The following fixed-time optimal control problem is considered: \[ \text{minimize}\quad F(x,y):= \int^T_0 f(x(t), u(t),t)dt \] \[ \text{subject to}\quad x(0)= x_0,\;x(t)= m(x(t), u(t), t),\;u(t)\in \Gamma(t),\quad 0\leq t\leq T, \] where \(f\), \(m\) are functions from \(C^1\), \(\Gamma(t)\) is a given set, \(x(.)\) is piecewise smooth, \(u(.)\) is piecewise continuous. Under a suitable invexity hypothesis, the sufficiency of Karush-Kuhn-Tucker conditions for a local and under certain additional conditions even for a global minimum is proved. Some duality results are obtained.