Lyapunov optimization of a damped system

A Spatio-temporal Design Problem for a Damped Wave Equation

“…Similarly, using that s(t, x) is the volume fraction of the α-material at point (t, x), we associate with s lim a sequence of bivalued functions s pen N ∈ L ∞ ((0, T ) × Ω, {0, 1}) (see [20]). For (α, β, d) = (1,4,10) and (a α , a β ) = (α, β), Figure 5.7 represents the function r pen M =30 associated with the density r lim of Figure 5.2(right). Similarly, Figure 5.8 represents the function s pen N =30 associated with the optimal density s lim of Figure 5.1(bottom right).…”

“…where I designates the cost function (1.3). Another approach may consist of averaging the cost function over all initial data of unit energy (we refer to [10] in a similar context). Finally, at the numerical level, it seems important to investigate the numerical approximation of the fully relaxed problem (RP) and compare it with the simplified formulation ( RP).…”

Optimal damping of the infinite-dimensional vibrational systems: commutative case

“…The main result of this paper, the construction of the optimal damping, is a well-known result in the matrix case. This result has been generalized in [5] to the non-modal damping, obtaining the same set of optimal matrices. The results for the optimal damping without restrictions on the structure of damping matrices usually are not very useful in practical applications, but there is some recent progress in treating such problems (see [2,18,19]).…”

Damping optimization in mechanical systems with external force

“…[21][22][23]16,18,24]) which is constructed using appropriate linearization of system (1). Within this category, one can consider the following three criteria: minimization of the trace the solution of the corresponding Lyapunov equation, minimization of the 2-norm or the Frobenius norm of the solution of the corresponding Lyapunov equation.…”