Please help me how to deal with maximization of functional like this: F { a ( s ) } = ∫ 0 t ( g ( a ( s ) ) − α v ( s ) 2 ) d s , a ( s ) ∈ [ 0 , ∞ ) where g ( x ) = x e − x , α = c o n s t , and v ( t ) = ∫ 0 t a ( s ) d s And I don't understand how to deal with v ( t ) 2

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Please help me how to deal with maximization of functional like this:  F  {  a  (  s  )  }  =      ∫    0    t        (    g    (    a    (    s    )    )    −    α        v    (    s          )      2        )    d  s  ,     a  (  s  )  ∈      [    0    ,    ∞    )  where   g  (  x  )  =  x      e          −      x      ,   α  =      c    o    n    s    t  , and       v    (    t    )    =          ∫      0      t        a    (    s    )    d    s  And I don&#039;t understand how to deal with   v  (  t      )    2

Cristopher Barrera · Accepted Answer

I don&#039;t understand how to deal with       v    2  Reformulate the problem in terms of   v. That is, you seek the maximum of                    (1)                    F        {        v        }        =                  ∫          0          t                          (          g          (                      v            ′                    (          s          )          )          −          α                    v          (          s                      )            2                    )                d        s        ,                 a        (        s        )        ∈                  [          0          ,          ∞          )                    over increasing differentiable functions   v with   v  (  0  )  =  0. The Euler-Lagrange equation for (2) is easy to state:                    (2)                    −                  d                      d            s                          (                  g          ′                (                  v          ′                (        s        )        )        )        −        2        α        v        (        s        )        ≡        0            Since       g    ″    (  x  )  =  (  x  −  2  )      e          −      x      , the equation (2) becomes                    (3)                    (        2        −                  v          ′                (        s        )        )                  e                      −                          v              ′                        (            s            )                                    v          ″                (        s        )        −        2        α        v        (        s        )        ≡        0            I wouldn&#039;t expect an explicit solution of (3), but a numeric solution should not be hard to obtain.

Please help me how to deal with maximization of functional like this: F <mo fence="false" stre

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