1
vote
0answers
177 views

An optimization problem on the sphere

Let $S$ be a sphere centered at origin in $\Bbb R^{2n}$ of radius $\sqrt{2n}$. Let $D$ be a diagonal matrix. Let $U$ be an orthogonal matrix. Let $r\in\Bbb Z_+$ be a fixed integer. Let vector ...
0
votes
0answers
181 views

A tricky optimization problem over matrices

Hi I have the following problem whose solution has lured me for some months now.... All matrices are complex $N\times N$. Let $A$ be a positive definite matrix with all eigenvalues strictly smaller ...
1
vote
0answers
90 views

null controllability of linear wave equation

Consider the linear wave equation : $$z_{tt}=\Delta z + k(x) z + h(t) , \; in \; \Omega\times (0,T)$$ Are there sufficient conditions on the functions $k(x)$ and $h(t)$ for which $(z,z_t)$ vanish ...
1
vote
0answers
54 views

strong stability for the wave equation

Consider the $n-$dimensional wave equation $$z_{tt}=\Delta z + k(x) z - \epsilon {1}_\omega z_t, \; in \; \Omega\times (0,T)$$ where $\omega\subset \Omega.$ Can I have $z(t) \to 0,$ as $t\to+\infty$ ...
1
vote
0answers
221 views

Matrix conditions under which spectral radius is smaller than 1?

Hello everyone, I would like to find out which conditions are necessary so that the spectral radius $\rho(M)<1$ where $M$ represents the following matrix: $M = \left( \begin{array}{ccc} W & 0 ...
3
votes
2answers
237 views

Analogue of PSD matrices for permanents?

Let's begin with a few observations. Suppose we consider the set of $N\times N$ matrices and consider the matrices with positive determinant. There are several connected components in this set; let ...