Optimal Boundary Control in Flood Management
In active flood hazard mitigation, lateral flow withdrawal is used to reduce the impact of flood waves in rivers. Through emergency side channels, lateral outflow is generated. The optimal outflow controls the flood in such a way that the cost of the crea
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ptimal Boundary Control in Flood Management Martin Gugat Abstract. In active flood hazard mitigation, lateral flow withdrawal is used to reduce the impact of flood waves in rivers. Through emergency side channels, lateral outflow is generated. The optimal outflow controls the flood in such a way that the cost of the created damage is minimized. The flow is governed by a networked system of nonlinear hyperbolic partial differential equations, coupled by algebraic node conditions. Two types of integrals appear in the objective function of the corresponding optimization problem: Boundary integrals (for example, to measure the amount of water that flows out of the system into the floodplain) and distributed integrals. For the evaluation of the derivative of the objective function, we introduce an adjoint backwards system. For the numerical solution we consider a discretized system with a consistent discretization of the continuous adjoint system, in the sense that the discrete adjoint system yields the derivatives of the discretized objective function. Numerical examples are included. Mathematics Subject Classification (2000). 35L45 35L50 35L65 93C20 . Keywords. St. Venant equations, subcritical states, adjoint system, optimal boundary control, necessary optimality conditions, classical solutions.
1. Introduction In flood management, the aim is to minimize the damage caused by a flood by active flow control (see [29]). In the present paper, we consider a problem where the aim is to find a compromise with minimal cost between flood in a city that occurs if the water level rises above a certain upper bound and the cheaper outflow through emergency side channels, for example to a floodplain. Often damage is caused if the water level in a certain area rises above a given upper bound. The control function in the corresponding problem of optimal boundary control is the This work was supported by DFG-research cluster: real-time optimization of complex systems; grant number Le595/13-1.
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discharge from the emergency side channel; in practice the control is realized by opening underflow gates. The flow in the considered channel network is described by a networked system of conservation laws. On each edge of the corresponding graph, the flow is described by a hyperbolic quasilinear system of partial differential equations, namely de St. Venant’s equations (see [11]), that model conservation of mass and the evolution of momentum. On the vertices, the flow variables are coupled by algebraic node conditions. We work with continuously differentiable solutions of the system equations. Our strategy is to control the flow in such a way that no singularities are generated. This implies that also the boundary control functions have to be continuously differentiable. Questions of boundary controllability within the class of continuously differentiable solutions have been studied in [8], [9], [12], [16], [26]. The results in these papers show that a large class or states can be reached with continuously differentiable solutions of the system equations.
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