Searcharxiv⌕ Search

arXiv subjects

Gabriela Marinoschi

Publications and source records attributed to Gabriela Marinoschi.

At least 19 recordsLinked to original sources
arXiv2026

An operatorial approach of the well-posedness of an algebraic Riccati equation

Finding the state feedback control in an $% H^{\infty }$-optimal control problem involves a challenging approach of the associated algebraic Riccati equation of the generic form $A^{\ast }P+PA+PΓP=F$. In view of this objective, we explore in this paper the existence of the solution to this algebraic Riccati equation by a direct operatorial approach in the space of Hilbert-Schmidt operators. The proofs are provided, under certain assumptions on the operators $Γ$ and $F,$ for the cases with $A$ coercive and $A\geq 0,$ respectively. They develop a constructive approach, possibly indicating a method for finding the numerical solution. Next, relying on the existence of the solution to the Riccati equation, we provide then a result concerning the associated $% H^{\infty }$-optimal control problem. An example regarding the application of the existence proof for the solution to the Riccati equation is given for a parabolic equation with a singular potential of Hardy type.

math.OC↗
arXiv2025

Chemotaxis-inspired PDE model for airborne infectious disease transmission: analysis and simulations

Partial differential equation (PDE) models for infectious disease have received renewed interest in recent years. Most models of this type extend classical compartmental formulations with additional terms accounting for spatial dynamics, with Fickian diffusion being the most common such term. However, while diffusion may be appropriate for modeling vector-borne diseases, or diseases among plants or wildlife, the spatial propagation of airborne diseases in human populations is heavily dependent on human contact and mobility patterns, which are not necessarily well-described by diffusion. By including an additional chemotaxis-inspired term, in which the infection is propagated along the positive gradient of the susceptible population (from regions of low- to high-density of susceptibles), one may provide a more suitable description of these dynamics. This article introduces and analyzes a mathematical model of infectious disease incorporating a modified chemotaxis-type term. The model is analyzed mathematically and the well-posedness of the resulting PDE system is demonstrated. A series of numerical simulations are provided, demonstrating the ability of the model to naturally capture important phenomena not easily observed in standard diffusion models, including propagation over long spatial distances over short time scales and the emergence of localized infection hotspots

q-bio.PE↗
arXiv2024

Optimal epidemic control by social distancing and vaccination of an infection structured by time since infection: the covid-19 case study

Motivated by the issue of COVID-19 mitigation, in this work we tackle the general problem of optimally controlling an epidemic outbreak of a communicable disease structured by time since exposure, by the aid of two types of control instruments namely, social distancing and vaccination by a vaccine at least partly effective in protecting from infection. Effective vaccines are assumed to be made available only in a subsequent period of the epidemic so that - in the first period - epidemic control only relies on social distancing, as it happened for the COVID-19 pandemic. By our analyses, we could prove the existence of (at least) one optimal control pair, we derived first-order necessary conditions for optimality, and proved some useful properties of such optimal solutions. A worked example provides a number of further insights on the relationships between key control and epidemic parameters.

q-bio.PE↗
arXiv2023

Global solution and optimal control of an epidemic propagation with a heterogeneous diffusion

In this paper, we explore the solvability and the optimal control problem for a compartmental model based on reaction-diffusion partial differential equations describing a transmissible disease. The nonlinear model takes into account the disease spreading due to the human social diffusion, under a dynamic heterogeneity in infection risk. The analysis of the resulting system provides the existence proof for a global solution and determines the conditions of optimality to reduce the concentration of the infected population in certain spatial areas.

math.OC↗
arXiv2023

Optimal control of a reaction-diffusion model related to the spread of COVID-19

This paper is concerned with the well-posedness and optimal control problem of a reaction-diffusion system for an epidemic Susceptible-Infected-Recovered-Susceptible (SIRS) mathematical model in which the dynamics develops in a spatially heterogeneous environment. Using as control variables the transmission rates $u_{i}$ and $u_{e}$ of contagion resulting from the contact with both asymptomatic and symptomatic persons, respectively, we optimize the number of exposed and infected individuals at a final time $T$ of the controlled evolution of the system. More precisely, we search for the optimal $u_{i}$ and $u_{e}$ such that the number of infected plus exposed does not exceed at the final time a threshold value $Λ$, fixed a priori. We prove here the existence of optimal controls in a proper functional framework and we derive the first-order necessary optimality conditions in terms of the adjoint variables.

