0
votes
0answers
27 views

An error in least square optimization problem in Matlab

I am new to MATLAB and I want to formulate the following lease square expression in Matlab. I have some codes that I am typing here. But the optimization problem solution seems not to be correct. Does ...
0
votes
0answers
28 views

Minimizing a multivariable function in several variables

I would like to show that a certain function is negative, to help establish asymptotic stability via a Lyapunov function for a system of differential equations. This is exactly what I need help on: ...
1
vote
3answers
39 views

Calculate minimum perimeter of a rectangle with an extra constraint.

I have been set this problem, and although I can derive a minimum perimeter using calculus, I now need to add an extra constraint to one side of the rectangle and I am having problems deriving a ...
1
vote
1answer
37 views

Constrained Optimization Problem with system of differential equations

thanks in advance for any help. My question relates to a homework problem. I have been given a system of generalized differential equations with the goal of creating an algorithm that will solve ...
0
votes
2answers
73 views

Minimizing the following objective function with matrices

Suppose $A$ and $B$ are known matrices, and we are to find matrix $X$ that minimizes the following function, $$\frac{1}{2}||X||^2+\frac{1}{2}||X^TAX-B||^2$$ Taking the relevant derivative w.r.t $X$ ...
1
vote
0answers
84 views

Algebraic manipulation of Lyapunov function

I have a problem I would like some feedback on. I have spent 6 hours on it examining various techniques (numerically and analytically). I need to find the values of $k$ for which $x^2+ky^2$ is a ...
0
votes
2answers
1k views

How to plot a phase portrait for this system of differential equations?

I beg your help.. I'd like the phase portrait for this system. I don't know how to use Mathematica/Matlab ... :( If anyone can make this portrait and post a print screen here, I would thank you ...
1
vote
1answer
75 views

A System of differential Equations

How can I analyze the phase diagram for this system of differential eqs? This field is not my area of my expertise, so please be generous with the answers. I appreciate quick references as well. ...
0
votes
1answer
62 views

Is it possible to solve or approximate this second order nonlinear system of differential equations.?

Given initial values $d[0]$ and $k[0]$, I would like to solve for the initial rate of change, $\dot d[0]$, and compare this value against some data. I have the following profit function, which I ...
0
votes
0answers
23 views

Proving existence of a solution to multidimensional ODE

I have the following ODE: $$ \frac{\partial u}{\partial t} + \min_h A(h, u) = 0, $$ where $h$ is a function of $u$, and both $h$ and $u$ are vectors of fixed dimension. The boundary condition is given ...
1
vote
1answer
422 views

Optimisation of a rectangles area under a function curve

I have a questions asking for the dimensions of the rectangle with the largest area that has two bottom corners on the x axis and two top corners on the curve $y=12-x^2$. I have plotted the curve and ...
0
votes
1answer
65 views

Differentiation optimisation

I have a question stating that an insulation strip is to be sealed completely around three edges of a rectangular solar panel. The length of this strip is 200cm. It is asking what dimension of the ...
0
votes
2answers
420 views

Second Order Differential Equation - finding maximum and minimum values of particular integrals

Given that $y$ = $\frac{3}{4}\cos3x + \frac{1}{4}\sin3x$ is a particular integral of the differential equation $$\frac{d^2y}{dx^2}+ 4\frac{dy}{dx}+13y = 6 \cos3x-8\sin3x$$ how do I show that ...
0
votes
2answers
242 views

Newtons method and finding stationary points

I have an equation $l(x) = \sqrt{(x - 0.2)^2 + (x^2 - 2.7)^2}$. Now I basically want to find at which x coordinate that $l(x)$ will be it's smallest. I have differentiated the equation to find ...
0
votes
0answers
37 views

Does this problem have an optimal solution

The problem is as follows, $\max_{g(\cdot)} -\int_0^{g(0)}(g(0)-x)g(x)dx$ s.t. $g(\cdot)$ is from the class of continuous strictly decreasing functions on $[0,1]$ and $0<g(0)<g(1)<1$. ...
0
votes
1answer
809 views

Fundamental matrices

Find the fundamental matrix for the two-dimensional system defined by $x_1' = x_1 + tx_2$, and $x_2'=x_2$. And determine the solution for which $x_1(0)=c_1$, and $x_2(0)=c_2$. I am stuck because of ...
2
votes
0answers
51 views

Finding the best real value for $C$.

