Martin Gugat

Friedrich-Alexander-University of Erlangen-Nürnberg | FAU · Department of Data Science

We analyze the subcritical gas flow through fan-shaped networks of pipes, that is, through tree-shaped networks with exactly one node where more than two pipes meet. The gas flow in pipe networks is modeled by the isothermal Euler equations, a hyperbolic PDE system of balance laws. For this system we analyze stationary states and classical nonstati...

For vibrating systems, a delay in the application of a feedback control can destroy the stabilizing effect of the control. In this paper we consider a vibrating string that is fixed at one end and stabilized with a boundary feedback with delay at the other end. We show that for certain feedback parameters the system is exponentially stable with con...

In many applications, in systems that are governed by a linear hyperbolic partial differential equations some of the problem parameters are uncertain. If information about the probability distribution of the parametric uncertainty distribution is available, the uncertain state of the system can be described using an intrinsic formulation through a...

We consider a state estimation problem for gas flows in pipeline networks where hydrogen is blended into the natural gas. The flow is modeled by the quasi-linear isothermal Euler equations coupled to an advection equation on a graph. The flow through the vertices where the pipes are connected is governed by algebraic node conditions. The state is a...

We consider a state estimation problem for gas pipeline flow modeled by the one-dimensional barotropic Euler equations. In order to reconstruct the system state, we construct an observer system of Luenberger type based on distributed measurements of one state variable. First, we show the existence of Lipschitz-continuous semi-global solutions of th...

We study the turnpike phenomenon for optimal control problems with mean field dynamics that are obtained as the limit $N\rightarrow \infty$ of systems governed by a large number $N$ of ordinary differential equations. We show that the optimal control problems with with large time horizons give rise to a turnpike structure of the optimal state and t...

While the quasilinear isothermal Euler equations are an excellent model for gas pipeline flow, the operation of the pipeline flow with high pressure and small Mach numbers allows us to obtain approximate solutions by a simpler semilinear model. We provide a derivation of the semilinear model that shows that the semilinear model is valid for suffici...

Uncertainty often plays an important role in dynamic flow problems. In this paper, we consider both, a stationary and a dynamic flow model with uncertain boundary data on networks. We introduce two different ways how to compute the probability for random boundary data to be feasible, discussing their advantages and disadvantages. In this context, f...

We present a positive and negative stabilization results for a semilinear model of gas flow in pipelines. For feedback boundary conditions, we obtain unconditional stabilization in the absence and conditional instability in the presence of the source term. We also obtain unconditional instability for the corresponding quasilinear model given by the...

We analyse the turnpike properties for a general, infinite dimensional, linear-quadratic (LQ) optimal control problem, both in the deterministic and in the stochastic case. The novelty of the paper is twofold. Firstly, it obtains positive turnpike results for systems that are (partially) uncontrollable. Secondly, it provides turnpike results for av...

In this chapter we survey recent progress on mathematical results on gas flow in pipe networks with a special focus on questions of control and stabilization. We briefly present the modeling of gas flow and coupling conditions for flow through vertices of a network. Our main focus is on gas models for spatially one-dimensional flow governed by hype...

In this paper we discuss an approach to the stability analysis for classical solutions of closed loop systems that is based upon the tracing of the evolution of the Riemann invariants along the characteristics. We consider a network where several edges are coupled through node conditions that govern the evolution of the Riemann invariants through t...

The operation of gas pipeline flow with high pressure and small Mach numbers allows to model the flow by a semilinear hyperbolic system of partial differential equations. In this paper we present a number of transient and stationary analytical solutions of this model. They are used to discuss and clarify why a PDE model is necessary to handle certa...

The flow of gas through networks of pipes can be modelled by coupling hyperbolic systems of partial differential equations that describe the flow through the pipes that form the edges of the graph of the network by algebraic node conditions that model the flow through the vertices of the graph. In the network, measurements of the state are availabl...

