Suppose that your optimization problem is $(P1) \quad \min_{x} f(x)$ such that $x \in \Omega \qquad$, where $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is a continuous real-valued function and $\Omega \... View answer 3 votes A short proof can be found in Problems 17 and 18 from 7.2 of Horn, Roger A., and Charles R. Johnson. Matrix Analysis. Cambridge University Press, 1990. Proof. We first prove the following inequality: ... View answer 3 votes We have that$f: \mathbb{R} \rightarrow \mathbb{R}$given by$f(x)=x^2$is not absolutely continuous (not even uniformly continuous), yet$f$is Fréchet differentiable with its Fréchet derivative ... View answer 2 votes To check the stability of dynamical systems, one can try to search for a real-valued positive function named Lyapunov function. If the dynamical system is LTI, that is, of the format $$\dot{x}(t) = ... View answer 2 votes Suppose that (A,B) is stabilizable and Q is Hermitian. Define the operator \mathcal{R}(X) as$$ \mathcal{R}(X) = A^TX + XA - XBB^TX+ Q. $$If there exists a Hermitian solution X of the ... View answer 2 votes There is a more straightforward way by using that a function is lower semicontinuous if and only if for each c \in \mathbb{R}, the set \{ x \in X : f(x) \le c \} is closed. For the proof of this ... View answer Accepted answer 2 votes This is also known as the modal decomposition in engineering applications, specially in vibrations. It is very useful for the decoupling of differential equations. A good example of its use in this ... View answer Accepted answer 1 votes The higher order differential equation you proposed to analyze is known in the literature as a Fuchsian differential equation. According to pp. 75-77 of the thesis Differential equations and group ... View answer 1 votes 1) It is defined for all kinds of cone, even for an arbitrary set (see this link for more information). 2) A figure is shown in p. 143 of Dattorro, Jon. Convex optimization & Euclidean distance ... View answer 1 votes Routh–Hurwitz stability criterion is a necessary and sufficient criterion for the Hurwitz stability of a polynomial, meaning that the real part of all the roots of the polynomial are negative. Note ... View answer 0 votes As the other answers have already stated, the concepts of stochastic differential equations (SDE) and random differential equations (RDE) are different. I would like to complement the answers with a ... View answer 0 votes Suppose first that p(z)=\sum_{k=0}^{n} a_k z^k is a polynomial without zeros on the imaginary axis and let \gamma(t):= R e^{\frac{\pi}{2}jt}, for t\in[-1,1]. By argument principle,$$\int_{... View answer 0 votes Recently, (1) proposed novel formulations of solution algorithms for differential matrix equations based on an$LDL^{T}\$ decomposition that keep the computations in real arithmetic. According to the ...