Titles and abstracts
Sebastian Aniţa, "Octav Mayer" Institute of Mathematics
Regional Control of a Prey-Predator Model
We investigate the problem of minimizing the total cost of the damages produced by an alien predator population and of the regional control paid to reduce this population. The dynamics of the predators is described by a prey-predator system with either local or nonlocal reaction terms. A sufficient condition for the zero-stabilizability (eradicability) of predators is given in terms of the sign of the principal eigenvalue of an appropriate operator that is not self-adjoint, and a stabilizing feedback control with a very simple structure is indicated. The minimization related to such a feedback control is treated for a closely related minimization problem viewed as a regional control problem. The level set method is a key ingredient.
Lucian Beznea,"Simion Stoilow" Institute of Mathematics
Path continuity of Markov processes and locality of Kolmogorov operators
We prove that if we are given a generator of a right Markov process with càdlàg paths and an open domain G in the state space, on which the generator has the local property expressed in a suitable way on a class of test functions that is sufficiently rich, then the Markov process has continuous paths when it passes through G. The approach uses potential theoretic tools and covers Markov processes associated with (possibly time-dependent) second order integro-differential operators defined on domains in Hilbert spaces or on spaces of measures.
The talk is based on joint works with Iulian Cîmpean (Bucharest) and Michael Röckner (Bielefeld).
Iulian Cîmpean, "Simion Stoilow" Institute of Mathematics
Ergodicity of Markov semigroups and application to singular SDEs
We prove existence and uniqueness of an invariant distribution for Markov semigroups under some general weak drift and coupling conditions. We study the stability of this approach under Girsanov transformation and we look at some applications to singular SDEs.
Pierluigi Colli, University of Pavia
Analysis and optimal control for a Cahn--Hilliard--Oono system with control in the mass term
In this talk I will present some results on the well-posedness and optimal control for the Cahn--Hilliard--Oono system when the control u is located in the mass term. First, the initial-boundary value problem and the dependence of the solution on the control u are discussed, and the analysis involves general potentials for the phase variable; here, both the cases of regular potentials and singular potential are considered. Next, the so-called separation property is investigated: it holds for everywhere defined regular potentials in 2D and 3D and for the logarithmic potential in 2D. In this framework, for the control problem it is shown that an optimal control exists; moreover, the control-to-state map is Frechet differentiable and a suitable variational inequality involving the solution of the adjoint system turns out to be a necessary condition for optimality. The results reported in this talk have been obtained in a collaboration with G. Gilardi, E. Rocca and J. Sprekels.
Valeriu Guţu, Moldova State University
On attractors of a type of iterated systems with condensation
Abstract.pdf file
Petru Jebelean, West University of Timişoara
Dirichlet systems with relativistic operator: differences vs differential
Abstract.pdf file
Étienne Pardoux, Aix-Marseille University
Recent results on epidemic models
In 1927, two Scottish epidemiologists, Kermack and McKendrick, published a paper on a SIR epidemic model, where each infectious individual has an age of infection dependent infectivity, and a random infectious period whose law is very general. This paper was quoted a huge number of times, but almost all authors who quoted it considered the simple case of a constant infectivity, and a duration of infection following the exponential distribution, in which case the integral equation model of Kermack and McKendrick reduces to an ODE.
It is classical that an ODE epidemic model is the Law of Large Numbers limits, as the size of the population tends to infinity, of finite population stochastic Markovian epidemic models.
One of our main contributions in recent years has been to show that the integral equation epidemic model of Kermack and McKendrick is the law of large numbers limit of stochastic non Markovian epidemic models. It is not surprising that the model of Kermack and Mc Kendrick, unlike ODE models, has a memory, like non Markovian stochastic processes. One can also write the model as a PDE, where the additional variable is the age of infection of each infected individual.
Similar PDE models have been introduced by Kermack and Mc Kendrick in their 1932 and 1933 papers, where they add a progressive loss of immunity. We have also shown that this 1932-33 model is the Law of Large Numbers limit of appropriate finite population non Markovian models.
Bibliography
[1] R. Forien, G. Pang and É Pardoux, Estimating the state of the Covid-19 epidemic in France using a non-Markov model, Royal Soc. Open Science 8, 202327, 2021.