math.OC↗
arXiv2020

Identification and control of SARS-CoV-2 epidemic model parameters

We propose a mathematical model with five compartments for the SARS-CoV-2 transmission: susceptible $S$, undetected infected asymptomatic $A$, undetected infected symptomatic $I$, confirmed positive and isolated $L$, and recovered $ R$, for which we have a twofold objective. First, we formulate and solve an inverse problem focusing mainly on the identification of the values $A_{0}$ and $ I_{0}$ of the undetected asymptomatic and symptomatic individuals, at a time $t_{0}$, by available measurements of the isolated and recovered individuals at two succeeding times, $ t_{0}$ and $ T>t_{0}.$ Simultaneously, we identify the rate standing for the average number of individuals infected in unit time by an infective symptomatic individual. Then, we propose a control problem aiming at controlling the infected classes by improving the actions in view of isolating as much as possible the populations $ A$ and $ I$ in the class $ L$. These objectives are formulated as minimization problems, the second one including a state constraint, which are treated by an optimal control technique. The existence of optimal controllers is proved and the first order necessary conditions of optimality are determined. For the second problem, they are deduced by passing to the limit in the conditions of optimality calculated for an appropriately defined approximating problem. In this case, the dual system is singular and has a component in the space of measures. The discussion of the asymptotic stability of the system done for the case when life immunity is gained reveals an asymptotic extinction of the disease, with a well determined reproduction number.

math.OC↗
arXiv2020

Optimal control of cytotoxic and antiangiogenic therapies on prostate cancer growth

Prostate cancer can be lethal in advanced stages, for which chemotherapy may become the only viable therapeutic option. While there is no clear clinical management strategy fitting all patients, cytotoxic chemotherapy with docetaxel is currently regarded as the gold standard. However, tumors may regain activity after treatment conclusion and become resistant to docetaxel. This situation calls for new delivery strategies and drug compounds enabling an improved therapeutic outcome. Combination of docetaxel with antiangiogenic therapy has been considered a promising strategy. Bevacizumab is the most common antiangiogenic drug, but clinical studies have not revealed a clear benefit from its combination with docetaxel. Here, we capitalize on our prior work on mathematical modeling of prostate cancer growth subjected to combined cytotoxic and antiangiogenic therapies, and propose an optimal control framework to robustly compute the drug-independent cytotoxic and antiangiogenic effects enabling an optimal therapeutic control of tumor dynamics. We describe the formulation of the optimal control problem, for which we prove the existence of at least a solution and determine the necessary first order optimality conditions. We then present numerical algorithms based on isogeometric analysis to run a preliminary simulation study over a single cycle of combined therapy. Our results suggest that only cytotoxic chemotherapy is required to optimize therapeutic performance and we show that our framework can produce superior solutions to combined therapy with docetaxel and bevacizumab. We also illustrate how the optimal drug-naïve cytotoxic effects computed in these simulations may be successfully leveraged to guide drug production and delivery strategies by running a nonlinear least-square fit of protocols involving docetaxel and a new design drug.

math.OC↗
arXiv2020

Exact controllability in minimal time of the Navier-Stokes periodic flow in a 2D-channel

This work is concerned with the necessary conditions of optimality for a minimal time control problem $(P)$ for the linearized Navier-Stokes periodic flow in a 2D-channel, subject to a boundary input which acts on the transversal component of the velocity. The objective in this problem is the reaching of the laminar regime in a minimum time, as well as its preservation after this time. The determination of the necessary conditions of optimality relies on the analysis of intermediate minimal time control problems $(P_{k})$ for the Fourier modes $"k"$ associated to the Navier-Stokes equations and on the proof of the maximum principle for them. Also it is found that one can construct, on the basis of the optimal controllers of problems $(P_{k}),$ a small time called here \textit{quasi minimal} and a boundary controller which realizes the required objective in $% (P).$

math.AP↗
arXiv2020

Minimal time sliding mode control for evolution equations in Hilbert spaces

This work is concerned with the time optimal control problem for evolution equations in Hilbert spaces. The attention is focused on the maximum principle for the time optimal controllers having the dimension smaller that of the state system, in particular for minimal time sliding mode controllers, which is one of the novelties of this paper. We provide the characterization of the controllers by the optimality conditions determined for some general cases. The proofs rely on a set of hypotheses meant to cover a large class of applications. Examples of control problems governed by parabolic equations with potential and drift terms, porous media equation or reaction-diffusion systems with linear and nonlinear perturbations, describing real world processes, are presented at the end.

math.AP↗
arXiv2020

Solvability and sliding mode control for the viscous Cahn-Hilliard system with a possibly singular potential

In the present contribution we study a viscous Cahn-Hilliard system where a further leading term in the expression for the chemical potential $ μ$ is present. This term consists of a subdifferential operator $S$ in $L^2(Ω)$ (where $Ω$ is the domain where the evolution takes place) acting on the difference of the phase variable $φ$ and a given state $φ^* $, which is prescribed and may depend on space and time. We prove existence and continuous dependence results in case of both homogeneous Neumann and Dirichlet boundary conditions for the chemical potential $μ$. Next, by assuming that $S=ρ\,$sign, a multiple of the sign operator, and for smoother data, we first show regularity results. Then, in the case of Dirichlet boundary conditions for $μ$ and under suitable conditions on $ρ$ and $Ω$, we also prove the sliding mode property, that is, that $φ$ is forced to join the evolution of $φ^* $ in some time $T^*$ lower than the given final time $T$. We point out that all our results hold true for a very general and possibly singular multi-well potential acting on $φ$.

