COMPUTATIONS FOR SECTION 8 OF
STRONG REGULARITY OF TIME AND NORM OPTIMAL CONTROLS
H. O. Fattorini
Department of Mathematics, University of California,
Los Angeles, California

EXAMPLE 8.1

y[x_] = (2/3)*((1 + x)^(3/2) - 1)/(1 + x)

Simplify[y'[x]]

Clear[ω, ω1, y]

SOL1 = NDSolve[{ω'[x] == ω1[x],
      ω1'[x] ==  y[x]^2*ω1[x]^3/(2*ω[x]^2),
      ω[0] == 10^(-8), ω1[0] == 1}, {ω, ω1}, {x, 1}]

ω = ω /. SOL1[[1]]
ω1 = ω1 /. SOL1[[1]]

Plot[{ω[x], a[x], x}, {x, 0, 1}]

Plot[{ω1[x], a'[x], 1}, {x, 0, 1}, PlotRange -> {0, 1.5}]

b =(2/3)*(2^(3/2)- 1)

F[x_] := NIntegrate[ y[t]^2, {t, x, 1}] + NIntegrate[b^2/(1 + t)^2, {t, 1, Infinity}]

F[0]

Plot[{F[x],(ω[x]/ω1[x])^2}, {x, 0, 1}, Ticks -> {{0.5, 1},{0.9}}, PlotStyle -> Thickness[0.003]]

F[1]

(ω[1]/ω1[1])^2

z[x_] =y[x]/ω[x]

z0[x_] = 1/(1 + x)

Plot[{z0[x], z[x]}, {x, 0, 1}, PlotRange -> {0, 2}]

Plot[z0[x]-z[x], {x, 0, 1}, PlotRange -> {-10^(-6), 10^(-6)}]

EXAMPLE 8.2

Clear[ω, ω1, y, z]

y[x_] = x/(1 + x^2)

Plot[y[x], {x, 0, 5}]

Simplify[y'[x]]

Integrate[(y[x]/x)^2, {x, 0, Infinity}]

N[%]

NIntegrate[(y[x]/x)^1.9, {x, 0, 2}] + NIntegrate[((1/1.9)^2)*y[x]^2, {x, 1, Infinity}]

Integrate[y[x]^2, {x, 0, Infinity}]

SOL1 = NDSolve[{ω'[x] == ω1[x],
      ω1'[x] ==  y[x]^2*ω1[x]^3/(2*ω[x]^2),
      ω[0] == 10^(-6), ω1[0] == 1}, {ω, ω1}, {x, 2}]

ω = ω /. SOL1[[1]]
ω1 = ω1 /. SOL1[[1]]

Plot[{ω[x], x}, {x, 0, 2}]

Plot[{ω1[x], 1}, {x, 0, 2}, PlotRange -> {0, 2}]

z[x_] = y[x]/ω[x]

Plot[z[x], {x, 0, 2}, PlotRange -> {0, 1}]

NIntegrate[z[x]^2, {x, 0, 1}]

F[x_] = Integrate[y[t]^2, {t, x, Infinity}]

Plot[{F[x],(ω[x]/ω1[x])^2}, {x, 0, 2}]

FindRoot[F[x]==(ω[x]/ω1[x])^2, {x, 1}]

Plot[{z[x]*ω[x],y[x]},{x,0,2}]

z[x_] = Piecewise[{{ω[0.942178]*(y[x]/ω[x]),x<0.942178},{y[x],x>0.942178}}]

Plot[z[x], {x, 0, 2}]

EXAMPLE 8.3

Clear[y, ω, ω1, z]

y[x_] =1.69* x/(1 + x^2)

NIntegrate[(y[x]/x)^2, {x, 0, Infinity}]

(1.69)^2*Pi/4

Integrate[y[x]^2, {x, 0, Infinity}]

Integrate[(y[x]/x)^2, x]

Integrate[y[x]^2, x]

SOL1 = NDSolve[{ω'[x] == ω1[x],
      ω1'[x] ==  y[x]^2*ω1[x]^3/(2*ω[x]^2),
      ω[0] == 10^(-8), ω1[0] == 1}, {ω, ω1}, {x, 5}]

ω = ω /. SOL1[[1]]
ω1 = ω1 /. SOL1[[1]]

Plot[{ω[x], x}, {x, 0, 5}]

Plot[{ω1[x], 1}, {x, 0, 5}]

F[x_] = Integrate[y[t]^2, {t, x, Infinity}]

Plot[{F[x],(ω[x]/ω1[x])^2}, {x, 0,5}]

FindRoot[F[x]==(ω[x]/ω1[x])^2, {x, 2.5}]

EXAMPLE 8.4.

Clear[y, ω, ω1, z]

y[x_] =1.695* x/(1 + x^2)

SOL1 = NDSolve[{ω'[x] == ω1[x],
      ω1'[x] ==  y[x]^2*ω1[x]^3/(2*ω[x]^2),
      ω[0] == 10^(-6), ω1[0] == 1}, {ω, ω1}, {x, 5}]

ω = ω /. SOL1[[1]]
ω1 = ω1 /. SOL1[[1]]

F[x_] = Integrate[y[t]^2, {t, x, Infinity}]

Plot[{F[x],(ω[x]/ω1[x])^2},{x, 0, 1.674433}]