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We investigate the interior turnpike phenomenon for discrete-time multi-agent optimal control problems. While for continuous systems the turnpike property has been established, we focus here on first-order discretizations of such systems. It is shown that the resulting time-discrete system inherits the turnpike property with estimates of the same type as in the continuous case. In particular, we prove that the discrete time optimal control problem is strictly dissipative and the cheap control assumption holds.

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 approximated by an observer system that uses nodal measurements. We prove that the state of the observer system converges to the original system state exponentially fast in the $L^2$-norm if the measurements are exact. If measurement errors are present we show that the observer state approximates the original system state up to an error that is proportional to the maximal measurement error. The proof of the synchronization result uses Lyapunov functions with exponential weights.

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 the observer system and of the original system for initial and boundary data satisfying smallness and compatibility conditions for a single pipe and for general networks. Second, based on an extension of the relative energy method we prove that the state of the observer system converges exponentially in the long time limit towards the original system state. We show this for a single pipe and for star-shaped networks.

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 the optimal control. For the proof, we use the fact that the turnpike structure for the problems on the level of ordinary differential equations is preserved under the corresponding mean-field limit.

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 at certain points in space. Based upon these nodal observations, the complete system state can be approximated using an observer system. In this paper we present a nodal observer, and prove that the state of the observer system converges to the original state exponentially fast. Numerical experiments confirm the theoretical findings.

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 hyperbolic balance laws. We survey results on classical solutions as well as weak solutions. We present results on well--posedness, controllability, feedback stabilization, the inclusion of uncertainty in the models and numerical methods.

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, feasible means, that the flow corresponding to the random boundary data meets some box constraints at the network junctions. The first method is the spheric radial decomposition and the second method is a kernel density estimation. In both settings, we consider certain optimization problems and we compute derivatives of the probabilistic constraint using the kernel density estimator. Moreover, we derive necessary optimality conditions for the stationary and the dynamic case. Throughout the paper, we use numerical examples to illustrate our results by comparing them with a classical Monte Carlo approach to compute the desired probability., Comment: 47 pages, 13 figures

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 prescribed. We assume that the system is either exactly controllable in the classical sense or nodal profile controllable. We show that both for systems that are governed by ordinary differential equations and for infinite-dimensional systems, for example for boundary control systems governed by the wave equation, under certain assumptions the optimal system state is steered exactly to the desired state after finite time.

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 optimal boundary control problem can be approximated by the solution of a steady state optimization problem. We show that for $T\rightarrow\infty$ the approximation error converges to zero in the sense of the norm in the Hilbert space $L^2(0,\,1)$ with the rate $1/T$, if the time interval $(0,\, T)$ is transformed to the fixed interval $(0,\, 1)$. Moreover, we show that also for optimal boundary control problems with integer constraints for the controls the turnpike phenomenon occurs. In this case the steady state optimization problem also has the integer constraints. If $T$ is sufficiently large, the integer part of each solution of the dynamic optimal boundary control problem with integer constraints is equal to the integer part of a solution of the static problem. A numerical verification is given for a control problem in gas pipeline operations., Comment: 3 Figures, 1 Table

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 is a classical Neumann velocity feedback with a feedback parameter $k>0$. We show that as long as the $H^1$-norm of the function that describes the noise in the customer's behavior decays exponentially with a rate that is sufficiently large, the velocity of the gas can be stabilized exponentially fast in the sense that a suitably chosen Lyapunov function decays exponentially. For the exponential stability it is sufficient that the feedback parameter $k$ is sufficiently large and the stationary state to which the system is stabilized is sufficiently small. The stability result is local, that is it holds for initial states that are sufficiently close to the stationary state. This result is an example for the exponential boundary feedback stabilization of a quasilinear hyperbolic system with uncertain boundary data. The analysis is based upon the choice of a suitably Lyapunov function. The decay of this Lyapunov function implies that also the $L^2$-norm of the difference of the system state and the stationary state decays exponentially.

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 which is proportional to the logarithm of the amplitude of the source term. The result is stated for a system with dynamical boundary conditions in order to deal with initial data that are free of any compatibility condition.

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 costs under rather natural assumptions. We prove a similar result for perturbations of the initial data, the constraints and the costs for problems involving linear dynamics, convex costs and convex constraints under a Slater-type constraint qualification. We show by an example that these results are in a sense sharp., Comment: The final publication is available at http://link.springer.com/article/10.1007/s00498-016-0183-4

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, we consider non-stationary solutions locally around a stationary state on a finite time interval and discuss the well-posedness of this kind of problem. We introduce a strict $H^2$-Lyapunov function and show that the boundary feedback constant can be chosen such that the $H^2$-Lyapunov function and hence also the $H^2$-norm of the difference between the non-stationary and the stationary state decays exponentially with time.

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 = +$\infty$ (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.

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 present a concept to obtain robust control schemes by the combination of the optimal control with a stabilizing feedback law. In this way, also for perturbed initial states the system is controlled in a reasonable way. In the talk we focus on the boundary control of the wave equation. However, our concept is applicable to the control of general time-dependent systems.

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 constant delays of the form 4L/c, 8L/c, 12L/c ... Moreover, we show that the system is exponentially stable with piecewise constant delays that attain the values 4L/c and 8L/c., Comment: 18 pages