  A Tikhonov regularized inertial primal\mbox{-}dual dynamical system with time
scaling and vanishing damping is proposed for solving a linearly constrained
convex optimization problem in Hilbert spaces. The system under consideration
consists of two coupled second order differential equations and its convergence
properties depend upon the decaying speed of the product of the time scaling
parameter and the Tikhonov regularization parameter (named the rescaled
regularization parameter) to zero. When the rescaled regularization parameter
converges rapidly to zero, the system enjoys fast convergence rates of the
primal-dual gap, the feasibility violation, the objective residual, and the
gradient norm of the objective function along the trajectory, and the weak
convergence of the trajectory to a primal-dual solution of the linearly
constrained convex optimization problem. When the rescaled regularization
parameter converges slowly to zero, the generated primal trajectory converges
strongly to the minimal norm solution of the problem under suitable conditions.
Finally, numerical experiments are performed to illustrate the theoretical
findings.
