Optimal control of exterior forces

Let F c (Q) be a bounded, closed and convex set. As it was proved in the previous subsection, for every f G F there exists a solution w = Wf of the problem (2.7)-(2.8). We consider the cost functional 

Let / G F be a minimizing sequence. It is bounded in H Q). Without loss of generality, we can assume that 

The above convergence of w , / allows us to pass to the lower limit in the last inequality. The resulting relation can be written as follows 

Since w G K we therefore have w = Wf. Now it is easy to complete the proof. Indeed, 

the function / solves the optimal control problem (2.19). Theorem 2.2 is proved. 

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