## Stabilization of non-autonomuous 1-d wavs equation

I want to ask two questions about the stabilization of the equation \eqalign{ & {y_{tt}} = k(t,x){y_{xx}}+a(t,x){y_t}+ b(t,x){y_x}+ c(t,x){y_x} +d(t,x)y \ \ (t,x) \in {\text{ }}(0,\infty ) \times (0 ,1) \cr & y(t,0) = y(t,1) = 0 \cr} The first one: Is there exists an exponential stabilization result concerning the above equation?. The second question: Is it interesting from mathematical or physical applications of this kind of equations?. Thanks.

## On equation $\Delta \circ \partial/\partial X=\partial/\partial X \circ \Delta$ on a Riemannian manifold

Assume that $$M$$ is a compact Riemannian manifold whose Laplacian is denoted by $$\Delta$$. Assume that the Euler characteristic of $$M$$ is zero. Does $$M$$ admit a non vanishing vector field $$X$$ which satisfy $$(*) \qquad \Delta \circ \partial/\partial X=\partial/\partial X \circ \Delta$$

What can be said about the structure of the Lie algebra of all vector fields $$X$$ with the property $$(*)$$?

## Is this a nonlinear algebraic equation?

$$\sum_{n=1}^{N} \alpha_n e^{i\beta_n f_m} X(f_m)=X_e(f_m)$$

If only $$\alpha_n$$ and $$\beta_n$$ are unknowns and $$1 \leq n \leq N$$, how can we find alpha’s and beta’s where $$1 \leq m \leq M$$ and $$M$$ is the number of total equations in the equation set.

So we are left with $$M$$ complex equations and $$2N$$ real unknowns. How can we solve this?

## Differential equation slope field and cauchy solution

Given the differential equation

$$2y’=\frac{yx}{x^2+4}+\frac{x}{y}$$

• I have to draw the slope field in a rectangle called $$P$$, containing the point (2, 1).
• Then in appropriate interval I have to find the solution of the Cauchy problem for the given differential equation with initial equation $$y(x_{0})=y_{0}$$ where $$(x_{0}, y_{0})$$ is inputed by clicking in the rectangle $$P$$
• and in the same rectangle $$P$$ the graph of the found approximation of the Cauchy problem with given initial equation given above .

Here is my solution:

 function Plotslope x=-5:0.6:5; y=-6:0.6:6; delta=0.2;  hold on axis([-5,5,-6,6]) daspect([1,1,1])  for k=1:length(x)         for m=1:length(y)           eps=delta/(sqrt(1+ff(x(k),y(m))^2));           plot([x(k)-eps, x(k)+eps],...          [y(m)-eps*ff(x(k),y(m)),...            y(m)+eps*ff(x(k),y(m))],'k');         plot(x(k),y(m),'k.','LineWidth',0.2)          end end  [x0,y0]=ginput(1); plot(x0,y0,'bo') [T,Y]=ode45(@ff,[x0,5],y0); [T1,Y1]=ode45(@ff,[x0,-6],y0); plot(T,Y,'r',T1,Y1,'r')  function z=ff(x,y)      z=(y*x)/(2*(x^2+4))+x/2; end end   

I am sure I am finding the first 2 bullets, but how can I solve the 3rd one?

## Five equation to solve 6-variables question, is it possible?

I was given a task to find the value of variable a,b,c,d,e and f. But I’m not sure it is even possible, given that only 5 equations are available. Can anybody point out how to solve these:

a+b+c=164.35; d+e+f=94.44;  a^2+d^2=20.06^2;  b^2+e^2=74.34^2;  c^2+f^2=123.27^2 

Thank you.

## PIDTune and charaistic equation of zero

I have a BLDC electric motor, I’m currently trying to control via a PIDTune. This is mostly an attempt to reduce (remove) a small run away drift that ends up showing up in the motor signal u[t].

