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VariationalCalculus

  

Convex

  

determine whether an integrand is convex

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

Convex(f, t, x(t))

Parameters

f

-

expression in t, x(t), and x'(t)

t

-

independent variable

x(t)

-

unknown function (or list of functions)

Description

• 

The Convex(f, t, x(t)) command determines if the integrand is convex.

• 

If the integrand is convex, the functional J=abft,x,x,'ⅆt is globally minimized by extremals (solutions of the Euler-Lagrange equations).

• 

For a convex integrand, the output is an expression sequence containing two items:

– 

Hessian matrix 2xx'f 

– 

Logical expression that is true iff the Hessian is positive semidefinite, which proves that J is a minimum

• 

If the integrand is not convex, Maple returns false.

• 

If LinearAlgebra[IsDefinite] cannot determine the convexity, the output is an expression sequence containing two items:

– 

Hessian matrix 2xx'f 

– 

unevaluated call to IsDefinite

• 

If an error occurs in the execution of LinearAlgebra[IsDefinite], only the Hessian matrix is returned.

• 

The arithmetic negation makes the Hessian negative semidefinite.

Examples

withVariationalCalculus

ConjugateEquation,Convex,EulerLagrange,Jacobi,Weierstrass

(1)

fⅆⅆtxt2+ⅆⅆtyt212

f:=ⅆⅆtxt2+ⅆⅆtyt2

(2)

Convexf,t,xt,yt

0000000000ⅆⅆtxt2ⅆⅆtxt2+ⅆⅆtyt23/2+1ⅆⅆtxt2+ⅆⅆtyt2ⅆⅆtxtⅆⅆtytⅆⅆtxt2+ⅆⅆtyt23/200ⅆⅆtxtⅆⅆtytⅆⅆtxt2+ⅆⅆtyt23/2ⅆⅆtyt2ⅆⅆtxt2+ⅆⅆtyt23/2+1ⅆⅆtxt2+ⅆⅆtyt2,01ⅆⅆtxt2+ⅆⅆtyt2

(3)

See Also

LinearAlgebra[IsDefinite]

VariationalCalculus

VariationalCalculus[EulerLagrange]

 


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