Ishmukhametov, A. Z. Synthesis of optimal control for systems described by a hyperbolic equation. (Russian) Zbl 0569.49014 Differ. Uravn. 21, No. 4, 597-604 (1985). The author deals with the optimal control problem for the minimum of the functional \(J(u)=| w(T)-y_ 0|^ 2_ 0+| w'(T)-y_ 1|^ 2_ 0\), \(u\in U_ R\), under conditions \(w''+b(t)w'+Aw+c(t)w=u(t)+f^ 0(t)\), \(t_ 0<t\leq T\), \(w(t_ 0)=\phi_ 0\), \(w'(t_ 0)=\phi_ 1\), where A is a linear selfadjoint coercive operator in a Hilbert space H. The differentiability of the functional J in the space \(L_ 2(t_ 0,T;H)\) is proved, necessary and sufficient optimality and controllability conditions are derived. Reviewer: I.Bock Cited in 1 ReviewCited in 1 Document MSC: 49K20 Optimality conditions for problems involving partial differential equations 49J50 Fréchet and Gateaux differentiability in optimization 93B05 Controllability 35B37 PDE in connection with control problems (MSC2000) 35L10 Second-order hyperbolic equations 93B50 Synthesis problems 47B25 Linear symmetric and selfadjoint operators (unbounded) 46C99 Inner product spaces and their generalizations, Hilbert spaces 93C05 Linear systems in control theory 93C20 Control/observation systems governed by partial differential equations 49K27 Optimality conditions for problems in abstract spaces Keywords:hyperbolic equation; differentiability of the functional; necessary and sufficient optimality and controllability conditions PDFBibTeX XMLCite \textit{A. Z. Ishmukhametov}, Differ. Uravn. 21, No. 4, 597--604 (1985; Zbl 0569.49014)