<p>We consider the shape and topology optimization for steady-state linear elasticity. Two models are considered. The first model is linear elasticity; the cost functional is given by an optimal control problem. The second model is the nonsmooth contact problem with a given friction. The cost functional for shape and topology optimization is also given by an optimal control problem. In the first case, the steady-state control problem is related to a dynamic optimal control problem. The so-called <i>Turnpike Property</i> of the dynamic control problem is exploited. The numerical results are presented for two control problems. Topology optimization is considered within the topological derivative method. The form of the topological derivative is obtained by using the domain decomposition method. The Steklov–Poincaré pseudodifferential operator is employed in the truncated domain for the purposes of topological sensitivity analysis.</p>