TY - INPR A1 - Gugat, Martin A1 - Sokolowski, Jan T1 - On Problems of Dynamic Optimal Nodal control for Gas Networks N2 - We consider a dynamic ptimal control problem for gas pipeline systems. The flow is governed by a quasilinear hyperbolic model. Since in the operation of the gas networks regular solutions without shocks are desirable, we impose appropriate state and control constraint in order to guarantee that a classical solution is generated. Due to a W^{2;inf}-regularization term in the objective function, we can show the existence of an optimal control. Moreover, we give conditions that guarantee that the control becomes constant a the end of the control time interval if the weight of the regularization term is suffciently large. KW - optimal nodal control KW - gas network KW - turnpike property KW - quasilinear hyperbolic problem KW - dynamic control Y1 - 2021 ER - TY - INPR A1 - Gugat, Martin A1 - Schuster, Michael A1 - Sokolowski, Jan T1 - Location Problem for Compressor Stations in Pipeline Networks N2 - In the operation of pipeline networks, compressors play a crucial role in ensuring the network’s functionality for various scenarios. In this contribution we address the important question of finding the optimal location of the compressors. This problem is of a novel structure, since it is related with the gas dynamics that governs the network flow. That results in non-convex mixed integer stochastic optimization problems with probabilistic constraints. Using a steady state model for the gas flow in pipeline networks including compressor control and uncertain loads given by certain probability distributions, the problem of finding the optimal location for the control on the network, s.t. the control cost is minimal and the gas pressure stays within given bounds, is considered. In the deterministic setting, explicit bounds for the pipe length and the inlet pressure, s.t. a unique optimal compressor location with minimal control cost exists, are presented. In the probabilistic setting, an existence result for the optimal compressor location is presented and the uniqueness of the solution is discussed depending on the probability distribution. For Gaussian distributed loads a uniqueness result for the optimal compressor location is presented. Further the problem of finding the optimal compressor locations on networks including the number of compressor stations as variable is considered. Results for the existence of optimal locations on a graph in both, the deterministic and the probabilistic setting, are presented and the uniqueness of the solutions is discussed depending on probability distributions and graph topology. The paper concludes with an illustrative example demonstrating that the compressor locations determined using a steady state approach are also admissible in transient settings. KW - Gas Networks KW - Compressor Control KW - Weber Problem KW - Optimal Location KW - Uncertain Boundary Data Y1 - 2024 ER - TY - JOUR A1 - Gugat, Martin A1 - Qian, Meizhi A1 - Sokolowski, Jan T1 - Network Design and Control: Shape and Topology Optimization for the Turnpike Property for the Wave Equation N2 - The optimal control problems for the wave equation are considered on networks. The turnpike property is shown for the state equation, the adjoint state equation as well as the optimal cost. The shape and topology optimization is performed for the network with the shape functional given by the optimality system of the control problem. The set of admissible shapes for the network is compact in finite dimensions, thus the use of turnpike property is straightforward. The topology optimization is analysed for an example of nucleation of a small cycle at the internal node of network. The topological derivative of the cost is introduced and evaluated in the framework of domain decomposition technique. Numerical examples are provided. KW - Turnpike Property KW - Wave Equation KW - Optimal Control KW - Spectral Method KW - Network Optimum Design Y1 - 2024 VL - J. Geom. Anal. IS - 34 ER - TY - JOUR A1 - Schuster, Michael A1 - Gugat, Martin A1 - Sokolowski, Jan T1 - The Location Problem for Compressor Stations in Pipeline Networks JF - Mathematics and Mechanics of Complex Systems N2 - In the operation of pipeline networks, compressors play a crucial role in ensuring the network’s functionality for various scenarios. In this contribution we address the important question of finding the optimal location of the compressors. This problem is of a novel structure, since it is related to the gas dynamics that governs the network flow. That results in nonconvex mixed integer stochastic optimization problems with probabilistic constraints. Using a steady state model for the gas flow in pipeline networks including compressor control and uncertain loads given by certain probability distributions, we consider the problem of finding the optimal location for the control on the network such that the control cost is minimal