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A class of optimal control problem for a semilinear elliptic partial differential equation
with control constraints is considered. It is well known that
sufficient second-order conditions ensure the stability of optimal solutions, the convergence of
numerical methods. Otherwise, such conditions are very difficult to verify (analytically or numerically).
We will propose a new approach: Starting with a numerical solution for a fixed mesh we will
show the existence of a local minimizer of the continuous problem. Moreover, we will prove that
this minimizer satisfies the sufficient second-order conditions.