My main research area is the analysis and control of partial differential equations, with a focus on well-posedness, controllability, observability, and stabilization of dispersive, parabolic, and hyperbolic equations, as well as coupled systems, transmission problems, and PDEs posed on networks.
Robust stabilization of PDEs: stabilization in the presence of nonlinear or non-ideal effects such as actuator saturation, time delays and disturbances.
Microlocal and spectral methods in control and inverse problems: Carleman estimates, semiclassical and microlocal analysis, resolvent estimates, and their applications to controllability, stabilization, and inverse problems.
Geometric, bilinear, and quantum control: geometric and saturation-based methods for nonlinear PDEs on manifolds, Schrödinger-type equations and related quantum control problems.
Initial-boundary value problems for dispersive PDEs: existence, uniqueness, regularity, and boundary-value problems for nonlinear dispersive equations, with particular interest in KdV-type equations and related systems
Numerical methods for kinetic and plasma models more recently, numerical and spectral methods for kinetic equations arising in plasma physics.
Publications
Preprints:
Saturation and small-time control on compact manifolds. These three preprints belong to one project. We study when repeatedly applying a nonlinearity to a finite family of controls generates a dense set of directions. This, is the starting point of the Agrachev–Sarychev approach. The first two build saturating families on compact manifolds for power nonlinearities and for the squared gradient |∇φ|². The third treats bilinear Schrödinger equations on compact homogeneous spaces, where the group structure gives explicit families of controls.
2026 R. Buffe, A. Duca, H. Parada, Saturation on compact manifolds via Stone–Weierstrass type theorem, and forced nonlinear PDEs. Submitted. Preprint version.
2026 R. Buffe, A. Duca, H. Parada, Saturating spaces for the squared-gradient nonlinearity on compact Riemannian manifolds via Nash embeddings. Submitted. Preprint version.
2026 H. Parada, Small-time approximate controllability of bilinear Schrödinger equations on compact normal homogeneous spaces. Submitted, Preprint version.