I’ve modelled this via:

ssm = StateSpaceModel[\[ScriptCapitalJ] \[Phi]''[t] + \[ScriptCapitalR] \[Phi]'[t] == \[ScriptCapitalT] u[t], {{\[Phi][t], 0}, {\[Phi]'[t], 0}, {u[t], 1}}, u[t], \[Phi]'[t], t] 

And simulated:

params = { \[ScriptCapitalJ] -> 4.63 10^-5, \[ScriptCapitalR] -> 1 10^-5, \[ScriptCapitalT] -> 0.0335}; Plot[Evaluate[OutputResponse[ssm /. params, 1, {t, 0, 12}]], {t, 0, 12}] 

This is a nice model response and mirrors the response of the real motor almost exactly.

So I tried to create a control system to add to the control signal and bring the system relatively quickly back to zero.

control = PIDTune[ssm /. params , {"PID"}] 

But I continue to get the following error:

PIDTune::infgains: Unable to compute finite controller parameters because a denominator in the tuning formula is effectively zero. 

I have tried all tuning methods within the documentation, however I continue to get errors.

Changing to a “PD” control

control = PIDTune[ssm /. params , {"PD"}] 

Gives me control system, however when adding it to the feedback and then seeing the response I get a different error:

simul = SystemsModelFeedbackConnect[ssm, control] /. params  OutputResponse[simul, UnitStep[t - 3], {t, 0, 12}]  OutputResponse::irregss: A solution could not be found for the irregular state-space model with a characteristic equation of zero. 

The error messages don’t really make any sense to me…or explain what the issue is with the model…being that it simulates reality quite well….How can I relieve these errors, or create a feedback loop via PIDTune for my system?

Thank you for the help!

There is a similar example with a dcmotor within the documentation for PIDTune for reference which works fine (albeit a different tfm):

dcMotor = TransferFunctionModel[Unevaluated[{{k/(s ((j s + b) (l s + r) + k^2))}}], s, SamplingPeriod ->None, SystemsModelLabels -> {{None}, {None}}] /. pars;  PIDTune[dcMotor, "PID", "PIDData"] 

## How to find a $3$-form that satisfies an integral equation

Q) Let $$(x_1,x_2,x_3,x_4)$$ be positively oriented coordinates. Let $$f(x_1,x_2,x_3,x_4)$$ be smooth on a bounded domain $$B\subset \mathbb{R}^4$$ with a smooth boundary $$\partial B$$ in the induced orientation. Find a $$3-$$form $$w$$ on $$B$$ s.t. $$\int_{B} \Delta fdx_1dx_2dx_3dx_4 = \int_{\partial B}w$$ where $$\Delta$$ is the Laplacian i.e. $$\Delta f = (\partial_{x_1})^2+(\partial_{x_2})^2+(\partial_{x_3})^2+(\partial_{x_4})^2$$.

My attempt: Since $$B$$ is bounded with boundary it is thus compact. Using Stoke’s $$\int_{B} \Delta fdx_1dx_2dx_3dx_4 = \int_{\partial B}d(\Delta fdx_1dx_2dx_3dx_4) = \int_{\partial B} \sum_{i}\partial_{x_1}(\Delta f)dx_i\wedge dx_1dx_2dx_3dx_4 = 0$$

Am I correct in the previous argument? If yes, I’m not sure how to go about finding the $$3-$$ form $$w$$. Does any $$dx_i\wedge dx_j\wedge dx_k$$ work? Thanks.

## Differential equation tangent and slope field method

Given the equation $$y’=\frac{y}{3x}+\frac{x^2}{3y}$$ I have to find :

1.The tangent to the integral curve of this equation which passes through the point $$(x_{0},y_{0}) ∈ R^2$$

2.Describe a method for building slope field of the given equation.

