The powerful Hamilton-Jacobi theory is used for
constructing regularizations and error estimates for optimal design
problems. The constructed Pontryagin method is a simple and general
method for optimal design and reconstruction: the first, analytical,
step is to regularize the Hamiltonian; next the solution to its
stationary Hamiltonian system, a nonlinear partial differential
equation, is computed with the Newton method. The method is
efficient for designs where the Hamiltonian function can be
explicitly formulated and when the Jacobian is sparse, but becomes
impractical otherwise (e.g. for non local control constraints). An
error estimate for the difference between exact and approximate
objective functions is derived, depending only on the difference of
the Hamiltonian and its finite dimensional regularization along the
solution path and its L2 projection, i.e. not on the difference of
the exact and approximate solutions to the Hamiltonian systems.