Stackelberg-Nash Controllability for a Quasi-linear Parabolic Equation in Dimension 1D, 2D, or 3D Dany Nina Huaman Journal of Dynamical and Control Systems, 2022 This paper deals with the application of Stackelberg-Nash strategies to the control to quasi-linear parabolic equations in dimensions 1D, 2D, or 3D. We consider two followers, intended to solve a Nash multi-objective equilibrium; and one leader satisfying the controllability to the trajectories.
Exact controllability to the trajectories for parabolic PDEs with nonlocal nonlinearities Enrique Fernández-Cara, J. Límaco, Dany Nina-Huaman, Miguel R. Núñez-Chávez Mathematics of Control Signals and Systems, 2019 This paper deals with the analysis of the internal control of a parabolic PDE with nonlinear diffusion, nonlocal in space. In our main result, we prove the local exact controllability to the trajectories with distributed controls, locally supported in space. The main ingredients of the proof are a compactness–uniqueness argument and Kakutani’s fixed-point theorem in a suitable functional setting. Some possible extensions and open problems concerning other nonlocal systems are presented.
Local null controllability of the N-dimensional Ladyzhenskaya-Smagorinsky with N-1 scalar controls Dany Nina Huaman, Juan Límaco, Miguel R. Nuñez Chávez Sema Simai Springer Series, 2018 This paper deals with the null controllability of a differential turbulence model of the Ladyzhenskaya-Smagorinsky kind. In the equations, we find local and nonlocal nonlinearities: the usual transport terms and a turbulent viscosity that depends on the global in space energy dissipated by the mean flow. We prove that the N-systems are locally null-controllable with N-1 scalar controls in an arbitrary control domain.
On the Theoretical and Numerical Control of a One-Dimensional Nonlinear Parabolic Partial Differential Equation Enrique Fernández-Cara, Dany Nina-Huamán, Miguel R. Nuñez-Chávez, Franciane B. Vieira Journal of Optimization Theory and Applications, 2017 This paper deals with the analysis of the internal and boundary control of a one-dimensional parabolic partial differential equation with nonlinear diffusion. First, we prove a local null controllability result with distributed controls, locally supported in space. The proof relies on local inversion (more precisely, we use Liusternik’s Inverse Function Theorem), together with some appropriate specific estimates. We also establish a similar result with controls on one side of the boundary. Then, we consider an iterative algorithm for the computation of null controls, we prove the convergence of the iterates, and we perform some numerical experiments.
