Papers by Gabriela Marinoschi
Nonlinear Analysis: Real World Applications, 2008
Coupling hysteretic hydraulic laws with the pressure form of Richards' equation, a mathematical m... more Coupling hysteretic hydraulic laws with the pressure form of Richards' equation, a mathematical model of a hysteretic wettingdrying cycle of a soil is settled. The particularity of the model resides in the blowing-up diffusion coefficient characterizing a strongly nonlinear behavior of the porous medium and in certain relationships between the hydraulic functions accounting for a sufficiently realistic hysteretic evolution of the envisaged process. The hysteretic effect of the hydraulic laws can be regained in the hysteretic behavior of the multivalued function defined as an antiderivative of the diffusivity function. We investigate the well-posedness of the model in appropriate functional spaces.
In this paper we deal with a feedback control design for the action potential of a neuronal membr... more 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.
A nonlinear divergence parabolic equation with dynamic boundary conditions of Wentzell type is st... more A nonlinear divergence parabolic equation with dynamic boundary conditions of Wentzell type is studied. The existence and uniqueness of a strong solution is obtained as the limit of a finite difference scheme, in the time dependent case and via a semigroup approach in the time-invariant case.
In this paper we study a singular control problem for a system of PDEs describing a phase-field m... more In this paper we study a singular control problem for a system of PDEs describing a phase-field model of Penrose-Fife type. The main novelty of this contribution consists in the idea of forcing a sharp interface separation between the states of the system by using heat sources distributed in the domain and at the boundary. We approximate the singular cost functional with a regular one, which is based on the Legendre-Fenchel relations. Then, we obtain a regularized control problem for which we compute the first order optimality conditions using an adapted penalization technique. The proof of some convergence results and the passage to the limit in these optimality conditions lead to the characterization of the desired optimal controller.
Springer INdAM Series
This article deals with the internal feedback stabilization of a phase field system of Cahn-Hilli... more This article deals with the internal feedback stabilization of a phase field system of Cahn-Hilliard type involving a logarithmic potential F, and extends the recent results provided in Barbu et al. (J Differ Equ 262:2286-2334, 2017) for the double-well potential. The stabilization is searched around a stationary solution, by a feedback controller with support in a subset ! of the domain. The controller stabilizing the linearized system is constructed as a finite combination of the unstable modes of the operator acting in the linear system and it is further provided in a feedback form by solving a certain minimization problem. Finally, it is proved that this feedback form stabilizes the nonlinear system too, if the stationary solution has not large variations. All these results are provided in the three-dimensional case for a regularization of the singular potential F, and allow the same conclusion for the singular logarithmic potential in the one-dimensional case.
Functional Approach to Nonlinear Models of Water Flow in Soils
Discrete & Continuous Dynamical Systems - S
We study a problem of a parameter identification related to a linear evolution equation in a Bana... more We study a problem of a parameter identification related to a linear evolution equation in a Banach space, using an additional information about the solution. For sufficiently regular data we provide an exact solution given by a Volterra integral equation, while for less regular data we obtain an approximating solution by an optimal control approach. Under certain hypotheses, the characterization of the limit of the sequence of the approximating solutions reveals that it is a solution to the original identification problem. An application to an inverse problem arising in population dynamics is presented.
Analele Universitatii "Ovidius" Constanta - Seria Matematica
In this paper we review some results obtained for a distributed con- trol problem regarding a cla... more In this paper we review some results obtained for a distributed con- trol problem regarding a class of phase field systems of Caginalp type with logarithmic potential. The aim of the control problem is forcing the location of the diffuse interface to be as close as possible to a pre- scribed set. However, due to some discontinuity in the cost functional, we have to regularize it and solve the related control problem for the approximation. We discuss the necessary optimality conditions.
Revue Roumaine de Physique
ABSTRACT
Nonlinear Analysis: Real World Applications, 2009
The movement of the metabolites crossing biological membranes is essentially controlled by the ac... more The movement of the metabolites crossing biological membranes is essentially controlled by the action of the membrane potential. In biophysics, the interest is on the way in which the membrane potential drives the particle motion and makes more probable its translocation, i.e., the transit through the channel and the escape through the opposite side with respect to the entrance. Our investigation will focus on the determination of the membrane potential which allows the particle escape from the channel and may possibly limit the particle lifetime inside it. In mathematical terms, the purpose of this work is to study some mathematical nonlinear control problems, consisting in the determination of an optimal potential which makes the channel more efficient in a certain sense. Some physically significant variations of this basic problem are also investigated and a discussion of the possible extension of the model to time-dependent situations is made. Finally, numerical results are presented.
Journal of Optimization Theory and Applications, 2014
We provide existence results for nonlinear diffusion equations with multivalued time-dependent no... more We provide existence results for nonlinear diffusion equations with multivalued time-dependent nonlinearities related to convex continuous not coercive potentials. The results in this paper, following a variational principle, state that a generalized solution of the nonlinear equation can be retrieved as a solution of an appropriate minimization problem for a convex functional involving the potential and its conjugate. In the not coercive case, this assertion is conditioned by the validity of a relation between the solution and the nonlinearity. A sufficient condition, under which this relation is true, is given. At the end, we present a discussion on the solution existence for a particular equation describing a self-organized criticality model.
Mathematical Control & Related Fields
In the present contribution we study a viscous Cahn–Hilliard system where a further leading term ... more In the present contribution we study a viscous Cahn–Hilliard system where a further leading term in the expression for the chemical potential \begin{document}$ \mu $\end{document} is present. This term consists of a subdifferential operator \begin{document}$ S $\end{document} in \begin{document}$ L^2(\Omega) $\end{document} (where \begin{document}$ \Omega $\end{document} is the domain where the evolution takes place) acting on the difference of the phase variable \begin{document}$ \varphi $\end{document} and a given state \begin{document}$ {\varphi^*} $\end{document}, which is prescribed and may depend on space and time. We prove existence and continuous dependence results in case of both homogeneous Neumann and Dirichlet boundary conditions for the chemical potential \begin{document}$ \mu $\end{document}. Next, by assuming that \begin{document}$ S = \rho\;{\rm{sign}} $\end{document}, a multiple of the \begin{document}$ \;{\rm{sign}} $\end{document} operator, and for smoother data, ...
ESAIM: Control, Optimisation and Calculus of Variations
This work is concerned with the time optimal control problem for evolution equations in Hilbert s... more 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.
Journal of mathematical biology, Jan 16, 2016
We consider a model with age and space structure for the epidermis evolution. The model, previous... more We consider a model with age and space structure for the epidermis evolution. The model, previously presented and analyzed with respect to the suprabasal epidermis, includes different types of cells (proliferating cells, differentiated cells, corneous cells, and apoptotic cells) moving with the same velocity, under the constraint that the local volume fraction occupied by the cells is constant in space and time. Here, we complete the model proposing a mechanism regulating the cell production in the basal layer and we focus on the stationary case of the problem, i.e. on the case corresponding to the normal status of the skin. A numerical scheme to compute the solution of the model is proposed and its convergence is studied. Simulations are provided for realistic values of the parameters, showing the possibility of reproducing the structure of both "thin" and "thick" epidermis.
The paper is meant to study the behaviour of a passive pollutant, downstream the pollution source... more The paper is meant to study the behaviour of a passive pollutant, downstream the pollution source, depending on the Danube river runoff, using the mathematical modelling as a study instrument. For the mathematical models proposed, certain physical and hydrological hypotheses were taken into account, allowing obtaining analytic solutions. Simulations with these models were performed under various conditions of the Danube runoff and considering the Argeș River as a pollutant source.
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Papers by Gabriela Marinoschi