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?

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}] 

plot

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!!