Stackelberg-Nash Exact Controllability for the Benney--Lin type Equation with Mixed Boundary Conditions
Résumé
This paper deals with a bi-objective control problem for a strongly dissipative fifth-order Korteweg-de Vries (KdV) equation with Dirichlet-periodic mixed boundary conditions by means of distributed controls.
More precisely, we address a Stackelberg-Nash control problem, where our first priority is to achieve the ``controllability'' type objective with a secondary objective of ``optimal control'' type.
The control is assumed to be localized in three mutually disjoint subsets of the spatial domain and the control restricted to each region has different objectives to accomplish. One of the localized controls is responsible for local controllability to a free trajectory of the concerned nonlinear model, while the other two controls have to act as a Nash equilibrium for a pair of given functionals.
To study such a problem, we give a hierarchy to the localized controls. More precisely, assuming the existence of the localized control {\em(called leader)} responsible for the ``controllability'' type objective, we establish the existence of the other two controls {\em(called followers)}, which depend on the choice of leader and whose goal is to keep the state close to two given trajectories on two different regions of the domain.
In order to find the leader control, a controllability result is first established for a linearized forward-backward coupled system by deriving a weighted observability estimate for the associated adjoint system. The local inversion mapping theorem is then employed to guarantee the existence of the leader control for the nonlinear problem.
To obtain the desired weighted observability estimate, a new Carleman estimate is derived.
Mots clés
Mathematics Subject Classification. 35Q30 35Q53 93B05 93B07 93C10 91A65 49J20 Fifth-order Korteweg-de Vries equation Benney-Lin type equation Hierarchical controls Exact controllability Carleman estimate
Mathematics Subject Classification. 35Q30
35Q53
93B05
93B07
93C10
91A65
49J20 Fifth-order Korteweg-de Vries equation
Benney-Lin type equation
Hierarchical controls
Exact controllability
Carleman estimate
Domaines
Mathématiques [math]Origine | Fichiers produits par l'(les) auteur(s) |
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