The flow of gas through a pipeline network can be modelled by a coupled system of 1-d quasilinear hyperbolic equations. Often for the solution of control problems it is convenient to replace the quasilinear model by a simpler semilinear model. We analyze the behavior of such a semilinear model on a star-shaped network. The model is derived from the...

In this paper the turnpike phenomenon is studied for problems of optimal control where both pointwise-in-time state and control constraints can appear. We assume that in the objective function, a tracking term appears that is given as an integral over the time-interval $$[0,\, T]$$ [ 0 , T ] and measures the distance to a desired stationary state....

The flow of gas through networks of pipes can be modeled by coupling hyperbolic systems of partial differential equations that describe the flow through the pipes that form the edges of the graph of the network by algebraic node conditions that model the flow through the vertices of the graph. In the network, measurements of the state are available...

In this paper, problems of optimal control are considered where in the objective function, in addition to the control cost, there is a tracking term that measures the distance to a desired stationary state. The tracking term is given by some norm, and therefore it is in general not differentiable. In the optimal control problem, the initial state i...

This contribution focuses on the analysis and control of friction-dominated flow of gas in pipes. The pressure in the gas flow is governed by a partial differential equation that is a doubly nonlinear parabolic equation of p-Laplace type, where p = 3 2 . Such equations exhibit positive solutions, finite speed of propagation and satisfy a maximum p...

Optimization problems under uncertain conditions abound in many real-life applications. While solution approaches for probabilistic constraints are often developed in case the uncertainties can be assumed to follow a certain probability distribution, robust approaches are usually applied in case solutions are sought that are feasible for all realiz...

In this article we survey recent progress on mathematical results on gas flow in pipe networks with a special focus on questions of control and stabilization. We briefly present the modeling of gas flow and coupling conditions for flow through vertices of a network. Our main focus is on gas models for spatially one-dimensional flow governed by hype...

Uncertainty often plays an important role in dynamic flow problems. In this paper, we consider both, a stationary and a dynamic flow model with uncertain boundary data on networks. We introduce two different ways how to compute the probability for random boundary data to be feasible, discussing their advantages and disadvantages. In this context, f...

In this paper, problems of optimal control are considered where in the objective function, in addition to the control cost there is a tracking term that measures the distance to a desired stationary state. The tracking term is given by some norm and therefore it is in general not differentiable. In the optimal control problem, the initial state is...

The Schlögl system is governed by a nonlinear reaction-diffusion partial differential equation with a cubic nonlinearity. In this paper, feedback laws of Pyragas-type are presented that stabilize the system in a periodic state with a given period and given boundary traces. We consider the system both with boundary feedback laws of Pyragas type and...

We consider the flow of gas through networks of pipelines. A hierarchy of models for the gas flow is available. The most accurate model is the pde system given by the 1-d Euler equations. For large-scale optimization problems, simplifications of this model are necessary. Here we propose a new model that is derived for high-pressure flows that are c...

In optimal control problems, often initial data are required that are not known exactly in practice. In order to take into account this uncertainty, we consider optimal control problems for a system with an uncertain initial state. A finite terminal time is given. On account of the uncertainty of the initial state, it is not possible to prescribe a...

In this paper the turnpike phenomenon is studied for problems of optimal boundary control. We consider systems that are governed by a linear $2\times 2$ hyperbolic partial differential equation with a source term. Turnpike results are obtained for problems of optimal Dirichlet boundary control for such systems with a strongly convex objective funct...

An example by Bastin and Coron illustrates that the boundary stabilization of 1-d hyperbolic systems with certain source terms is only possible if the length of the space interval is sufficiently small. We show that related phenomena also occur for networks of vibrating strings that are governed by the wave equation with a certain source term. It...

We study problems of optimal boundary control with systems governed by linear hyperbolic partial differential equations. The objective function is quadratic and given by an integral over the finite time interval (0, T) that depends on the boundary traces of the solution. If the time horizon T is sufficiently large, the solution of the dynamic optim...

