Structural properties of solutions, approximation and control for conservation laws with discontinuous flux and bioinspired PDE models

In this thesis we study theoretical and control type properties of three different classes of PDE: Scalar Conservation Laws with flux discontinuous, respectively, in the space and in the conserved quantity, and a Partial Integro-Differential Equation. In the first chapter we analyze the set of attainable profiles, at fixed time (with initial datum regarded as a control), by solutions to conservation laws with flux having a single discontinuity in the space and strictly convex or strictly concave behaviour outside the discontinuity. This analysis yields compactness properties of such a set which is instrumental to study many variational problems involving the profiles of the solutions. In the second chapter we examine a class of conservation laws with flux discontinuous in the conserved quantity that emerges in a model of industrial conveyor belt and in supply chains. We first introduce an appropriate notion of pair of entropic solution-flux, we provide existence of entropic solution-fluxes by front-tracking and we show a Kruzhkov’s type stability of such solutions. Next, we analyze the associated Hamilton-Jacobi equation, we derive an Hopf-Lax type representation formula of the solutions and we show how to recover the pair of entropic solution-flux of the conservation law from the gradient of the solution of the Hamilton-Jacobi equation. Finally, we consider the problem of a junction with a buffer (to store processed products) and with incoming and outgoing belts modeled by the class of conservation laws analyzed beforehand. Existence and uniqueness of the solution to the junction problem is established. The last chapter describes a mathematical model for a robotic root to be used in rescue technology. The root movement is described by two different Partial Integro-Differential Equations, one for the body and one for the tip. When the root encounters an obstacle in a "good" configuration, it moves around it by bending through a determined angular velocity minimizing the elastic deformation, the cost of moving sand and digging where the soil is dense. A restarting procedure is instead introduced to handle the crossing of obstacles in "bad" configuration. Some numeric simulation are also produced.