math.AP↗
arXiv2019

Mathematical analysis and simulation study of a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects

Cytotoxic chemotherapy is a common treatment for advanced prostate cancer. These tumors are also known to rely on angiogenesis, i.e., the growth of local microvasculature via chemical signaling produced by the tumor. Thus, several clinical studies have been investigating antiangiogenic therapy for advanced prostate cancer, either as monotherapy or combined with standard cytotoxic protocols. However, the complex genetic alterations promoting prostate cancer growth complicate the selection of the best chemotherapeutic approach for each patient's tumor. Here, we present a mathematical model of prostate cancer growth and chemotherapy that may enable physicians to test and design personalized chemotherapeutic protocols in silico. We use the phase-field method to describe tumor growth, which we assume to be driven by a generic nutrient following reaction-diffusion dynamics. Tumor proliferation and apoptosis (i.e., programmed cell death) can be parameterized with experimentally-determined values. Cytotoxic chemotherapy is included as a term downregulating tumor net proliferation, while antiangiogenic therapy is modeled as a reduction in intratumoral nutrient supply. Another equation couples the tumor phase field with the production of prostate-specific antigen, which is an extensively used prostate cancer biomarker. We prove the well-posedness of our model and we run a series of representative simulations using an isogeometric method to explore untreated tumor growth as well as the effects of cytotoxic chemotherapy and antiangiogenic therapy, both alone and combined. Our simulations show that our model captures the growth morphologies of prostate cancer as well as common outcomes of cytotoxic and antiangiogenic mono and combined therapy. Our model also reproduces the usual temporal trends in tumor volume and prostate-specific antigen evolution observed in previous studies.

math.AP↗
arXiv2019

Sliding mode control of the Hodgkin-Huxley mathematical model

In this paper we deal with a feedback control design for the action potential of a neuronal membrane in relation with the non-linear dynamics of the Hodgkin-Huxley mathematical model. More exactly, by using an external current as a control expressed by a relay graph in the equation of the potential, we aim at forcing it to reach a certain manifold in finite time and to slide on it after that. From the mathematical point of view we solve a system involving a parabolic differential inclusion and three nonlinear differential equations via an approximating technique and a fixed point result. The existence of the sliding mode and the determination of the time at which the potential reaches the prescribed manifold is proved by a maximum principle argument. Numerical simulations are presented.

math.OC↗
arXiv2017

Rescaling approach for a stochastic population dynamics equation perturbed by a linear multiplicative Gaussian noise

We are concerned with a nonlinear nonautonomous model represented by an equation describing the dynamics of an age-structured population diffusing in a space habitat $O,$ governed by local Lipschitz vital factors and by a stochastic behavior of the demographic rates possibly representing emigration, immigration and fortuitous mortality. The model is completed by a random initial condition, a flux type boundary conditions on $\partial O$ with a random jump in the population density and a nonlocal nonlinear boundary condition given at age zero. The stochastic influence is expressed by a linear multiplicative Gaussian noise perturbation in the equation. The main result proves that the stochastic model is well-posed, the solution being in the class of path-wise continuous functions and satisfying some particular regularities with respect to the age and space. The approach is based on a rescaling transformation of the stochastic equation into a random deterministic time dependent hyperbolic-parabolic equation with local Lipschitz nonlinearities. The existence and uniqueness of a strong solution to the random deterministic equation is proved by combined semigroup, variational and approximation techniques. The information given by these results is transported back via the rescaling transformation towards the stochastic equation and enables the proof of its well-posedness.

math.AP↗
arXiv2017

Sliding mode control for a phase field system related to tumor growth

In the present contribution we study the sliding mode control (SMC) problem for a diffuse interface tumor growth model coupling a viscous Cahn-Hilliard type equation for the phase variable with a reaction-diffusion equation for the nutrient. First, we prove the well-posedness and some regularity results for the state system modified by the state-feedback control law. Then, we show that the chosen SMC law forces the system to reach within finite time the sliding manifold (that we chose in order that the tumor phase remains constant in time). The feedback control law is added in the Cahn-Hilliard type equation and leads the phase onto a prescribed target $ϕ^*$ in finite time.

math.AP↗
arXiv2017

Sliding mode control for a nonlinear phase-field system

In the present contribution the sliding mode control (SMC) problem for a phase-field model of Caginalp type is considered. First we prove the well-posedness and some regularity results for the phase-field type state systems modified by the state-feedback control laws. Then, we show that the chosen SMC laws force the system to reach within finite time the sliding manifold (that we chose in order that one of the physical variables or a combination of them remains constant in time). We study three different types of feedback control laws: the first one appears in the internal energy balance and forces a linear combination of the temperature and the phase to reach a given (space dependent) value, while the second and third ones are added in the phase relation and lead the phase onto a prescribed target. While the control law is non-local in space for the first two problems, it is local in the third one, i.e., its value at any point and any time just depends on the value of the state.

math.AP↗