In this paper, we introduce a stationary model for gas flow based on simplified isothermal Euler equations in a non-cycled pipeline network. Especially the problem of the feasibility of a random load vector is analyzed. Feasibility in this context means the existence of a flow vector meeting these loads, which satisfies the physical conservation la...

We investigate the long-time behavior of solutions of quasilinear hyperbolic systems with transparent boundary conditions when small source terms are incorporated in the system. Even if the finite-time stability of the system is not preserved, it is shown here that an exponential convergence towards the steady state still holds with a decay rate wh...

In the context of gas transportation, analytical solutions are helpful for the understanding of the underlying dynamics governed by a system of partial differential equations. We derive traveling wave solutions for the one-dimensional isothermal Euler equations, where an affine linear compressibility factor is used to describe the correlation betwe...

We study problems of optimal boundary control with systems governed by linear hyperbolic partial differential equations. The objective function is quadratic and given by an integral over the finite time interval $(0,\, T)$ that depends on the boundary traces of the solution. If the time horizon $T$ is sufficiently large, the solution of the dynamic...

We study the transient optimization of gas transport networks including both discrete controls due to switching of controllable elements and nonlinear fluid dynamics described by the system of isothermal Euler equations, which are partial differential equations in time and 1-dimensional space. This combination leads to mixed-integer optimization pr...

The flow of gas through networks of pipes can be modeled by the isothermal Euler equations and algebraic node conditions that model the flow through the vertices of the network graph. This motivates our analysis of the well-posedness of a coupled system of (Formula presented.) conservation laws on a network. We consider initial data and control fun...

We propose a decomposition based method for solving mixed-integer nonlinear optimization problems with “black-box” nonlinearities, where the latter, for example, may arise due to differential equations or expensive simulation runs. The method alternatingly solves a mixed-integer linear master problem and a separation problem for iteratively refinin...

Consider a star-shaped network of strings. Each string is governed by the wave equation. At each boundary node of the network there is a player that performs Dirichlet boundary control action and in this way influences the system state. At the central node, the states are coupled by algebraic conditions in such a way that the energy is conserved. W...

In a gas transport system, the customer behavior is uncertain. Motivated by this situation, we consider a boundary stabilization problem for the flow through a gas pipeline, where the outflow at one end of the pipe that is governed by the customer's behavior is uncertain. The control action is located at the other end of the pipe. The feedback law...

We discuss coupling conditions for the p-system in case of a transition from supersonic states to subsonic states. A single junction with adjacent pipes is considered where on each pipe the gas ow is governed by a general p-system. By extending the notion of demand and supply known from traffic flow analysis we obtain a constructive existence resul...

The one–dimensional isothermal Euler equations are a well-known model for the flow of gas through a pipe. An essential part of the model is the source term that models the influence of gravity and friction on the flow. In general the solutions of hyperbolic balance laws can blow-up in finite time. We show the existence of initial data with arbitrar...

The relaxation approximation for systems of conservation laws has been studied intensively. In this paper the corresponding relaxation approximation for $2\times 2$ systems of balance laws is studied. Our driving example is gas flow in pipelines described by the isothermal Euler equations. We are interested in the limiting behavior as the relaxatio...

We study the optimal value function for control problems on Banach spaces that involve both continuous and discrete control decisions. For problems involving semilinear dynamics subject to mixed control inequality constraints, one can show that the optimal value depends locally Lipschitz continuously on perturbations of the initial data and the cos...

For the management of gas transportation networks, it is essential to know how the stationary states of the system are determined by the boundary data. The isothermal Euler equations are an accurate pde-model for the gas flow through each pipe. A compressibility factor is used to model the nonlinear relationship between density and pressure that oc...

