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Optimization of semilinear hyperbolic systems with smooth boundary controls. (English, Russian) Zbl 1008.49014
Russ. Math. 45, No. 2, 1-9 (2001); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2001, No. 2, 3-12 (2001).
The authors consider the following optimal control problem for semilinear hyperbolic systems with smooth boundary controls $$u(s)$$: $J(u)= \int_S \varphi(x(s, t_1), s)\,ds+ \iint_P F(x,s,t)\,ds\,dt\to \text{minimum},$ ${\partial x\over\partial t}+ A(s,t) {\partial x\over\partial s}= f(x,s,t),$ $x(s, t_0)= p(u(s), s),\quad x^+(s_0, t)= M(t) x^-(s_0, t)+ g^{(1)}(t),$ $x^-(s_1, t)= N(t) x^+(s_1, t)+ g^{(2)}(t).$ For these problems, a necessary optimality condition is derived and a numerical method is given, which is based on the optimality condition. A numerical test is given.

##### MSC:
 49K20 Optimality conditions for problems involving partial differential equations 49M05 Numerical methods based on necessary conditions 35Q93 PDEs in connection with control and optimization 35L40 First-order hyperbolic systems