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        Questions tagged [oc.optimization-and-control]

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

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        Can the partial concavity of the following decomposable objective function be used for optimization?

        The problem I am trying to solve is the following: $$\begin{array}{ll} \min & f(x)+g(y) \\ \mathrm{s.t.} & y\ge x\ge 0,\\ \ & p\le ax+by\le q, \end{array}$$ where $a,b,p,q$ are ...
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        1answer
        36 views

        Measuring how suboptimal control is

        Suppose I have a linear dynamical system to control. I use PMP to find necessary conditions for the optimal control of the system wrt to some objective function. Now, suppose that the trajectory I ...
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        0answers
        108 views

        Rigorous proof of the good regulator theorem

        As an applied mathematician with an interest in control theory, I have read several research papers that explicitly use the good regulator theorem of Conant and Ashby 1 which states that: Every ...
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        13 views

        Simple examples of the Chen–Fliess series and the Fliess operator applications

        I am looking for simple examples of the application of the Chen–Fliess series to input–output systems. In the best case scenario, something like linear systems affine in control for which the result ...
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        votes
        1answer
        182 views

        Best approximation of a compactly supported density by a single Gaussian

        Note: This is a follow-up question inspired by a previous (more difficult) question I asked on MathOverflow. Let $f:\mathbb{R}\to\mathbb{R}$ be a (sufficiently regular, e.g. smooth) probability ...
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        1answer
        61 views

        Conditioning on an irrelevant variable in a martingale control problem

        Suppose I have two independent Brownian motions $B^1_t, B^2_t$ and $\mathbb F_t$ be the natural filtration generated by them. Let $T > 0$ be a fixed finite number. Let $q_t$ be a $[-1,1]$ valued $\...
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        1answer
        112 views

        Trajectory leaving a set

        Consider the differential equation $\dot{x}=f(x)$, where $f: \mathbb{R}^2 \to \mathbb{R}^2$ is smooth. Given a set $A \subset \mathbb{R}^2$, are there some results saying that whenever $x(0) \in A$, ...
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        2answers
        82 views

        Expected minimum of a linear function on the unit cube

        Let $c\in\mathbb{R}^n$ and let $X_1,X_2$ be two independent uniform samples on the unit cube in $\mathbb{R}^n$. Is there anything at all (in an analytic sense) we can say about the expectation $E\min\...
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        33 views

        Convex optimization problem with really simple submodular structure

        I am trying to characterize the solutions to the below convex optimization problem as concisely as possible, where we are given as input a probability vector $\mathbf{p}\in\mathbb{R}^n$ and a positive ...
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        21 views

        Optimal control problem with spike source and “split” state

        For $p \in \mathbb{R}$, consider the following problem: \begin{equation} \label{1} \begin{cases} \operatorname{div}(a \nabla u ) = p\delta_{x_0} \quad \text{in } \Omega \\ u=0 \quad \text{on } \...
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        161 views

        Equal principal minors of matrix plus rank-1 and inverse

        Given an invertible real matrix $A$ and real column vectors $b$ and $c$. For which $A$, $b$ and $c$ are all corresponding principal minors of $B = A-bc^T$ and $A^{-1}$ equal? According to a ...
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        237 views

        Geometric bang-bang theorem for nonlinear optimal control

        The classical bang-bang theorem is usually stated for linear systems (e.g. Control Theory from the Geometric Viewpoint by Agrachev-Sachkov, p. 209). Sussman proved a nice generalization for systems ...
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        30 views

        Single step analysis of Augmented Lagrangian Method

        I am wondering if there is any single step analysis for the Augmented Lagrangian method. Specifically, the problem is $$\min f(x) \text { s.t. } A x=b$$ where $f$ is convex, smooth. Such an objective ...
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        117 views

        How to choose phase to give a desired Fourier transform

        Cross posted from MSE. I have a mathematical problem arising from a physics application, which I feel must have been solved before, but I don't know the terminology associated with it. I am looking ...
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        Parametric research for the global minimum in a family of polynomial multivariable functions on closed domains

        Consider the family of functions: $$ V(\{x_j\},\{y_j\})=\sum_{j=1}^L\left[\frac{1}{2} x_j^2+ \frac{\beta^2}{2}y_j^2 + \alpha\beta\, x_jy_j \right] $$ Each member of the family is therefore ...

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