Unbounded Control Operators: Hyperbolic Equations With Control on the Boundary.

We use here the notation of Chapter 3 in Part IV. We assume that $$ (\mathcal{H}\mathcal{P})\infty \left\{ \begin{gathered} ({\text{i}}){\text{ }}A{\text{ generates an analytic semigroup e}}^{tA} {\text{ in }}H \hfill \\ {\text{ of type }}\omega _{\text{0}} {\text{ and }}\lambda _0 {\text{ is a real number in }}\rho {\text{(}}A{\text{)such}} \hfill \\ {\text{ that }}\omega _{\text{0}} < \lambda _0 , \hfill \\ {\text{(ii) }}E \in \mathcal{L}(U;H), \hfill \\ {\text{(iii) }}\forall T {\text{ > 0, }}\exists K_T {\text{ > 0 such that}} \hfill \\ {\text{ }}\int_{\text{0}}^t {{\text{<INNOPIPE>}}E{\text{*}}A{\text{*}}e^{sA{\text{*}}} x<INNOPIPE>^2 ds \leqslant K_T^2 <INNOPIPE>x<INNOPIPE>^2 } ,{\text{ }}\forall x \in D(A*),t \geqslant 0, \hfill \\ {\text{(iv) }}C \in \mathcal{L}(H;Y). \hfill \\ \end{gathered} \right. $$ Clearly, if $$ (\mathcal{H}\mathcal{H})\infty $$ hold, then the hypotheses $$ (\mathcal{H}\mathcal{H}) $$ of Chapter 3 in Part IV are fulfilled with P0 = 0. We want to minimize the cost function: (1.1)$$ J_\infty (u) = \int_0^\infty {\{ <INNOPIPE>Cx(s)<INNOPIPE>^2 + <INNOPIPE>u(s)<INNOPIPE>^2 \} ds,} $$ over all controls u ∈ L2(0,∞;U) subject to the equation constraint (1.2)$$ \begin{array}{*{20}c} {x(t) = e^{tA} x_0 + G(u)(s),} \\ {G(u)(s) = (\lambda _0 - A)\int_0^t {e^{(t - s)A} Eu(s)ds.} } \\ \end{array} $$ Moreover, x0 ∈ H and u ∈ L2(0,∞;U). We recall that by Proposition 3.1 in Chapter 1 in Part II, x ∈ C([0, T];H) for all ∈ L2(0, T;U); more precisely $$ G \in \mathcal{L}(L^2 (0,T;U);C([0,T];H)),{\text{ }}\forall T > 0. $$ [ABSTRACT FROM AUTHOR]