Answer Happy

Consider the functional ∙tf =S₁ to J(x(.)) = c) d) (x(t), x(t), t) dt for to= 0 and tf = 2. a) State the first order necessary condition on the trajectory x* (.) that gives an extremum value to J. b) Let $(x(t), x(t), t) = x²(t)+4t²x(t). State the extremizing trajectory x*(-) for J. What is the value of J(x*(-)) given of task b)? State in your own words why, for small variations & > 0 and x* (t) + ε8x(t)) around x* (t), te [to, tf], dj(x* (t) + ε8x(t))| de ε=0 = 0 reduces to a necessary condition for x*(-). You may use graphics to support your explanation. e) Suppose x(t) = f(x(t)), for any x(to). State how the extremizing / is approa- ched what needs to be considered additionally?