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### Open Access Journals

$(*) \quad \quad u_t+[F(u)]_x=g(t,x,u),\quad u(0,x)=\overline{u}(x),$

for a nonlinear $2\times 2$ hyperbolic system of
inhomogeneous balance laws
in one space dimension. As usual,
we assume that the system is strictly hyperbolic and that each
characteristic field is either
linearly degenerate or genuinely nonlinear.

Under suitable assumptions on $g$, we prove that there exists
$T>0$ such that, for every $\overline{u}$ with sufficiently small
total variation, the Cauchy problem ($*$)
has a unique "viscosity solution",
defined for $t\in [0,T]$,
depending continuously on the initial data.

Dynamic flux balance analysis of a bioreactor is based on the coupling between a dynamic problem, which models the evolution of biomass, feeding substrates and metabolites, and a linear program, which encodes the metabolic activity inside cells. We cast the problem in the language of optimal control and propose a hybrid formulation to model the full coupling between macroscopic and microscopic level. On a given location of the hybrid system we analyze necessary conditions given by the Pontryagin Maximum Principle and discuss the presence of singular arcs. For the multi-input case, under suitable assumptions, we prove that generically with respect to initial conditions optimal controls are bang-bang. For the single-input case the result is even stronger as we show that optimal controls are bang-bang.

We focus on a controlled Cucker--Smale model in finite dimension. Such dynamics model self-organization and consensus emergence in a group of agents. We explore how it is possible to control this model in order to enforce or facilitate pattern formation or convergence to consensus. In particular, we are interested in designing control strategies that are componentwise sparse in the sense that they require a small amount of external intervention, and also time sparse in the sense that such strategies are not chattering in time. These sparsity features are desirable in view of practical issues.

We first show how very simple sparse feedback strategies can be designed with the use of a variational principle, in order to steer the system to consensus. These feedbacks are moreover optimal in terms of decay rate of some functional, illustrating the general principle according to which ``sparse is better''. We then combine these results with local controllability properties to get global controllability results. Finally, we explore the sparsity properties of the optimal control minimizing a combination of the distance from consensus and of a norm of the control.

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*Discrete and Continuous Dynamical Systems -- Series S*is devoted to

*Traffic Modeling and Management*. This subject dramatically developed in recent years. On one hand, the successes of the analytical theory of conservation laws have provided new tools to traffic researchers while, on the other hand, the requirements coming from the applications have grown dramatically. Remarkably, two of the papers that opened the way to this decades long development date the same year. In 1995

*``The Unique Limit of the Glimm Scheme''*by A. Bressan (Archive for Rational Mechanics and Analysis, 130, 3, 205--230) gave a basis for several well posedness results for 1D systems of conservation laws. In the same year,

*``Requiem for High-Order Fluid Approximations of Traffic Flow''*by C. Daganzo (Transportation Research Part B: Methodological, 29B, 4, 277--287) posed serious criticisms to models studied at that time and started to fix minimal requirements for a traffic model to be seriously considered.

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