[2] R. Forien, G. Pang and É Pardoux, Epidemic models with varying infectivity, SIAM J. Applied Math. 81, 1893-1930, 2021.
[3] R. Forien, G. Pang and É Pardoux, Multi-patch multi-group epidemic model with with varying infectivity, Probability, Uncertainty and Quantitative Risk 7, 333-364, 2022.
[4] R. Forien, G. Pang and É Pardoux, Recent advances in epidemic modeling: non Markov stochastic models and their scaling limits, Graduate J. Math. 7, 19-75, 2022.
[5] G. Pang and É Pardoux, Functional law of large numbers and PDEs for epidemic models with infection age dependent infectivity, Applied Math & Optimization 87, 2023.
[6] R. Forien, G. Pang, É Pardoux and A.B. Zotsa-Ngoufack, Stochastic epidemic models with varying infectivity and susceptibility, submitted.
Joint work with R. Forien (INRAE Avignon, France), G. Pang (Rice Univ., Houston, Texas, USA) and A.B. Zotsa-Ngoufack (AMU and Univ. Yaoundé 1)
Andrei Perjan, Moldova State University
The singular perturbations for the abstract semi-linear evolution equations in Hilbert spaces
Abstract.pdf file
Cătălin Popa, "Octav Mayer" Institute of Mathematics
Exact controllability of the three-dimensional magnetohydrodynamic equations with four or three scalar control functions
We present two new results of exact controllability for the three-dimensional magnetohydrodynamic (MHD) equations: The target solution can be attained by solutions whose initial configurations are sufficiently "close" to that of the target by acting essentially on four or three scalar control functions in an arbitrary small region whose boundary, however, must gave an essential closed edge. In the first case, the goal is reached with three scalar control functions in the "fluid" part of the system but with three scalar control functions which depend linearly (in a specific way) on only one scalar "input" function in the "magnetic" part. In the second case, the way of action is identical in the "magnetic" part of the system but one can act with only two scalar control functions in the "fluid" part provided that the control subdomain touches the boundary along a portion with non-void interior. In this way, previous controllability results for the MHD equations obtained four years ago with Teodor Havarneanu (where the control action in the "magnetic" part of the system is exerted through two scalar control functions) are improved. The new results seem surprising but, physically, they are an effect of the action of the magnetic field combined with the fact that the involved fluid is electrically conducting. The proofs are subordinate to the duality strategy based on the essential use of appropriate Carleman estimates for the adjoint linearized (around the target) equations.
Aurel Răşcanu, "Octav Mayer" Institute of Mathematics
$Lˆ{p}$-variational, $p ≥ 1$, solutions of BSDEs driven by a subdifferential operator
Abstract.pdf file
Elisabetta Rocca, University of Pavia
Direct and Inverse problems for a prostate cancer growth model
In this talk we present a mathematical model for prostate cancer growth which couples three nonlinear PDEs for the tumor phase field variable, the nutrient and the prostate index. The system is subjected to combined cytotoxic and antiangiogenic therapies, and we 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. 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 the inverse identification of the initial data, starting from a measurement at the final time. This can be useful in medical applications, after some diagnostic images are obtained, to locate with more precision the areas where the tumour started growing, as well as some information on the initial distribution of nutrient and PSA, which could still give more insight to the practitioners. These pieces of information can then be used to better calibrate therapies on the patients.
The talk is related to two recent collaborations with Pierluigi Colli, Guillermno Lorenzo, Gabriela Marinoschi, Alessandro Reali and Elena Beretta, Cecilia Cavaterra and Matteo Fornoni.
Michael Röckner, Bielefeld University
Nonlinear Fokker–Planck equations with fractional Laplacian and McKean-Vlasov SDEs with Lévy-noise
Abstract.pdf file
Gabriel Turinici, Université de Paris Dauphine -PSL
Reinforcement learning in finance: portfolio allocation, value functions and policy gradients flows
The reinforcement learning paradigm has stimulated model-free approaches in the numerical control area and in finance in particular. We discuss how reinforcement learning can be invoked for the construction of portfolio strategies. We compare the meaning of the value function with the financial literature counterpart, including the situation when the reward distribution is fat tailed ; finally, we compare the policy gradient methods with general gradient flow equations in metric spaces.