MinJ=int_0^T3y^2+u^2y'=y+uwith0<t<Ty(0)=y0y(T)=freeueU
–Vt=maxu[–3y^2–u^2+(y+u)Vy]
In[]:=
Maximize[–3y^2–u^2+(y+u)Vy,u]
Out[]=
+Vyy–3,u
2
Vy
4
2
y
Vy
2
In[]:=
Expand–1*Vt==(–3y^2–u^2+(y+u)Vy)/.u
Vy
2
Out[]=
–Vt+Vyy–3
2
Vy
4
2
y
In[]:=
{–Derivative[1,0][V][t,y]==1/4Derivative[0,1][V][t,y]^2+Derivative[0,1][V][t,y]y–3y^2,V[T,y]==0}
Out[]=
–[t,y]–3+y[t,y]+[t,y],V[T,y]0
(1,0)
V
2
y
(0,1)
V
1
4
2
(0,1)
V
In[]:=
FullSimplify[DSolve[{–Derivative[1,0][V][t,y]==1/4Derivative[0,1][V][t,y]^2+Derivative[0,1][V][t,y]y–3y^2,V[T,y]==0},V[t,y],{t,y}]]
Out[]=
{}
In[]:=
sol=DSolve[{v'[t]==3–v[t]^2–2v[t],v[T]==0},v,t]
Out[]=
vFunction{t},
3(–)
4t
4T
3+
4t
4T
In[]:=
v[t]=First[Evaluate[v[t]/.sol]]
Out[]=
3(–)
4t
4T
3+
4t
4T
In[]:=
V[t,y]=v[t]*y[t]^2
Out[]=
3(–)
4t
4T
2
y[t]
3+
4t
4T
In[]:=
u[t]=1/2D[V[t,y],y[t]]
Out[]=
3(–)y[t]
4t
4T
3+
4t
4T
In[]:=
DSolve[{y'[t]==y[t]+u[t],y[0]==y0},y,t]
Out[]=
yFunction{t},(3+)
(3+)y0
4t
4T
4t
4T
In[]:=
Plot(3+)/.{y0–>1,T–>1},{t,0,1}
(3+)y0
4t
4T
4t
4T
Out[]=
In[]:=
PlotTable(3+)/.{T–>1},{y0,0,1,1/10},{t,0,1}
(3+)y0
4t
4T
4t
4T
Out[]=
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