Flow optimization in vascular networks
Radu C. Cascaval Ciro D'Apice Maria Pia D'Arienzo Rosanna Manzo
Mathematical Biosciences & Engineering 2017, 14(3): 607-624 doi: 10.3934/mbe.2017035

The development of mathematical models for studying phenomena observed in vascular networks is very useful for its potential applications in medicine and physiology. Detailed $3$D studies of flow in the arterial system based on the Navier-Stokes equations require high computational power, hence reduced models are often used, both for the constitutive laws and the spatial domain. In order to capture the major features of the phenomena under study, such as variations in arterial pressure and flow velocity, the resulting PDE models on networks require appropriate junction and boundary conditions. Instead of considering an entire network, we simulate portions of the latter and use inflow and outflow conditions which realistically mimic the behavior of the network that has not been included in the spatial domain. The resulting PDEs are solved numerically using a discontinuous Galerkin scheme for the spatial and Adam-Bashforth method for the temporal discretization. The aim is to study the effect of truncation to the flow in the root edge of a fractal network, the effect of adding or subtracting an edge to a given network, and optimal control strategies on a network in the event of a blockage or unblockage of an edge or of an entire subtree.

keywords: Blood flow network optimization discontinuous Galerkin scheme
A continuum-discrete model for supply chains dynamics
Gabriella Bretti Ciro D’Apice Rosanna Manzo Benedetto Piccoli
Networks & Heterogeneous Media 2007, 2(4): 661-694 doi: 10.3934/nhm.2007.2.661
This paper is focused on continuum-discrete models for supply chains. In particular, we consider the model introduced in [10], where a system of conservation laws describe the evolution of the supply chain status on sub-chains, while at some nodes solutions are determined by Riemann solvers. Fixing the rule of flux maximization, two new Riemann Solvers are defined. We study the equilibria of the resulting dynamics, moreover some numerical experiments on sample supply chains are reported. We provide also a comparison, both of equilibria and experiments, with the model of [15].
keywords: networks conservation laws fluid-dynamic models finite difference schemes Supply chains
On optimal controls in coefficients for ill-posed non-Linear elliptic Dirichlet boundary value problems
Olha P. Kupenko Rosanna Manzo
Discrete & Continuous Dynamical Systems - B 2018, 23(4): 1363-1393 doi: 10.3934/dcdsb.2018155

We consider an optimal control problem associated to Dirichlet boundary value problem for non-linear elliptic equation on a bounded domain $Ω$. We take the coefficient $u(x)∈ L^∞(Ω)\cap BV(Ω)$ in the main part of the non-linear differential operator as a control and in the linear part of differential operator we consider coefficients to be unbounded skew-symmetric matrix $A_{skew}∈ L^q(Ω;\mathbb{S}^N_{skew})$. We show that, in spite of unboundedness of the non-linear differential operator, the considered Dirichlet problem admits at least one weak solution and the corresponding OCP is well-possed and solvable. At the same time, optimal solutions to such problem can inherit a singular character of the matrices $A^{skew}$. We indicate two types of optimal solutions to the above problem and show that one of them can be attained by optimal solutions of regularized problems for coercive elliptic equations with bounded coefficients, using the two-parametric regularization of the initial OCP.

keywords: Generalized p-Laplace equations control in coefficients variational convergence
A fluid dynamic model for supply chains
Ciro D'Apice Rosanna Manzo
Networks & Heterogeneous Media 2006, 1(3): 379-398 doi: 10.3934/nhm.2006.1.379
The paper deals with a fluid dynamic model for supply chains. A mixed continuum-discrete model is proposed and possible choices of solutions at nodes guaranteeing the conservation of fluxes are discussed. Fixing a rule a Riemann solver is defined and existence of solutions to Cauchy problems is proved.
keywords: supply chains. Conservation laws
On relaxation of state constrained optimal control problem for a PDE-ODE model of supply chains
Ciro D'Apice Peter I. Kogut Rosanna Manzo
Networks & Heterogeneous Media 2014, 9(3): 501-518 doi: 10.3934/nhm.2014.9.501
We discuss the optimal control problem (OCP) stated as the minimization of the queues and the difference between the effective outflow and a desired one for the continuous model of supply chains, consisting of a PDE for the density of processed parts and an ODE for the queue buffer occupancy. The main goal is to consider this problem with pointwise control and state constraints. Using the so-called Henig delation, we propose the relaxation approach to characterize the solvability and regularity of the original problem by analyzing the corresponding relaxed OCP.
keywords: entropy solutions optimal control relaxation supply chains Conservation laws Henig dilating cone.
Unbounded perturbations of the generator domain
Said Hadd Rosanna Manzo Abdelaziz Rhandi
Discrete & Continuous Dynamical Systems - A 2015, 35(2): 703-723 doi: 10.3934/dcds.2015.35.703
Let $X,U$ and $Z$ be Banach spaces such that $Z\subset X$ (with continuous and dense embedding), $L:Z\to X$ be a closed linear operator and consider closed linear operators $G,M:Z\to U$. Putting conditions on $G$ and $M$ we show that the operator $\mathcal{A}=L$ with domain $D(\mathcal{A})=\big\{z\in Z:Gz=Mz\big\}$ generates a $C_0$-semigroup on $X$. Moreover, we give a variation of constants formula for the solution of the following inhomogeneous problem \begin{align*} \begin{cases} \dot{z}(t)=L z(t)+f(t),& t\ge 0,\cr G z(t)=Mz(t)+g(t),& t\ge 0,\cr z(0)=z^0. \end{cases} \end{align*} Several examples will be given, in particular a heat equation with distributed unbounded delay at the boundary condition.
keywords: unbounded perturbation closed-loop systems $C_0$--semigroup inhomogeneous boundary problem. regular linear systems Banach space
On boundary optimal control problem for an arterial system: First-order optimality conditions
Ciro D'Apice Olha P. Kupenko Rosanna Manzo
Networks & Heterogeneous Media 2018, 13(4): 585-607 doi: 10.3934/nhm.2018027

We discuss a control constrained boundary optimal control problem for the Boussinesq-type system arising in the study of the dynamics of an arterial network. We suppose that the control object is described by an initial-boundary value problem for $ 1D $ system of pseudo-parabolic nonlinear equations with an unbounded coefficient in the principle part and the Robin-type of boundary conditions. The main question we study in this part of the paper is about the existence of optimal solutions and first-order optimality conditions.

keywords: Boussinesq-type system existence result first-order optimality conditions optimal solutions

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