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Control Systems Equation Solving Wolfram Language Wavelets and Fourier Series

Help!! I can't find the error in this program? (InverseFunction)

Lina Lili

Lina Lili, faculty of science

Posted 3 years ago

Hello, please I 'm really stuck in this program, I can't find the error, I really need to solve this problem If anyone can help me, please T = 2 \[Pi]; l = \[Pi]; a = 1/2; w1 = 2 \[Pi]; (Définit la fonction indicatrice) indicator[x_] := Piecewise[{{1, 0 < x < \[Pi]/2}}, 0]; S[x_, t_] := Sum[ Exp[- t n^2] Integrate[x Sin[s n^2], {s, 0, \[Pi]}] Sin[x n^2], {n, 5}] (controllability map) G[x_, t_] := indicator[x] S[x, T - t] (Grammian operator Q) Q[x_, t_] := (indicator[x])^2 Integrate[(S[x, T - s])^2, {s, T - l, T}] (inverse function) v[x_, t_] := InverseFunction[a + Q][x, t]; (Test) v[w1 - S[1, T],T]

Hello, please I 'm really stuck in this program, I can't find the error, I really need to solve this problem If anyone can help me, please

T = 2 \[Pi];
l = \[Pi];
a = 1/2;
w1 = 2 \[Pi];
(*Définit la fonction indicatrice*)
indicator[x_] := Piecewise[{{1, 0 < x < \[Pi]/2}}, 0];
S[x_, t_] := Sum[ Exp[- t  n^2] Integrate[x Sin[s n^2], {s, 0, \[Pi]}]  Sin[x n^2], {n, 5}]
(*controllability map*)
G[x_, t_] := indicator[x] S[x, T - t]
(*Grammian operator Q*)
Q[x_, t_] := (indicator[x])^2  Integrate[(S[x, T - s])^2, {s, T - l, T}]
(*inverse function*)
v[x_, t_] := InverseFunction[a + Q][x, t];
(*Test*)
v[w1 - S[1, T],T]

POSTED BY: Lina Lili

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Hans Dolhaine

Hans Dolhaine, retired

Posted 3 years ago

Hmmmm, what do you understand by InverseFunction[a + Q] ???? I assume that this is nonsense. Look at ContourPlot[a + Q[x, y], {x, .`5, 1}, {y, 4, 5}] and you see that there are really many combinations of x and y giving the same value of a + Q [x, y], so the inverse function is not really existing. For example a + Q[.5397, 4.651] a + Q[.87921, 4.243] So what do expect as result of InverseFunction[ a + Q ] ?

POSTED BY: Hans Dolhaine

Hans Dolhaine

Hans Dolhaine, retired

Posted 3 years ago

InverseFunction[ a + Q ] for me is the inverse of 1/2+Q. OK, but this does (in general) not exist. So - what do you expect of a non-existing function in your equations? S looks indeed somewhat peculiar, but I can't see why there should be a problem. S[x_, t_] := Sum[-Exp[-t n^2] (\[Pi] Cos[\[Pi] n^2])/(n^2) Sin[x n^2], {n, 5}] Plot3D[S[x, t], {x, 0, \[Pi]}, {t, 0, T}, PlotRange -> All, PlotPoints -> 50] Why don't you tell us about the physical problem you want to describe. I got the feeling that your equations are somewhat strange.

POSTED BY: Hans Dolhaine

Rohit Namjoshi

Posted 3 years ago

What is `indicator[x]`?

POSTED BY: Rohit Namjoshi

Lina Lili

Lina Lili, faculty of science

Posted 3 years ago

Sorry, I forget to add indicator[x]. I edit the question: (Définit la fonction indicatrice) indicator[x_] := Piecewise[{{1, 0 < x < \[Pi]/2}}, 0]; do you have any idea why it doesn't work ? this is error i get : NDSolve::underdet: There are more dependent variables, {f[x,t],InverseFunction[1/2+Q][x,t]}, than equations, so the system is underdetermined.

Sorry, I forget to add indicator[x]. I edit the question:

(*Définit la fonction indicatrice*)
indicator[x_] := Piecewise[{{1, 0 < x < \[Pi]/2}}, 0];

do you have any idea why it doesn't work ? this is error i get :

NDSolve::underdet: There are more dependent variables, {f[x,t],InverseFunction[1/2+Q][x,t]}, than equations, so the system is underdetermined.

POSTED BY: Lina Lili

Lina Lili

Lina Lili, faculty of science

Posted 3 years ago

Hi Hans, As you said InverseFunction[ a + Q ] for me is the inverse of 1/2+Q. But it doesn't have any sense I think the problem is in S[x,t], I just test It's not working !! can you see it, please! Best regards,

POSTED BY: Lina Lili

Lina Lili

Lina Lili, faculty of science

Posted 3 years ago

Hi Mr. Hans, I just edit my question, I've just noticed that there are some mistakes in my original question, I've changed it completely. but I still have in the inverse !!