Consider the recurrence $f_{n+1}=f_n + \ln(f_n)$ with $f_0=2$. Also consider differential equations of type $g(0)=2$ and $\dfrac{d g}{d x}=\ln(g(x)- C \cdot \ln(g(x)))$. Lets call the solution ...
3
votes
1answer
128 views

minimization problem on differential equations - optimal control

I am trying to minimize an time-integral of a linear function with respect to differential equations. The problem is formally defined as follows: Given $\lambda< \mu_1, \mu_2$ fixed ...
1
vote
1answer
160 views

Enigmatic optimization problem

My problem, which I proposed to myself months ago is based on the simple optimization problem in which you find the best path for a lifeguard to rescue a drowning victim. Obviously the shortest ...
4
votes
1answer
294 views

Optimizing a functional with a differential equation as a constraint

I am working on solving the following optimization problem. I think it is well-poised but, if not, please give me some pointers that could make the question make more sense. We have a parametric ...
0
votes
1answer
37 views

Numeric Differentiation of Analytic funtion

Can anyone validate if my understanding regarding numeric differentiation is correct?? $z = f(x,y)$ is an analytic function. $$\frac{\partial z}{\partial x} = \frac{f(x+h,y)-f(x,y)}{h}$$ ...
1
vote
0answers
152 views

A dynamic Stackelberg game - general characterization

my question is about general representation of a dynamic Stackelberg game which is played in continuous time. We have maximization problems of two agents who play this game. Agents are 'Leader' and ...
1
vote
1answer
91 views

2nd Order Optimal Control Problem

I'm working on a homework problem in optimal controls and my plant model is described as: $$\ddot{x}(t) = u(t)$$ The performance index (cost function) is described by: $$J = 1/2\int_0^5u^{2}(t)dt\,$$ ...
1
vote
1answer
115 views

how to obtain Euler equation for smoothing spline minimization problem?

The question might be trivial, but I don't understand why this minimization problem in Sobolev space $$ \min_{g}\int_{0}^{1}\left\{ f(x)-g(x)\right\}^{2} dx+\lambda\int_{0}^{1}\left\{ ...
3
votes
1answer
282 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 ...
3
votes
1answer
72 views

Maximizing $f(x,y)$

Could somebody please shed some light on this problem? Let $x,y \in \mathbb R$, we wish to maximize $f(x,y)=\frac{x^2-y^2}{(x^2+y^2)^2}$ by finding suitable values of $x,y$. Setting $\partial f\over ...
2
votes
1answer
296 views

Find equation of line such that area formed by line & positive coordinate axis is minimal

Find equation of line passing through $(20,12)$ such that the area of the triangle formed by the line and the positive axis is smallest possible. Also: $\frac{x}{a}+\frac{x}{b}=1$ where $a, b$ are ...
7
votes
2answers
1k views

Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising

For an image denoising problem, the author has a functional $E$ defined $$E(u) = \iint_\Omega F \;\mathrm d\Omega$$ which he wants to minimize. $F$ is defined as $$F = \|\nabla u \|^2 = u_x^2 + ...
5
votes
2answers
291 views

How to Interpret Time Scales in a Dynamic System

Here I have a question about time scales in dynamic systems - for reference you can look at a previous question that spurs this one: Minimizing the cost of a path in a dynamic system That question ...
2
votes
1answer
143 views

Minimizing the cost of a path in a dynamic system

So suppose I want a path from 0 to $c>0$ on the real line, and I am going to use the function $S(t)$ to get there in (discrete) time $T$. That is, my position at time 0 is 0, my position at time $T$ ...
3
votes
4answers
2k views

Karush-Kuhn-Tucker condition - Lagrange multiplier

I was maths student but now I'm a software engineer and almost all my knowledge about maths formulas is perished. One of my client wants to calculate optimal price for perishable products. He has ...