For 1) my solution is the following :

so $$y'(x)=tg(\alpha)$$ and let $$f(x_{0},y_{0})= \frac{y_{0}}{3x_{0}}+\frac{x_{0}^2}{3y_{0}}$$ ,however $$tg(\alpha)=\frac{y-y_{0}}{x-x_{0}}$$ so $$\frac{y-y_{0}}{x-x_{0}} = \frac{y_{0}}{3x_{0}}+\frac{x_{0}^2}{3y_{0}}$$ . Then $$y= \frac{\frac{y_{0}}{3x_{0}}+\frac{x_{0}^2}{3y_{0}}}{\frac{y-y_{0}}{x-x_{0}}} + y_{0}$$ which gives us the desired answer

For 2) my solution is the following : for the right side of the resulting equation we assosciate vector field in the plane, for which for every point $$(x,y) \in R$$ of the domain of $$f(x,y)$$ corresponds to the vector $$(1,f(x,y))$$ ,however it is better to use lines – they are parts of the tangent throught every point $$(x,y) \in R^2$$ from the domain of $$f(x,y)$$.The resulting lines make the slope field, so the method will be :

1.choose $$\delta$$ >0

2.for each point $$(x_{k},y_{k})$$ we compute $$\epsilon$$ and we draw a line connecting the points $$(x_{k}-\epsilon,\bar{y_{m}})$$ and $$(x_{k}+\epsilon,\underline{y_{m}})$$

so let $$2\delta$$ be the length of the line ,then: $$tg(\alpha)=y'(x_{k})=f(x_{k},y_{m})$$

$$tg(\alpha)=\frac{\sqrt{\delta^2-\epsilon^2}}{\epsilon}=f(x_{k},y_{m})$$ from which we directly get that $$\epsilon=\frac{\delta}{\sqrt{1+f^2(x_{k},y_{m})}}$$

Now $$y_{m}-\bar{y_{m}}=\sqrt{\delta^2-\epsilon^2}=\delta f(x_{k},y_{m})=>\underline{y_{m}}=y_{m}-\delta f(x_{k},y_{m})$$

Do you think I have any mistakes in my solution and if yes please correct me.

## Help to understand Goldstain equation for Lagrangian and hamiltonian

I was looking for help in order to proove 2 relations that Goldstein has put in his book.

$$L(q, \dot{q}, t)=L_{0}(q, t)+\tilde{\mathbf{\dot{q}}} \mathbf{a}+\frac{1}{2} \tilde{\boldsymbol{\dot{q}}} \mathbf{T} \dot{\mathbf{q}}$$ $$H(q, p, t)=\frac{1}{2}(\tilde{\mathbf{p}}-\tilde{\mathbf{a}}) \mathbf{T}^{-1}(\mathbf{p}-\mathbf{a})-L_{0}(q, t)$$ I understand how they came mathematically, but Im not sure about what does this relations implies physically, i dont get it how can I interpret the vector generalizaed velocities and also the momenta vectors. Also, I assume that my lagrangian its in the form:

$$L\left(q_{i}, \dot{q}_{i}, t\right)=L_0(q, t)+\dot{q}_{i} a_{i}(q, t)+\dot{q}_{i}^2T_{i}(q, t)$$ I will very grateful for any advice in the proove, or interpretation of the vectors.

Pd: I know how to proove the second relationship from the first one, so Im more interested on the first one. Also, this eqs came from Classical Mechanics by H Goldstein 3rd Edition (8.22), (8.23), (8.27)

## Solve integer equation $2^m.m^2=9n^2-12n+19$

Problem: Find $$m,n\in \mathbb{N}^*$$ satisfied: $$2^m.m^2=9n^2-12n+19$$.

This is my attemp: We have $$9n^2-12n+19\equiv 1(\text{ mod 3})$$, so $$2^{m}.m^2\equiv 1(\text{ mod }3)(1)$$.

In addition, we have: $$m^2\equiv 0\text{ or }1(\text{ mod }3)$$

So $$(1)\implies m\equiv 1(\text{ mod }3)$$.

Suppose that: $$m=3k+1(k\in \mathbb{N})$$.

We have: $$2^{m}.m^2=2^{3k+1}.(3k+1)^2=2.8^{k}.(3k+1)^2\equiv 2^{k+1}.(3k+1)^{2}(\text{ mod }3)$$.

$$\implies k\equiv 1(\text{ mod 2})$$. I can only come here!!