Title: Bilinear control of parabolic evolution equations
Abstract: Despite the importance of control systems governed by bilinear controls for the description of phenomena that cannot be realistically modeled by additive controls, the action of multiplicative controls is generally not so widely studied as it happens for boundary and locally distributed controls. The main reasons of this fact might be found in the intrinsic nonlinear nature of such problems and furthermore, for controls that are scalar functions of time, in an ineluctable obstruction for proving results of exact controllability, which is contained in the celebrated work of Ball, Marsden and Slemrod, 1982.
In this talk, I will present results on the stabilization and controllability of parabolic-type evolution equations using a scalar input bilinear control. Specifically, I will focus on steering the system toward particular target trajectories known as eigensolutions. I will also introduce an extension of this work that enables us to control the Fokker-Planck equation by the drift term.
Finally, if time permits, I will discuss some recent results on small-time reachability properties of a nonlinear parabolic equation, controlled via a bilinear control, defined on a torus of arbitrary dimension. Assuming a saturation condition on the potentials, we establish the small-time approximate controllability between states that share the same sign. Furthermore, in the one-dimensional case, we extend this result by combining it with a local exact controllability property. This approach allows us to demonstrate the small-time exact controllability of any positive state to the ground state of the evolution operator.
References:
J. M. Ball, J. E. Marsden, M. Slemrod. “Controllability for distributed bilinear systems.” SIAM Journal on Control and Optimization vol. 20, n. 4, pp. 575-597 (1982)
F. Alabau-Boussouira, P. Cannarsa, C. Urbani “Superexponential stabilizability of evolution equations of parabolic type via bilinear control”, Journal of Evolution Equations, vol. 21, pp. 941-967 Springer (2021)
F. Alabau-Boussouira, P. Cannarsa, C. Urbani “Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control”, Nonlinear Differ. Equ. Appl., vol. 29, pp. 38 (2022)
F. Alabau-Boussouira, P. Cannarsa, C. Urbani “Bilinear control of evolution equations with unbounded lower order terms. Application to the Fokker-Planck equation”, Comptes Rendus Mathématiques, vol. 362, pp. 511-545 (2024)
A. Duca, E. Pozzoli, C. Urbani “On the small-time bilinear control of a nonlinear heat equation: global approximate controllability and exact controllability to trajectories”, Journal de Mathématique Pures et Appliquées, vol. 203, pp. 103758, (2025)