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Let $D\in\mathbb{R}^{n\times n}$ be a diagonal positive definite matrix s.t. $D\leq I$ ($I$ denotes the $n$-dim. identity matrix) and let $\alpha$ be a strictly positive real number. Consider the opti …

Let $\mathrm{diag}(A)$ denote the diagonal matrix with diagonal entries of $A\in\mathbb{R}^{n\times n}$ and let $\succeq$ denote the standard partial order in the cone of (symmetric) positive definite …

Let $A\in\mathbb{R}^{n\times n}$ be a diagonalizable matrix with real and strictly negative eigenvalues. Furthermore, suppose that $\mathrm{tr}(A)=-1$. My question. I'm wondering whether it is pos …

Consider the linear time invariant system: $$\tag{1}\label{eq1} \dot{x}(t) = Ax(t) + Bu(t), \ \ x(0)=x_0\in\mathbb{R}^n, $$ where $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times m}$. Let $p_M(s …

Let $A\in\mathbb{R}^{n\times n}$ be a stable matrix (i.e., the eigenvalues of $A$ have negative real parts). Consider the following optimization problem in $X \in \mathbb{R}^{n \times n}$ $$\begin{ar …

Let $A_1$, $A_2$ be $n\times n$ real matrices. Suppose that $A_1$ and $A_2$ are Schur stable (i.e., their eigenvalues are strictly inside the unit circle in the complex plane). Let $B_1$, $B_2$ be two …

Let $A\in\mathbb{R}^{n\times n}$ be a matrix having eigenvalues with strictly negative real part (in other words, $A$ is supposed to be Hurwitz stable). Let $\mathrm{tr}(\cdot)$ denote the trace opera …

Let $A \in \mathbb{R}^{n \times n}$ be a Hurwitz matrix, i.e. $A$ satisfies $\mathrm{Re}\,\lambda_i< 0$, where $\{\lambda_i\}_{i=1}^n$ is the set of eigenvalues of $A$. Suppose that the trace of $A$ i …

Let $A$ be an Hurwitz stable matrix (i.e. the real part of the eigenvalues of $A$ lie in the left-half plane) and $Q$ be a positive semidefinite matrix ($Q\ge 0$, for short). Let $P>0$ be the solution …