Joint research project

Control and stabilization problems for phase field and biological systems

Project leaders
Pierluigi Colli, Gabriela Marinoschi
ROMANIA - RA - The Romanian Academy
CNR-RA 2017-2019
Engineering, ICT and technologies for energy and transportation
Thematic area
Engineering, ICT and technologies for energy and transportation
Status of the project

Research proposal

This project is aimed at continuing the successful collaboration of the members of these teams and turns out a natural development of the project "Nonlinear partial differential equations (PDE) with applications in modeling cell growth, chemotaxis and phase transition", ruled within the same framework of the collaboration agreement between CNR Italy and Romanian Academy, and concluded with an important number of articles (see [1-7]) recently published in top journals of Mathematics, indexed by Thomson Reuters and classified in the first 25% range of importance according to the impact factor and especially the more exclusive AIS factor. The project will address control and optimization problems related to some classes of nonlinear parabolic equations and systems that describe processes of major interest in phase transition/separation and biology (chemotaxis, population dynamics). Essentially we refer to optimal control problems [4-6], feedback stabilization [1,8], controllability and sliding mode control [2], and related well-posedness theories [3]. The classes of nonlinear systems under study bring two or more nonlinear diffusion equations intrinsically coupled (see e.g., [4, 6]), as in phase transition and separation, or dynamic population (cell growth) models and with nonlinear cross diffusion, that is of chemotaxis type. The topic proposed in this project has both theoretical and practical importance, for it is oriented towards a research line of current international interest. Dissemination of results will be done by publication in first class journals and presentation at international scientific meetings.

1. Optimal control, feedback control and stabilization of phase transition/separation systems
Phase transitions are thermodynamic processes in which under the action of some external conditions, such as temperature, pressure, or others, a transition between two or multiple phases of the same matter occurs (as solid to liquid, liquid to gaseous and conversely). A phase is a configuration in which matter has uniform physical properties. Phase separation is a process in which a mixture of two or more components spontaneously separates in its components. These models include equations for the order parameter and for energy and/or momentum balance, with initial and boundary conditions. Well-established models of phase field type will be considered. A certain richness of nonlinearities and somehow unpleasant terms in the equations makes the investigation of these problems particularly interesting. The present concern refers to optimal control problems and stabilization of various models of these types. Stabilization is a challenging problem, especially for systems of equations: it involves solvability issues, analysis of eigenvalues/eigenvectors of the linearized system, and accurate estimates. Special (nonlocal) boundary conditions may complicate the study requiring adapted mathematical techniques. We shall also consider systems of similar types which describe cell growth phenomena, or chemotaxis phenomena with a very complicate cross-diffusion. Especially, we will deal with the control of the model of epidermis growth which is represented by a system of many nonlinear hyperbolic equations with a free boundary and a nonlocal boundary conditions.

2. Sliding mode control
Sliding mode control (SMC) is one of the fundamental approaches for the systematic design of robust controllers for nonlinear complex dynamic systems that operate under uncertainty and it is considered a classical tool for the control of continuous or discrete time systems in finite-dimensional settings. It has the advantage of controlling the separation of the motion of the overall system in independent partial components of lower dimensions, thus reducing the complexity of the problem. The design of feedback control systems with sliding modes implies the design of suitable control functions enforcing motion along ad-hoc manifolds. The idea is first to identify a manifold of lower dimension (called the sliding manifold) where the control goal is fulfilled and such that the original system restricted to this sliding manifold has a desired behavior, and then to act on the system through the control in order to constrain the evolution on it, that is, to design a SMC-law that forces the trajectories of the system to reach the sliding surface and maintains them on it. This technique works especially for systems of equations describing certain processes, so that it perfectly fits to the models of nonlinear coupled equations we aim to study.

[1] V. Barbu, P. Colli, G. Gilardi, G. Marinoschi, Feedback stabilization of the Cahn-Hilliard type system for phase separation, J. Differential Equations, under review (early version in
[2] V. Barbu, P. Colli, G. Gilardi, G. Marinoschi, E. Rocca, Sliding mode control for a nonlinear phase-field system, SIAM J. Control Optim., under review (early version in
[3] V. Barbu, A. Favini, G. Marinoschi, Nonlinear parabolic flows with dynamic flux on the boundary, J. Differential Equations 258, 2160-2195, 2015
[4] P. Colli, G. Gilardi, G. Marinoschi, A boundary control problem for a possibly singular phase field system with dynamic boundary conditions, J. Math. Anal. Appl. 434, 432-463, 2016
[5] P. Colli, G. Gilardi, G. Marinoschi, E. Rocca, Optimal control for a phase field system with a possibly singular potential, Math. Control Relat. Fields 6, 95-112, 2016
[6] P. Colli, G. Marinoschi, E. Rocca, Sharp interface control in a Penrose-Fife model, ESAIM Control Optim. Calc. Var. 22, 473-499, 2016
[7] A. Gandolfi, M. Iannelli, G. Marinoschi, The steady state of epidermis: mathematical modeling and numerical simulation, J. Math. Biology, doi:10.1007/s00285-016-1006-4
[8] I. Munteanu, Boundary stabilization of the phase field system by finite-dimensional feedback controllers, J. Math. Anal. Appl. 412, 964-975, 2014

Research goals

From the mathematical viewpoint, we aim to develop methods for the study of nonlinear PDE, some of them with nonlocal boundary conditions, and describing phenomena with a free boundary evolution. The impact of the project in the main domain of research, that is mathematics, will be ensured by the publication of original results in high impact journals. At the same time, the investigated models will give the possibility of a better understanding of some peculiarities of the concerned physical and biological processes. We will obtain efficient procedures for solving in a different way the PDE and associated control problems, and for devising some numerical methods that lead to the development of computer codes.

In particular, we shall deal with
1. Feedback stabilization of phase transition systems (Allen-Cahn and Caginalp models) and of
phase separation models (Cahn-Hilliard and conserved models);
2. Sliding mode control of dynamical systems involving a phase variable;
3. Relevant optimal control problems in relation with phase field, cell growth and dynamic population models (existence of the control, well-posedness of the direct and dual systems, determination of the necessary optimality conditions);
4. Numerical computation of the cell growth system of equations in the nonstationary case: discretization, computation of the solutions along characteristics, exploration of simulation scenarios giving importance to the various parameters of the model.

Last update: 07/10/2022