Speaker
Description
We consider damped vibrational systems of the form $M\ddot{q}(t)+D(\nu)\dot{q}(t)+Kq(t)=0$, where $M$ and $K$ are positive
definite and $D=D_{\text{int}}+D_{\text{ext}}(\nu)$ with $D_{\text{int}}$ representing some Rayleigh damping and $D_{\text{ext}}(\nu)= \sum_{i=1}^{k}\nu_i^{}\; d_i d_i^T$ representing some external damping caused by $k$ dampers. Optimal damping consists of determining a viscosity vector $\nu\in\mathbb{R}^k_+$ that maximizes the rate of decay of the energy of the system as $t$ tends to infinity.
Several algorithms have been proposed to solve this problem but without the stability constraint on the vibrating system nor the nonnegative constraint on $\nu$. We present a new approach that addresses both constraints. We start with a test that checks a priori for stability of the system for all $\nu\ge 0$. Assuming that the system is stable, we derive the Karush-Kuhn-Tucker (KKT) conditions associated with the optimisation problem. We show that the linear independence constraint qualification (LICQ) holds, which is a crucial requirement for the validity of the KKT conditions at a feasible point. We also derive second order sufficient conditions.
We solve the KKT system with a residual minimization algorithm combined
with Barzilai-Borwein stepsize.
This is joint work with Qingna Li (Beijing Institute of Technology).
