Control and optimum design for networks with nonlinear state equations

Optimal control problem for semilinear hyperbolic equations is considered on 3 -star metric graph. The distributed controls on the edges are considered. The tracking type cost for steady state equation is considered for the control problem. Using the Turnpike Property for dynamic control problem, the control problem is solved for the steady state equations. The necessary and sufficient optimality conditions are established for static control problems. Then the optimum design problem is considered for the topological perturbation of the graph. The small cycle of size $\varepsilon \rightarrow 0$ replaces the central node of 3 -star graph. The distributed controls are introduced on the edges of the cycle. The topological derivative of the shape functional indicates the possibility of nucleation of the small cycle.