# Tagged Questions

Operations research, linear programming, control theory, systems theory, optimal control, game theory

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### Compactness of the set of solutions to an ODE

I am interested in optimal control problems with state constraints in the form of ordinary differential equations given by $\dot{x} = f(x,p), \quad t \in [0,T], \quad x(0) = x_{0}, \quad p \in K$, ...
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### Representation of all pass transfer functions/inner functions as Blaschke product.

What is the proof for: 'An all pass transfer function/inner function can be represented by a Blaschke Product' ?
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### Control a linear system to the kernal space of the output matrix.

Consider the following controllable and observable linear system $$\dot x=Ax+Bu, y=Cx,$$ where $x\in \mathbb{R}^n, u\in \mathbb{R}^m, y\in \mathbb{R}^p$. The observability of $(A,C)$ ...
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### Approximating solutions to minima of the discrete Lagrangian

I have been stuck on this problem for a week and I'm not sure whether or not it is hard or I'm just missing something obvious. General gist of the problem I have a variational problem on a ...
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### Sufficiency of stationary policy for negative stochastic dynamic programming

Consider a Markov Decision Process with Borel state space $X$ and Borel action space $U$, like the one defined in the book "Stochastic Optimal Control: Discrete-time case" by Bertsekas and Shreve. All ...
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### Optimal Control

Consider the optimal control problem with an optimal trajectory $x^*(t)$ and an initial point $(x_1^0,x_2^0)$ \begin{split} \dot{x}=A x + Bu \\\ J=\int^\infty_0(x_2^2+\epsilon u^2)dt ...
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### Riemannian metric on a space of “not-quite-smooth” (hyper)surfaces?

Apologies if the title's a bit vague; I'll try to explain everything below, and please let me know if I should clarify anything. I've been looking for a while at variational problems on polytopes. I'...
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### Upper bound concerning Snell envelope

Consider, on a filtred probability space $\left (\Omega, \mathcal F, \mathbb F , \mathbb P \right )$ where $\mathbb F = \left(\mathcal F_ t \right )_ {t\geq 0}$ is filtration satisfying the usuual ...
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### maximum of the sum of polynomials

Hi I have $n$ polynomials, each one is positive over a certain range and the maximum value each can attain is 1. Also each polynomial has atmost one peak(maximum). Is there some way that I can show ...
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### If “force” is periodic does it imply “velocity” is periodic ? (or decoding tail-bited conv. codes)

I'll try to translate certain problem about convolutional codes to more common language of ODE, hope my translation is correct, but welcome to criticize. Consider two given functions periodic ...
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### An S-lemma for polynomials of degree 4 in three variables

Might the following be true: Let $p,q\in\mathbb{R}[x,y,z]$ be homogeneous polynomials, with $\deg(p)\leq 4$ and $\deg(q)= 2$. Suppose $q(x,y,z)>0$ for some $x,y,z\in\mathbb{R}$. Then the ...
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### Simultaneous maximization of two Generalized Rayleigh Ritz Ratios

Consider hermitian positive semi-definite matrices $A_1$ and $A_2$. Consider also positive definite matrices $B_1$ and $B_2$. I want to maximize the minimum of the two Generalized Rayleigh Ritz ratios ...
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### solving Lyapunov-like equation

The following matrix equation might be a Lyapunov-like equation, but it seems hard for me to develop a simpler way to solve it. From the computation effort, I need some help for solving the special ...
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### Gandhi's quote formalized [closed]

Hello, I hope this question is appropriate for Mathoverflow. Gandhi said, "Be the change that you wish to see in the world". I don't understand anything in Game/optimization theory (I don't know ...
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### (probably simple) optimization question

Suppose you have a concave function defined over a non-polyhedral convex cone and you are interested in the infimum. What would be standard approaches to tackle the question? (The cone is actually PSD ...
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### LP/QP with not-so-constant linear constaints

I have an otherwise standard LP or PSD QP problem as below: $\min\limits_x {c}' x$ subject to $Ax\leq b$ or $\min\limits_x \frac{1}{2}{x}' Qx + {c}' x$ subject to $Ax\leq b$ the only exception ...
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### A Function with Exactly $k$ Minima in a Bounded Space

Is it possible to have a function with the following properties? (i) The function maps a bounded $n$-dimensional space $\mathcal{X}$ (say $\left[0,1\right]^n$) onto a bounded interval $\mathcal{Y}$ (...
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Consider a closed Riemannian manifold $(M,g)$ and a positive function $\psi: M \to R$. Fix a point $p \in M$, I have been struggling to construct a solution to the heat equation, $\partial_t u = \... 1answer 222 views ### Minimizing action squared versus action I have a very basic question in the calculus of variations: Suppose I want to minimize the functional $$A[r, r'] = \int_\Omega L(r, r') dx$$ When is it possible to say that extremals of$A$agree ... 4answers 4k views ### “You can't push a rope” [closed] "You can't push a rope" is a wisdom saying that some engineering teachers pass along to their students. Since I'm not an engineer, I can only guess at what they mean, but it sounds to me like code ... 1answer 289 views ### Going in the direction of the gradient First, a motivating example. Suppose$f(x)$is convex, differentiable, with a single minimum$x^*$. Then the differential equation $$\dot{x}(t) = -\nabla f(x(t))$$ drives$x(t)$to$x^*$. Now my ... 0answers 136 views ### Convexified threshold of a function Upd. The question in a nutshell: find convex set on plane which is «closest» to a given non-convex set. It is given integrable function$0\leq f(x,y)\leq 1$with bounded support:$f(x,y)=0$when$x^2+...
Dear MO contributors, let $r > 0, L > 0$. I am interested in maximizing the integral: $$\int_0^{2\pi} \frac{f(\alpha)^2 f'(\alpha)^2}{\sqrt{f(\alpha)^2 + f'(\alpha)^2}} \ \mathrm{d} \alpha$$ ...
I have the following optimization problem: $$\text{find } x= \min_{a} \max_{\lambda\in\Lambda} |R(h\lambda)|$$ where $\Lambda$ is some finite, fixed set of complex numbers and $R(z)$ is a ...