Questions tagged [linearization]
On the many different ways to turn non-linear systems of equations into linear ones.
385 questions
1 vote
1 answer
57 views
Linearization Question for max-min|x| Bi-level Optimization Problem
I'm currently working on a bi-level optimization problem with the following structure: max min |x| I attempted to linearize this problem using the following approach: Introduce an auxiliary variable ...
- 13
0 votes
0 answers
55 views
Case Study with Problem in Optimization
I have a problem that I'm trying to solve involving a bicycle ride share program. Riders rent bikes of various kinds (electric or normal-non powered) at various stations in a city, and after they are ...
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2 votes
1 answer
191 views
Why is the linearization of a differential equation still an equation?
Picture below is from Evans’ Partial differential equations. I want to know why the linearization of a differential equation still be an equation? In my view, the equation can be regarded as an ...
- 6,976
0 votes
1 answer
86 views
Why is the small strain hypothesis considered sufficient for finite strain linearization?
In continuum mechanics, the infinitesimal strain tensor $ \varepsilon $ is introduced as: $$ \varepsilon_{ij} = \frac{1}{2} \left( u_{i,j} + u_{j,i} \right), \quad u_{i,j} = \frac{\partial u_i}{\...
3 votes
1 answer
302 views
Is there a differentiable function with a "slow" linearisation?
Suppose $f$ is a differentiable real-valued function of a real variable. By linearisation, we can write $$f(x)=f(0)+xf'(0)+xh(x)$$ where $\lim_{x\to 0} h(x)=0$. If $f$ is twice-differentiable then we ...
- 3,725
1 vote
1 answer
93 views
Deriving the Jacobi equation as linearization of the geodesic equation
Consider the geodesic equation $D_{\gamma'}\gamma'=0$. In coordinates this reads as the non-linear ODE: $$(1)\hspace{2cm}(\gamma^i)''+(\gamma^j)'(\gamma^k)'\Gamma_{jk}^i=0.\hspace{5cm}$$ In Eschenburg'...
- 1,197
0 votes
0 answers
88 views
Linearization of the division of linear functions
I have an optimization problem in which I have the term $\frac{f(x)}{g(y)}$, where $f(x)$ and $g(y)$ are both linear functions and furthermore $g(y) >0$. I thought that I could define a variable $...
0 votes
1 answer
57 views
Seeking advice on how to "see" this derivation
I'm struggling to model this constraint for a problem: $$x_C^4 = 1 \implies (x_A^4 + x_B^4 \geq 1 \land x_A^1 + x_B^1 = 0) \;\lor\; x_A^2x_B^3 = 1 \;\lor\;x_A^3x_B^2=1.$$ where all variables are ...
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1 vote
1 answer
91 views
In the linearization of the Ricci Flow
I am trying to prove the following: Let $R_{ij}$ be the coordinates of the Ricci curvature tensor in a Riemannian manifold $(M,\,g(t))$ with time-dependent metric $g$. Suppose $\dfrac{\partial}{\...
- 361
0 votes
1 answer
76 views
Criteria on Jacobian for stability analysis of fixed points of autonomous system of ODE
I have two questions regarding stability analysis of a system of ODEs of the form: \begin{equation} \begin{cases} \frac{dx}{dt}=f(x,y)\\ \frac{dy}{dt}=g(x,y) \end{cases} \end{equation} If $(x^*,y^*)$ ...
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1 vote
1 answer
69 views
How to linearize a nonlinear state space model?
I have been studying the Feedback Control of Dynamic Systems 8th by G. F. Franklin. Here in chapter 9 I have been working on the Problem 9.3 I have a question regarding the point (b). I have ...
- 401
0 votes
0 answers
37 views
Approximation of Perturbations in a Non-Hyperbolic System
I'm going to present a simple example with $n=2$ to show the question I would like to ask. Consider the discrete map $ x' = f(x) $ with a fixed point $ (\bar{x}, \bar{y}) $. Suppose the Jacobian ...
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1 vote
1 answer
167 views
How can I linearize a state space model about non-fixed points?
I have an adequate understanding of state-space models. I've stabilized an inverted pendulum cart robot with a linear quadratic regulator by modeling the system, finding the Jacobian of the nonlinear ...
- 331
0 votes
0 answers
63 views
Transfer 1D heat-diffusion equation in continuous-time state-space model structure
I'm trying to transfer the 1D heat diffusion equation: $$ \frac{\partial T(t,\zeta)}{\partial t}=\alpha \frac{\partial^2 T(t,\zeta)}{\partial \zeta^2}$$ into the continuous-time state-space model ...
1 vote
1 answer
57 views
How to linearize the adiabatic pressure of a gas spring using binomial expansion?
I found the following formula in a textbook on Stirling engines and am struggling understanding the linearization. $$p_b=p_{mean}\left(\frac{V_B}{V_B-A_p(x_p+x_c)}\right)^{\gamma}$$ Linearization ...
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