This major milestone in research on the Stackelberg–Nash strategy for Partial Differential Equations (PDEs) has been attained by AIMS Cameroon Ph.D. Research Fellow LIONEL LANDRY DJOMEGNE NJOUKOUE in his new research paper titled “Hierarchic Control for a Nonlinear Parabolic Equation in an Unbounded Domain”. The paper was recently published on September 29e, 2021 in the international journal titled “Applicable Analysis”.
The Stackelberg leadership model is a multiple-objective optimization approach initiated by H. von Stackelberg. This model is a strategic game in Economics in which two firms compete on the market with the same product. The first to act must integrate the reaction of the other company in the choices it makes in the quantity of the product that it decides to put on the market.
DJOMEGNE NJOUKOUE notes that there is some literature on the Stackelberg strategy in the framework of Partial differential equations (PDEs). Literature also contains some results about Stackelberg–Nash strategy for partial differential equations. However, works on hierarchic control for partial differential equations were considered in bounded domains. The authors proved the Stackelberg strategy in the sense of null controllability for a linear parabolic equation in an unbounded domain.
”As far as we know, Stackelberg–Nash strategy has not yet been considered in an unbounded domain in the sense of null controllability,” the researcher explains.
“In this paper, we study the hierarchical control using the Stackelberg–Nash strategy for a nonlinear parabolic equation in an unbounded domain. We assume that we can act on the system by three controls hierarchically. Two controls called Followers that provide a Nash equilibrium for two cost functionals. The third control named Leader is supposed to bring the state of the system to rest at the final time. The results are achieved by means of observability inequality of Carleman type that we established for the adjoint systems and a fixed point theorem under the assumption that the uncontrolled domain is bounded.”
The motivation of the hierarchic control, says the researcher, comes for example from the environmental problems. “The system can be used to describe the diffusion of a pollutant (e.g. the chemical product) named y in a river. Our goal here is to bring the concentration of this pollutant to zero at the final time T >0 with appropriate control denoted k, trying meanwhile to keep the concentration of this pollutant to the desired state in Oi,d along the interval (0, T) with another controls named vi.”
Following the ideas of cited papers on the null controllability problem in unbounded domains, DJOMEGNE NJOUKOUE, therefore, studies the Stackelberg–Nash null controllability of a nonlinear heat equation in an unbounded domain. ”Using the fact that the control which brings the system to rest at the final time acts on an open unbounded set such that the uncontrolled domain is bounded, we prove using appropriate Carleman inequalities and fixed-point theorem that the system is Stackelberg–Nash null controllable. The novelty of this paper is that we extend Stackelberg–Nash strategy to an unbounded domain,” he states.
He further points out that, in the linear case, the quadratic functionals are convex and then a Nash equilibrium is looked for. “But, in the semi-linear case, we don’t have the convexity of those functionals in general. That is the reason why we redefine the concept of equilibrium and then, we now look for a Nash-quasi equilibrium. Next, we show that under certain conditions, there is an equivalence between Nash equilibrium and quasi-Nash equilibrium.”
Ph.D. Research Fellow LIONEL LANDRY DJOMEGNE NJOUKOUE was supported by the German Academic Exchange Service (D.A.A.D) under the AIMS Cameroon Ph.D. Scholarship Program. His research domain is specialized in Control Theory and its Application to PDEs and Pollutions.
Full Research Paper: https://bit.ly/3GsjDu4