Friedrich-Alexander-Universität Erlangen-Nürnberg
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- Mixed-Integer Nonlinear Programming (2)
- Robust Optimization (2)
- A posteriori error estimates (1)
- AC Optimal Power Flow (1)
- Adjustable Robustness (1)
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We introduce and study the turnpike property for time-varying shapes, within the viewpoint of optimal control. We focus here on second-order linear parabolic equations where the shape acts as a source term and we seek the optimal time-varying shape that minimizes a quadratic criterion. We first establish existence of optimal solutions under some appropriate sufficient conditions. We then provide necessary conditions for optimality in terms of adjoint equations and, using the concept of strict dissipativity, we prove that state and adjoint satisfy the measure-turnpike property, meaning that the extremal time-varying solution remains essentially close to the optimal solution of an associated static problem. We show that the optimal shape enjoys the exponential turnpike property in term of Hausdorff distance for a Mayer quadratic cost. We illustrate the turnpike phenomenon in optimal shape design with several numerical simulations.
Time-Domain Decomposition for Optimal Control Problems Governed by Semilinear Hyperbolic Systems
(2020)
In this article, we extend the time-domain decomposition method described by Lagnese and Leugering (2003) to semilinear optimal control problems for hyperbolic balance laws with spatio-temporal varying coefficients. We provide the design of the iterative method applied to the global first-order optimality system, prove its convergence, and derive an a posteriori error estimate. The analysis is done entirely on the continuous level. A distinguishing feature of the method is that the decomposed optimality system can be interpreted as an optimality system of a local "virtual" optimal control problem. Thus, the iterative time-domain decomposition of the optimality system can be interpreted as an iterative parallel scheme for virtual optimal control problems on the subintervals. A typical example and further comments are given to show the range of potential applications. Moreover, we provide some numerical experiments to give a first interpretation of the role of the parameters involved in the iterative process.
We present a new proof of the turnpike property for nonlinear optimal control problems, when the running target is a steady control-state pair of the underlying dynamics. Our strategy combines the construction of suboptimal quasi-turnpike trajectories via controllability, and a bootstrap argument, and does not rely on analyzing the optimality system or linearization techniques. This in turn allows us to address several optimal control problems for finite-dimensional, control-affine systems with globally Lipschitz (possibly nonsmooth) nonlinearities, without any smallness conditions on the initial data or the running target. These results are motivated by the large-layer regime of residual neural networks, commonly used in deep learning applications. We show that our methodology is applicable to controlled PDEs as well, such as the semilinear wave and heat equation with a globally Lipschitz nonlinearity, once again without any smallness assumptions.
Abstract. This paper deals with the averaged dynamics for heat equations in the degenerate case where the diffusivity coefficient, assumed to be constant, is allowed to take the null value. First we prove that the averaged dynamics is analytic. This allows to show that, most often, the averaged dynamics enjoys the property of unique continuation and is approximately controllable. We then determine if the averaged dynamics is actually null controllable or not depending on how the density of averaging behaves when the diffusivity vanishes. In the critical density threshold the dynamics of the average is similar to the $\frac{1}{2}$-fractional Laplacian, which is wellknown to be critical in the context of the controllability of fractional diffusion processes. Null controllability then fails (resp. holds) when the density weights more (resp. less) in the null diffusivity regime than in this critical regime.
Linear bilevel optimization problems have gained increasing attention both in theory as well as in practical applications of Operations Research (OR) during the last years and decades. The latter is mainly due to the ability of this class of problems to model hierarchical decision processes. However, this ability makes bilevel problems also very hard to solve. Since no general-purpose solvers are available, a "best-practice" has developed in the applied OR community, in which not all people want to develop tailored algorithms but "just use" bilevel optimization as a modeling tool for practice. This best-practice is the big-M reformulation of the Karush-Kuhn-Tucker (KKT) conditions of the lower-level problem - an approach that has been shown to be highly problematic by Pineda and Morales (2019). Choosing invalid values for M yields solutions that may be arbitrarily bad. Checking the validity of the big-Ms is however shown to be as hard as solving the original bilevel problem in Kleinert et al. (2019). Nevertheless, due to its appealing simplicity, especially w.r.t. the required implementation effort, this ready-to-use approach still is the most popular method. Until now, there has been a lack of approaches that are competitive both in terms of implementation effort and computational cost.
In this note we demonstrate that there is indeed another competitive ready-to-use approach: If the SOS-1 technique is applied to the KKT complementarity conditions, adding the simple additional root-node inequality developed by Kleinert et al. (2020) leads to a competitive performance - without having all the possible theoretical disadvantages of the big-M approach.
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
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.
Multi-modal distributed energy system planning is applied in the context of smart grids, industrial energy supply,and in the building energy sector. In real-world applications, these systems are commonly characterized by existing system structures of different age where monitoring and investment are conducted in a closed-loop, with the iterative possibility to invest. The literature contains two main approaches to approximate this computationally intensive multiperiod
investment problem. The first approach simplifies the temporal decision-making process collapsing the multistage decision to a two-stage decision, considering uncertainty in the second stage decision variables. The second approach considers multi-period investments under the assumption of perfect foresight. In this work, we propose a
multi-stage stochastic optimization problem that captures multi-period investment decisions under uncertainty and solves the problem to global optimality, serving as a first-best benchmark to the problem. To evaluate the performance of conventional approaches applied in a multi-year setup and to solve the multi-period problem at lower computational effort, we propose a rolling horizon heuristic that on the one hand reveals the performance of conventional approaches
applied in a multi-period set-up and on the other hand enables planners to identify approximate solutions to the original
multi-stage stochastic problem. Additionally, we consider an open-loop version of the rolling horizon algorithm to evaluate how single-period investments perform with respect to the entire scenario tree and compared to multi-period investments.
We conduct a real-world case study and investigate solution quality as well as the computational performance of the proposed approaches. Our findings indicate that the approximation of multi-period investments by two-stage stochastic approaches yield the best results regarding constraint satisfaction, while deterministic multi-period approximations yield
better economic and computational performance.
The use of electric fuels (e-fuels) enables CO2-neutral mobility and opens therefore an alternative to fossil-fuel-fired engines or battery-powered electric motors. This paper compares the cost-effectiveness of Fischer-Tropsch diesel, methanol, and hydrogen stored as cryogenic liquid (LH2) or in form of liquid organic hydrogen carriers (LOHCs). The production cost of those fuels are to a large extent driven by the energy-intensive electrolytic water splitting. The option of producing e-fuels in Germany competes with international locations with excellent conditions for renewable energy harvesting and thus very low levelized cost of electricity. We developed a mathematical model that covers the entire process chain. Starting with the production of the required resources such as fresh water, hydrogen, carbon dioxide, carbon monoxide, electrical and thermal energy, the subsequent chemical synthesis, the transport to filling stations in Germany and finally the energetic utilization of the fuels in the vehicle. We found that the choice of production site can have a major impact on the mobility cost using the respective fuels. Especially in case of diesel production, the levelized cost of electricity driven by the full load hours of the applied renewable energy source have a huge impact. An LOHC-based system is shown to be less dependent on the kind of electricity source compared to other technologies due to its comparatively low electricity consumption and the low cost for the hydrogenation units. The length of the transportation route and the price of the filling station infrastructure, on the other hand, clearly increase mobility cost for LOHC and LH2.
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.