Friedrich-Alexander-Universität Erlangen-Nürnberg
We consider non-overlapping domain decomposition methods for ordinary and partial differential equations and corresponding optimal control problems on metric graphs. As an exemplary context, we chose a semilinear approximation of the Euler system and a doubly nonlinear parabolic model that has come to be known as friction dominated flow in gas pipe networks. By this choice, we encounter hyperbolic, parabolic and elliptic linear and nonlinear problems in a single highly motivating application. We depart from the classical domain decomposition methods described by P.L. Lions and J.L. Lions and O. Pironneau and extend those to problems on metric graphs. The choice of methods is determined by the desire to use a control concept that has come to be known as virtual controls which, in turn, possibly lead to a fully parallel decomposition of the corresponding optimality systems. In a second step, we extend the methods to p-Laplace problems on networks. The analysis, due to space limitations, will appear in a forthcoming publication. See however J.E. Lagnese and G. Leugering. Furthermore, we then describe methods for space and time domain decomposition or optimal control problems in the spirit of J.E. Lagnese and G. Leugering. We finally provide some comments on PINN-based approximations of the methods described before. We provide numerical evidence for all algorithms discussed.
We consider a non-overlapping domain decomposition method for an optimal control problem related to the flow of gas in a pipe network. The equations of motions are taken to be represented by a friction dominated model derived from a semi-linear approximation of the fully nonlinear isothermal Euler gas equations. This involves a p-Laplace-type problem on the graph with p = 3/2. We continue the work by Leugering and Mophou where such a problem has been discussed in the context of an instantaneous control strategy. We provide a non-overlapping domain decomposition in the spirit of P.L. Lions for elliptic problems and extend the method to the first order optimality system.
Space-time-domain decomposition for optimal control problems governed by linear hyperbolic systems
(2021)
In this article, we combine a domain decomposition method in space and time for optimal control problems with PDE-constraints described by Lagnese and Leugering to a simultaneous space-time decomposition applied to optimal control problems for systems of linear hyperbolic equations with distributed control. We thereby extend the recent work by Krug et al. and answer a long standing open question as to whether the combination of time- and space domain decomposition for the method under consideration can be put into one single convergent iteration procedure. The algorithm is designed for a semi-elliptic system of equations obtained from the hyperbolic optimality system by the way of reduction to the adjoint state. The focus is on the relation to the classical procedure introduced by Lions for elliptic problems.
Space-Time-Domain Decomposition for Optimal Control Problems Governed by Linear Hyperbolic Systems
(2021)
In this article, we combine a domain decomposition method in space and time for optimal control problems with PDE-constraints described by Lagnese and Leugering to a simultaneous space-time decomposition applied to optimal control problems for systems of linear hyperbolic equations with distributed control. We thereby extend the recent work by Krug et al. and answer a long standing open question as to whether the combination of time- and space domain decomposition for the method under consideration can be put into one single convergent iteration procedure. The algorithm is designed for a semi-elliptic system of equations obtained from the hyperbolic optimality system by the way of reduction to the adjoint state. The focus is on the relation to the classical procedure introduced by Lions for elliptic problems.
In this article, we continue our work (Krug et al., 2021) on time-domain decomposition of optimal control problems for systems of semilinear hyperbolic equations in that we now consider mixed two-point boundary value problems and provide an in-depth well-posedness analysis. The more general boundary conditions significantly enlarge the scope of applications, e.g., to hyperbolic problems on metric graphs with cycles. We design an iterative method based on the optimality systems that can be interpreted as a decomposition method for the original optimal control problem into virtual control problems on smaller time domains.
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.
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.