solution to the Cauchy problem with initial and boundary data in $L^{\infty}$. We also obtain the stability (continuous dependence) of both the solution and the out-flux with respect to the initial and boundary data. Finally, we prove the existence of an optimal control that minimizes, in the $L^p$-sense with $p\in [1,\infty)$, the difference between the actual out-flux and a forecast demand over a fixed time period.Comment: 32 pages, 12 figure

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

Filtered gradient algorithms for inverse design problems of one-dimensional burgers equation

The final publication is available at Springer via https://doi.org/10.1007/978-3-319-49262-9_7Inverse design for hyperbolic conservation laws is exemplified through the 1D Burgers equation which is motivated by aircraft’s sonic-boom minimization issues. In particular, we prove that, as soon as the target function (usually a Nwave) isn’t continuous, there is a whole convex set of possible initial data, the backward entropy solution being possibly its centroid. Further, an iterative strategy based on a gradient algorithm involving “reversible solutions” solving the linear adjoint problem is set up. In order to be able to recover initial profiles different from the backward entropy solution, a filtering step of the backward adjoint solution is inserted, mostly relying on scale-limited (wavelet) subspaces. Numerical illustrations, along with profiles similar to F-functions, are presentedAcknowledgements This work was partially supported by the Advanced Grant 694126-DYCON (Dynamic Control) of the European Research Council Executive Agency, ICON of the French ANR (2016-ACHN-0014-01), FA9550-15-1-0027 of AFOSR, A9550-14-1-0214 of the EOARD-AFOSR, and the MTM2014-52347 Grant of the MINECO (Spain