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Mathematical Control & Related Fields

September 2021 , Volume 11 , Issue 3

Issue on system modeling and optimization

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Preface
Eduardo Casas, Christian Clason and Arnd Rösch
2021, 11(3): i-ii doi: 10.3934/mcrf.2021008 +[Abstract](882) +[HTML](298) +[PDF](88.08KB)
Abstract:
Optimal control of elliptic variational inequalities with bounded and unbounded operators
Livia Betz and Irwin Yousept
2021, 11(3): 479-498 doi: 10.3934/mcrf.2021009 +[Abstract](816) +[HTML](142) +[PDF](514.8KB)
Abstract:

This paper examines optimal control problems governed by elliptic variational inequalities of the second kind with bounded and unbounded operators. To tackle the bounded case, we employ the polyhedricity of the test set appearing in the dual formulation of the governing variational inequality. Based thereon, we are able to prove the directional differentiability of the associated solution operator, which leads to a strong stationary optimality system. The second part of the paper deals with the unbounded case. Due to the non-smoothness of the variational inequality and the unboundedness of the governing elliptic operator, the directional differentiability of the solution operator becomes difficult to handle. Our strategy is to apply the Yosida approximation to the unbounded operator, while the non-smoothness of the variational inequality is still preserved. Based on the developed strong stationary result for the bounded case, we are able to derive optimality conditions for the unbounded case by passing to the limit in the Yosida approximation. Finally, we apply the developed results to Maxwell-type variational inequalities arising in superconductivity.

Tracking aircraft trajectories in the presence of wind disturbances
Nikolai Botkin, Varvara Turova, Barzin Hosseini, Johannes Diepolder and Florian Holzapfel
2021, 11(3): 499-520 doi: 10.3934/mcrf.2021010 +[Abstract](706) +[HTML](314) +[PDF](3382.96KB)
Abstract:

A method of path following, utilized in the theory of position differential games as a tool for establishing theoretical results, is adopted in this paper for tracking aircraft trajectories under windshear conditions. It is interesting to note that reference trajectories, obtained as solutions of optimal control problems with zero wind, can very often be tracked in the presence of rather severe wind disturbances. This is shown in the present paper for rather realistic and highly nonlinear models of aircraft dynamics.

Optimal control of a non-smooth quasilinear elliptic equation
Christian Clason, Vu Huu Nhu and Arnd Rösch
2021, 11(3): 521-554 doi: 10.3934/mcrf.2020052 +[Abstract](914) +[HTML](237) +[PDF](762.17KB)
Abstract:

This work is concerned with an optimal control problem governed by a non-smooth quasilinear elliptic equation with a nonlinear coefficient in the principal part that is locally Lipschitz continuous and directionally but not Gâteaux differentiable. This leads to a control-to-state operator that is directionally but not Gâteaux differentiable as well. Based on a suitable regularization scheme, we derive C- and strong stationarity conditions. Under the additional assumption that the nonlinearity is a \begin{document}$ PC^1 $\end{document} function with countably many points of nondifferentiability, we show that both conditions are equivalent. Furthermore, under this assumption we derive a relaxed optimality system that is amenable to numerical solution using a semi-smooth Newton method. This is illustrated by numerical examples.

Optimal control of ODEs with state suprema
Tobias Geiger, Daniel Wachsmuth and Gerd Wachsmuth
2021, 11(3): 555-578 doi: 10.3934/mcrf.2021012 +[Abstract](712) +[HTML](152) +[PDF](500.5KB)
Abstract:

We consider the optimal control of a differential equation that involves the suprema of the state over some part of the history. In many applications, this non-smooth functional dependence is crucial for the successful modeling of real-world phenomena. We prove the existence of solutions and show that related problems may not possess optimal controls. Due to the non-smoothness in the state equation, we cannot obtain optimality conditions via standard theory. Therefore, we regularize the problem via a LogIntExp functional which generalizes the well-known LogSumExp. By passing to the limit with the regularization, we obtain an optimality system for the original problem. The theory is illustrated by some numerical experiments.

Performance estimates for economic model predictive control and their application in proper orthogonal decomposition-based implementations
Lars Grüne, Luca Mechelli, Simon Pirkelmann and Stefan Volkwein
2021, 11(3): 579-599 doi: 10.3934/mcrf.2021013 +[Abstract](705) +[HTML](151) +[PDF](701.68KB)
Abstract:

In this paper performance indices for economic model predictive controllers (MPC) are considered. Since existing relative performance measures, designed for stabilizing controllers, fail in the economic setting, we propose alternative absolute quantities. We show that these can be applied to assess the performance of the closed loop trajectories on-line while the controller is running. The advantages of our approach are demonstrated by simulations involving a convection-diffusion-system. The method is also combined with proper orthogonal decomposition, thus demonstrating the possibility for both efficient and performant MPC for systems governed by partial differential equations.