For a system that is governed by the isothermal Euler equations with friction for ideal gas, the corresponding field of characteristic curves is determined by the velocity of the flow. This velocity is determined by a second-order quasilinear hyperbolic equation. For the corresponding initial-boundary value problem with Neumann-boundary feedback, w...

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...

We study optimal control problems for linear systems with prescribed initial and terminal states. We analyze the exact penalization of the terminal constraints. We show that for systems that are exactly controllable, the norm-minimal exact control can be computed as the solution of an optimization problem without terminal constraint but with a nons...

We consider traffic flow governed by the LWR model. We show that a Lipschitz continuous initial density with free-flow and sufficiently small Lipschitz constant can be controlled exactly to an arbitrary constant free-flow density in finite time by a piecewise linear boundary control function that controls the density at the inflow boundary if the o...

We consider a system of scalar nonlocal conservation laws on networks that model a highly re-entrant multi-commodity manufacturing system as encountered in semiconductor production. Every single commodity is mod-eled by a nonlocal conservation law, and the corresponding PDEs are coupled via a collective load, the work in progress. We illustrate the...

Pipeline networks for gas transportation often contain circles. For such networks it is more difficult to determine the stationary states than for networks without circles. We present a method that allows to compute the stationary states for subsonic pipe flow governed by the isothermal Euler equations for certain pipeline networks that contain cir...

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 correct a technical error in the paper of Gugat, Herty, Schleper, Math. Methods Appl. Sci. 34 (2011), where a framework for controllability of quasi-linear hyperbolic systems has been studied. The application to the case of gas networks is specified in more detail in the current work.

We consider the problem of boundary feedback stabilization of a vibrating string that is fixed at one end and with control action at the other end. In contrast to previous studies that have required L 2-regularity for the initial position and H −1-regularity for the initial velocity, in this paper we allow for initial positions with L 1-regularity...

In optimal control loops delays can occur, for example through transmission via digital communication channels. Such delays influence the state that is generated by the implemented control. We study the effect of a delay in the implementation of L 2-norm minimal Neumann boundary controls for the wave equation. The optimal controls are computed as s...

This brief considers recent results on optimal control and stabilization of systems governed by hyperbolic partial differential equations, specifically those in which the control action takes place at the boundary. The wave equation is used as a typical example of a linear system, through which the author explores initial boundary value problems, c...

Up to now, we have considered linear systems. If for such a linear system the existence of a solution can be shown for a certain finite time interval, then the solution exists for all times provided that the control keeps its regularity. For nonlinear systems, the situation is completely different. In a nonlinear hyperbolic system, the solution can...

In optimal control problems, we choose the ‘best’ controls from the set of all admissible controls. In our case, the set of admissible controls consists of the set of all controls that steer the system to the desired terminal state at the given terminal time. In general, these exact controls are not uniquely determined. Therefore we can choose from...

We consider systems that are governed by hyperbolic partial differential equations (pdes). As a first example, we consider the wave equation $$\displaystyle\begin{array}{rcl} y_{tt} = c^{2}\,y_{ xx}.& & {}\\ \end{array}$$ Here c is a real number and | c | is called the wave speed. We will focus on the one-dimensional case, where we can present esse...

The aim of controls for boundary stabilization is to influence the system state from the points where the control action takes place in such a way that the states approaches a given desired state. Moreover, this should happen quite fast, if possible with an exponential rate. Often this is done using feedback laws, where the current observation and...

The question of exact controllability (see Lions, SIAM Rev. 30, 1–68, 1988; Russell, J. Math. Anal. Appl. 18, 542–560, 1967) is: Which states can be reached exactly at given control time T with a given set of control functions starting at time zero with an initial state from a prescribed set?

For the analysis of partial differential equations often derivatives in the sense of distributions are needed, since classical solutions do not exist. Therefore we present a very short introduction to the theory of distributions that has essentially been influenced by Laurent Schwartz (see Schwartz, Méthodes mathématiques pour les sciences physique...

