  The second part of our study is devoted to an analysis of the exactness of
penalty functions for optimal control problems with terminal and pointwise
state constraints. We demonstrate that with the use of the exact penalty
function method one can reduce fixed-endpoint problems for linear time-varying
systems and linear evolution equations with convex constraints on the control
inputs to completely equivalent free-endpoint optimal control problems, if the
terminal state belongs to the relative interior of the reachable set. In the
nonlinear case, we prove that a local reduction of fixed-endpoint and
variable-endpoint problems to equivalent free-endpoint ones is possible under
the assumption that the linearised system is completely controllable, and point
out some general properties of nonlinear systems under which a global reduction
to equivalent free-endpoint problems can be achieved. In the case of problems
with pointwise state inequality constraints, we prove that such problems for
linear time-varying systems and linear evolution equations with convex state
constraints can be reduced to equivalent problems without state constraints,
provided one uses the $L^{\infty}$ penalty term, and Slater's condition holds
true, while for nonlinear systems a local reduction is possible, if a natural
constraint qualification is satisfied. Finally, we show that the exact
$L^p$-penalisation of state constraints with finite $p$ is possible for convex
problems, if Lagrange multipliers corresponding to the state constraints belong
to $L^{p'}$, where $p'$ is the conjugate exponent of $p$, and for general
nonlinear problems, if the cost functional does not depend on the control
inputs explicitly.