A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system
Marita Holtmannspötter, Arnd Rösch and Boris Vexler
2021, 11(3): 601-624 doi: 10.3934/mcrf.2021014 +[Abstract](651) +[HTML](146) +[PDF](522.87KB)
Abstract:

In this paper we investigate a priori error estimates for the space-time Galerkin finite element discretization of an optimal control problem governed by a simplified linear gradient enhanced damage model. The model equations are of a special structure as the state equation consists of an elliptic PDE which has to be fulfilled at almost all times coupled with an ODE that has to hold true in almost all points in space. The state equation is discretized by a piecewise constant discontinuous Galerkin method in time and usual conforming linear finite elements in space. For the discretization of the control we employ the same discretization technique which turns out to be equivalent to a variational discretization approach. We provide error estimates of optimal order both for the discretization of the state equation as well as for the optimal control. Numerical experiments are added to illustrate the proven rates of convergence.

External polyhedral estimates of reachable sets of discrete-time systems with integral bounds on additive terms
Elena K. Kostousova
2021, 11(3): 625-641 doi: 10.3934/mcrf.2021015 +[Abstract](622) +[HTML](150) +[PDF](457.49KB)
Abstract:

We deal with the reachability problem for linear and bilinear discrete-time uncertain systems under integral non-quadratic constraints on additive input terms and set-valued constraints on initial states. The bilinearity is caused by an interval type uncertainty in coefficients of the system. Algorithms for constructing external parallelepiped-valued (shorter, polyhedral) estimates of reachable sets are presented. For linear time-invariant systems, two techniques for constructing touching external estimates with constant orientation matrices are described and compared. For time-dependant bilinear systems, parallelepiped-valued estimates are constructed. For bilinear systems with constant coefficients, nonconvex estimates are proposed in the form of unions of parallelepipeds. Evolution of all estimates is determined by systems of recurrence relations.

Modeling the pressure distribution in a spatially averaged cerebral capillary network
Andrey Kovtanyuk, Alexander Chebotarev, Nikolai Botkin, Varvara Turova, Irina Sidorenko and Renée Lampe
2021, 11(3): 643-652 doi: 10.3934/mcrf.2021016 +[Abstract](628) +[HTML](157) +[PDF](717.42KB)
Abstract:

A boundary value problem for the Poisson's equation with unknown intensities of sources is studied in context of mathematical modeling the pressure distribution in cerebral capillary networks. The problem is formulated as an inverse problem with finite-dimensional overdetermination. The unique solvability of the problem is proven. A numerical algorithm is proposed and implemented.

Second order directional shape derivatives of integrals on submanifolds
Anton Schiela and Julian Ortiz
2021, 11(3): 658-679 doi: 10.3934/mcrf.2021017 +[Abstract](573) +[HTML](159) +[PDF](464.49KB)
Abstract:

We compute first and second order shape sensitivities of integrals on smooth submanifolds using a variant of shape differentiation. The result is a quadratic form in terms of one perturbation vector field that yields a second order quadratic model of the perturbed functional. We discuss the structure of this derivative, derive domain expressions and Hadamard forms in a general geometric framework, and give a detailed geometric interpretation of the arising terms.

Uncertainty damping in kinetic traffic models by driver-assist controls
Andrea Tosin and Mattia Zanella
2021, 11(3): 681-713 doi: 10.3934/mcrf.2021018 +[Abstract](612) +[HTML](150) +[PDF](1740.37KB)
Abstract:

In this paper, we propose a kinetic model of traffic flow with uncertain binary interactions, which explains the scattering of the fundamental diagram in terms of the macroscopic variability of aggregate quantities, such as the mean speed and the flux of the vehicles, produced by the microscopic uncertainty. Moreover, we design control strategies at the level of the microscopic interactions among the vehicles, by which we prove that it is possible to dampen the propagation of such an uncertainty across the scales. Our analytical and numerical results suggest that the aggregate traffic flow may be made more ordered, hence predictable, by implementing such control protocols in driver-assist vehicles. Remarkably, they also provide a precise relationship between a measure of the macroscopic damping of the uncertainty and the penetration rate of the driver-assist technology in the traffic stream.

2020 Impact Factor: 1.284
5 Year Impact Factor: 1.345
2020 CiteScore: 1.9

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