In the design and computation of optimal controls for systems that evolve in time, usually the effect of delay is ignored. However in the implementation of the computed optimal controls in the control systems often delays occur, for example through transmission via digital communication channels. The question to be addressed is whether such small d...

We summarize recent theoretical results as well as numerical results on the feedback stabilization of first order quasilinear hyperbolic systems (on networks). For the stabilization linear feedback controls are applied at the nodes of the network. This yields the existence and uniqueness of a C 1-solution of the hyperbolic system with small C 1-nor...

The Schlögl system is governed by a nonlinear reaction-diffusion partial differential equation with a cubic nonlinearity that determines three constant equilibrium states. It is a classical example of a chemical reaction system that is bistable. The constant equilibrium that is enclosed by the other two constant equilibrium points is unstable. In t...

We consider the problem to control a vibrating string to rest in a given finite time. The string is fixed at one end and controlled by Neumann boundary control at the other end. We give an explicit representation of the L 2-norm minimal control in terms of the given initial state. We show that if the initial state is sufficiently regular, the same...

We study a semilinear mildly damped wave equation that contains the telegraph equation as a special case. We consider Neumann velocity boundary feedback and prove the exponential stability of the closed loop system. We show that for vanishing damping term in the partial differential equation, the decay rate of the system approaches the rate for the...

Audioslides to Boundary feedback stabilization of the telegraph equation: Decay rates for vanishing damping term

We present results on a method for infinite dimensional constrained optimization problems. In particular, we are interested in state constrained optimal control problems and discuss an algorithm based on penalization and smoothing. The algorithm contains update rules for the penalty and the smoothing parameter that depend on the constraint violatio...

We consider an exact boundary control problem for the wave equation with given initial and terminal data and Dirichlet boundary control. The aim is to steer the state of the system that is defined on a given domain to a position of rest in finite time. The optimal control that is obtained as the solution of the problem depends on the data that defi...

In the semianalytical models for squeeze film damping, the coefficient of damping torque is expressed as an infinite double series. To work with these models, methods for the efficient numerical evaluation of these double series are important, because, as has been pointed out, the results are given by complicated equations; the application of the r...

We consider a system that is exactly controllable. For given initial state, terminal state and objective function, an optimal control is often well-defined. Such an optimal control has the disadvantage that although it works perfectly well for the given initial state, for a perturbed initial state it often does not make sense. In this talk we prese...

We consider the subcritical gas flow through star-shaped pipe networks. The gas flow is modeled by the isothermal Euler equations with friction. We stabilize the isothermal Euler equations locally around a given stationary state on a finite time interval. For the stabilization we apply boundary feedback controls with time-varying delays. The delays...

Compressible squeeze film damping is a phenomenon of great importance for micromachines. For example, for the optimal design of an electrostatically actuated micro-cantilever mass sensor that operates in air, it is essential to have a model for the system behavior that can be evaluated efficiently. An analytical model that is based upon a solution...

We consider a water distribution network where at a finite number of nodes, contaminant injection can occur. We consider the problem of the identification of the contaminations. This problem can be considered as an optimal control problem with a networked system that is governed by a transport reaction equation. The identification is based upon obs...

We consider the isothermal Euler equations without friction that simulate gas flow through a pipe. We consider the problem of boundary stabilisation of this system locally around a given stationary state. We present a feedback law that is linear in the physical variables and yields exponential decay of the system state. For the numerical solution o...

We present an overview on recent results concerning hyperbolic systems on networks. We present a summary of theoretical results on existence, uniqueness and stability. The established theory extends previously known results on the Cauchy problem for nonlinear, 2×2 hyperbolic balance laws. The proofs are based on Wave-Front Tracking and therefore we...

We consider the feedback stabilization of quasilinear hyperbolic systems on star-shaped networks. We present boundary feedback controls with varying delays. The delays are given by C1-functions with bounded derivatives. We obtain the existence of unique C1-solutions on a given finite time interval. In order to measure the system evolution, we intro...

