  Solving equilibrium problems under constraints is an important problem in
optimization and optimal control. In this context an important practical
challenge is the efficient incorporation of constraints. We develop a
continuous-time method for solving constrained variational inequalities based
on a new penalty regulated dynamical system in a general potentially
infinite-dimensional Hilbert space. In order to obtain strong convergence of
the issued trajectory of our method, we incorporate an explicit Tikhonov
regularization parameter in our method, leading to a class of time-varying
monotone inclusion problems featuring multiscale aspects. Besides strong
convergence, we illustrate the practical efficiency of our developed method in
solving constrained min-max problems.
