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- Turnpike phenomenon (3)
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- Boundary feedback control, feedback stabilization, exponential stability, isothermal Euler equations, second-order quasilinear equation, Lyapunov function, stationary state, non-stationary state, gas pipeline. (1)
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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 sufficiently low Mach numbers and sufficiently high pressures. We prove an existence result for continuous solutions of the semilinear model that takes into account lower and upper bounds for the pressure and an upper bound for the magnitude of the Mach number of the gas flow. These state constraints are important both in the operation of gas pipelines and to guarantee that the solution remains in the set where the model is physically valid. We show the constrained exact boundary controllability of the system with the same pressure and Mach number constraints.
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=2/3. Such equations exhibit positive solutions, finite speed of propagation and satisfy a maximum principle. The pressure is fixed on one end (upstream), and the flow is specified on the other end (downstream). These boundary conditions determine a unique steady equilibrium flow. We present a boundary feedback flow control scheme, that ensures local exponential stability of the equilibrium in an L2-sense. The analysis is done both for the pde system and an ode system that is obtained by a suitable spatial semi-discretization. The proofs are based upon suitably chosen Lyapunov functions.
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.
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.
Boundary feedback stabilization of a semilinear model for the flow in star-shaped gas networks
(2020)
The flow of gas through a pipeline network
can be modelled by a
coupled system of 1-d quasilinear hyperbolic equations.
In this system, the influence of
certain source terms that model friction effects is essential.
Often for the solution of
control problems it is convenient to replace the quasilinear model
by a simpler semilinear model.
In this paper, we analyze the behavior of such
a semilinear model on a star-shaped network.
The model is derived from the diagonal form of the quasilinear model by replacing the eigenvalues by
the sound speed multiplied by
1 or -1 respectively.
Thus in the corresponding eigenvalues
the influence of the gas velocity is neglected,
which is justified in the applications
since it is much smaller than the sound speed in
the gas.
For a star-shaped network of horizontal pipes
for suitable coupling conditions we present boundary feedback laws
that stabilize the system state exponentially fast
to a position of rest
for sufficiently small initial data.
We show the exponential decay of
the $H^1$-norm
for arbitrarily long pipes.
This is remarkable since in general
even for linear systems, for certain source terms
the system can become exponentially unstable
if the space interval is too long.
Our proofs
are based upon
observability inequalities
for the $L^2$ and the $H^1$-norm.
We present a positive and a negative stabilization result for a semilinear
model of gas flow in pipelines. For feedback boundary conditions we obtain an
unconditional stabilization result 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 isothermal Euler equations
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.
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] and measures the distance to a desired stationary state. In the
optimal control problem, both the initial and the desired terminal state are prescribed. We assume that the system is exactly controllable in an abstract sense if the time horizon is long enough.
We show that that the corresponding optimal control problems on the time intervals [0, T] give rise to a turnpike structure in the sense that for natural numbers n if T is suciently large, the contribution
of the objective function from subintervals of [0, T] of the form
[t - t/2^n, t + (T-t)/2^n]
is of the order 1/min{t^n, (T-t)^n}. We also show that a similar result holds for epsilon-optimal solutions of the optimal control problems if epsilon > 0 is chosen suffciently small. At the end of the paper we present
both systems that are governed by ordinary differential equations and
systems governed by partial differential equations where the results can be applied.
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 certain dynamic situations in the operation of gas transportation networks. We show that adequate numerical discretizations can capture the dynamical behavior sufficiently accurate. We also present examples that show that in certain cases an optimization approach that is based upon multi-period optimization of steady states does not lead to approximations that converge to the optimal state.