and the gas pressure stays within given bounds. In the deterministic setting, we present explicit bounds for the pipe length and the inlet pressure such that a unique optimal compressor location with minimal control cost exists. In the probabilistic setting, we give an existence result for the optimal compressor location and discuss the uniqueness of the solution depending on the probability distribution. For Gaussian distributed loads a uniqueness result for the optimal compressor location is presented. We further present the problem of finding optimal compressor locations on networks including the number of compressor stations as a variable. Results for the existence of optimal locations on a graph in both the deterministic and the probabilistic setting are presented, and the uniqueness of the solutions is discussed depending on probability distributions and graph topology. The paper concludes with an illustrative example on a diamond graph demonstrating that the minimal number of compressor stations is not necessarily equal to the optimal number of compressor stations. KW - gas network KW - compressor control KW - Weber problem KW - uncertain boundary data KW - non convex mixed integer stochastic problem Y1 - 2024 U6 - https://doi.org/10.2140 VL - 12 IS - 4 SP - 507 EP - 546 ER - TY - INPR A1 - Gugat, Martin A1 - Qian, Meizhi A1 - Sokolowski, Jan T1 - Topological derivative method for control of wave equation on networks N2 - The dynamical, boundary optimal control problems on networks are considered. The domain of definition for the distributed parameter system is given by a graph G. The optimal cost function for control problem is further optimized with respect to the shape and topology of the graph Ω. The small cycle is introduced and the topological derivative of the cost with respect to the size of the cycle is determined. In this way, the singular perturbations of the graph can be analyzed in order to change the topology Ω. The topological derivative method in shape and topology optimization is a new tool which can be used to minimize the shape functionals under the Partial Differential Equations (PDEs) constraints. The topological derivative is used as well for solution of optimum design problems for graphs. In optimal control problems the topological derivative is used for optimum design of the domain of integration of the state equation. As an example, optimal control problems are considered on a cross with a small cycle. The state equation is the wave equation on the graph. The boundary control problem by Neumann conditions at a boundary vertex is solved for a tracking cost function. The shape functional is given by the optimal value of the control cost. The topological derivative of the shape functional is determined for the steady state model with the size of a cycle ε → 0. Numerical results for a model problem are presented. KW - distributed parameter system KW - optimal control KW - shape optimization KW - topological derivative KW - network modelling Y1 - 2023 ER - TY - INPR A1 - Schuster, Michael A1 - Sokolowski, Jan T1 - The Topological Derivative Method for Optimum Shape Design and Control of Gas Networks N2 - In this paper, topological derivatives are defined and employed for gas transport networks governed by nonlinear hyperbolic systems of PDEs. The concept of topological derivatives of a shape functional is introduced for optimum design and control of gas networks. First, the dynamic model for the network is considered. The cost for the control problem includes the deviations of the pressure at the inflow and outflow nodes. For dynamic control problems of gas networks when the turnpike property occurs, the synthesis of control and optimum design of the network can be simplified. That is, the design of the network can be performed for optimal control of the steady-state network model. The cost of design is defined by the optimal control cost for the steady-state network model. The topological derivative of the design cost, given by the optimal control cost with respect to the nucleation of a small cycle, is determined. Tree-structured networks can be decomposed into single network junctions. The topological derivative of the design cost is systematically evaluated at each junction of the decomposed network. This allows for the identification of internal nodes with negative topological derivatives, where replacing the node with a small cycle leads to an improved design cost. As the set of network junctions is finite, the iterative procedure is convergent. This design procedure is applied to representative examples and it can be generalized to arbitrary network graphs. A key feature of such modeling approach is the availability of exact steady-state solutions, enabling a fully analytical topological analysis of the design cost without numerical approximations. KW - Gas Networks KW - Optimum Design KW - Optimal Control KW - Topological Derivative KW - Turnpike Phenomenon Y1 - 2025 ER -