The authors consider the problem of boundary feedback stabilization of the 1D Euler gas dynamics locally around stationary states and prove the exponential stability with respect to the H 2-norm. To this end, an explicit Lyapunov function as a weighted and squared H 2-norm of a small perturbation of the stationary solution is constructed. The autho...

We consider the isothermal Euler equations with friction that model the gas flow through pipes. We present a method of time-delayed boundary feedback stabilization to stabilize the isothermal Euler equations locally around a given stationary subcritical state on a finite time interval. The considered control system is a quasilinear hyperbolic syste...

We study the isothermal Euler equations with friction and consider non-stationary solutions locally around a stationary subcritical state on a finite time interval. The considered control system is a quasilinear hyperbolic system with a source term. For the corresponding initial-boundary value problem we prove the existence of a continuously differ...

In optimal control problems frequently pointwise control constraints appear. We consider a finite string that is fixed at one end and controlled via Dirichlet conditions at the other end with a given upper bound M for the L ∞-norm of the control. The problem is to control the string to the zero state in a given finite time. If M is too small, no fe...

For vibrating systems, a delay in the application of a feedback control may destroy the stabilizing effect of the control. In this paper we consider a vibrating string that is fixed at one end and stabilized with a boundary feedback with delay at the other end.We show that certain delays in the boundary feedback preserve the exponential stability o...

We consider general, not necessarily convex, optimization problems with inequality constraints. We show that the smoothed penalty algorithm generates a sequence that converges to a stationary point. In particular, we show that the algorithm provides approximations of the multipliers for the inequality constraints. The theoretical analysis is illust...

We consider the subcritical flow in gas networks consisting of a finite linear sequence of pipes coupled by compressor stations. Such networks are important for the transportation of natural gas over large distances to ensure sustained gas supply. We analyse the system dynamics in terms of Riemann invariants and study stationary solutions as well a...

In the application of feedback controls, the computation of the controls may cause a delay. For vibrating systems, a constant delay can destroy the stabilizing effect of the control. To avoid this problem we consider a feedback where a certain delay is a part of the control law and not a perturbation. We consider a string that is fixed at one end a...

We present an algorithm for the solution of general inequality constrained optimization problems. The algorithm is based upon an exact penalty function that is approximated by a family of smooth functions. We present convergence results. As numerical examples we treat state constrained optimal control problems for elliptic partial differential equa...

We consider a star-shaped network consisting of a single node with N � 3 connected arcs. The dynamics on each arc is governed by the wave equation. The arcs are coupled at the node and each arc is controlled at the other end. Without assumptions on the lengths of the arcs, we show that if the feedback control is active at all exterior ends, the sys...

In the application of feedback controls, a delay may appear as a perturbation caused by the computation of the controls. For vibrating systems, this delay can destroy the stabilizing effect of the control. To avoid this problem, we consider feedback laws where a certain delay is included as a part of the control law and not as a perturbation. We co...

We consider the approximation of semigroups $e^{\tau A}$ and of the functions $\varphi_j(\tau A)$ that appear in exponential integrators by resolvent series. The interesting fact is that the resolvent series expresses the operator functions $e^{\tau A}$ and $\varphi_j(\tau A)$, respectively, in efficiently computable terms. This is important for se...

A highly efficient decoding algorithm for the REMOS (REverberation MOdeling for Speech recognition) con-cept for distant-talking speech recognition as proposed in [1] is suggested to reduce the computational com-plexity by about two orders of magnitude and thereby allowing for first real-time implementations. REMOS is based on a combined acoustic m...

A Lavrentiev prox-regularization method for optimal control problems with pointwise state constraints is introduced. The convergence of the controls generated by the iterative Lavrentiev prox-regularization algorithm is studied. For a sequence of regularization parameters that converges to zero, strong convergence of the generated control sequence...