Abstract.We consider the flow of gas through pipelines controlled by a compressor station. Under a subsonic flow assumption we prove the existence of classical solutions for a given finite time interval. The existence result is used to construct Riemannian feedback laws and to prove a stabilization result for a coupled system of gas pipes with a compressor station. We introduce a Lyapunov function and prove exponential decay with respect to the L 2 -norm.Mathematics Subject Classification. 76N25, 35L50, 93C20.

We consider the problem of exactly controlling the states of the de St. Venant equations from a given constant state to another constant state by applying nonlinear boundary controls. During this transition the solution stays in the class of C 1-solutions. There are no restrictions on the distance between the initial state and the target state, so our result is a global controllability result for a nonlinear hyberbolic system.

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We consider a vibrating string that is fixed at one end with Neumann control action at the other end. We investigate the optimal control problem of steering this system from given initial data to rest, in time T , by minimizing an objective functional that is the convex sum of the L 2 -norm of the control and of a boundary Neumann tracking term.We provide an explicit solution of this optimal control problem, showing that if the weight of the tracking term is positive, then the optimal control action is concentrated at the beginning and at the end of the time interval, and in-between it decays exponentially. We show that the optimal control can actually be written in that case as the sum of an exponentially decaying term and of an exponentially increasing term. This implies that, if the time T is large the optimal trajectory approximately consists of three arcs, where the first and the third short-time arcs are transient arcs, and in the middle arc the optimal control and the corresponding state are exponentially close to 0. This is an example for a turnpike phenomenon for a problem of optimal boundary control. If T = +∞ (infinite horizon time problem), then only the exponentially decaying component of the control remains, and the norms of the optimal control action and of the optimal state decay exponentially in time. In contrast to this situation if the weight of the tracking term is zero and only the control cost is minimized, then the optimal control is distributed uniformly along the whole interval [0, T ] and coincides with the control given by the Hilbert Uniqueness Method.

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We consider a network of pipelines where the flow is controlled by a number of compressors. The consumer demand is described by desired boundary traces of the system state. We present conditions that guarantee the existence of compressor controls such that after a certain finite time the state at the consumer nodes is equal to the prescribed data. We consider this problem in the framework of continuously differentiable states. We give an explicit construction of the control functions for the control of compressor stations in gas distribution networks.

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SUMMARYWe consider a tree-like network of open channels with out ow at the root. Controls are exerted at the boundary nodes of the network except for the root. In each channel, the ow is modelled by the de St. Venant equations. The node conditions require the conservation of mass and the conservation of energy. We show that the states of the system can be controlled within the entire network in ÿnite time from a stationary supercritical initial state to a given supercritical terminal state with the same orientation. During this transition, the states stay in the class of C 1 -functions, so no shocks occur.