Space-Time-Domain Decomposition for Optimal Control Problems Governed by Linear Hyperbolic Systems

In this article, we combine a domain decomposition method in space and time for optimal control problems with PDE-constraints described in [2] 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 [31, 32] 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 P. L. Lions [25] for elliptic problems.In this article, we combine a domain decomposition method in space and time for optimal control problems with PDE-constraints described in [2] 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 [31, 32] 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 P. L. Lions [25] for elliptic problems

Boundary Controllability and Observability of a Viscoelastic String

In this paper we consider an integrodifferential system, which governs the vibration of a viscoelastic one-dimensional object. We assume that we can act on the system at the boundary and we prove that it is possible to control both the position and the velocity at every point of the body and at a certain time $T$, large enough. We shall prove this result using moment theory and we shall prove that the solution of this problem leads to identify a Riesz sequence which solves controllability and observability. So, the result as presented here are constructive and can lead to simple numerical algorithms

Topological derivatives for networks of elastic strings

International audienceWe consider second order problems on metric graphs under given boundary and nodal conditions. We consider the problem of changing the topology of the underlying graph in that we replace a multiple node by an imported subgraph, or, in reverse, concentrate a subgraph to a single node or delete or add edges, respectively. We wish to do so in some optimal fashion. More precisely, given a cost function we may look at such operations in order to find an optimal topology of the graph. Thus, finally we are looking into the topological gradient of an elliptic problem on a graph

Dynamic domain decomposition of optimal control problems for networks of Euler-Bernoulli beams

We consider planar networks of Euler-Bernoulli beams subject to Neumann-type boundary controls at simple nodes. The object is to minimize a cost functional along some part of the beam structure by the way of dynamic domain decomposition

On boundary observability estimates for semi-discretizations of a dynamic network of elastic strings

We consider a tripod as an exemplaric network of strings. We know that such a network is exactly controllable in the natural finite energy space, if, e.g., the simple nodes are controlled by Dirichlet controls in H^l (0, T ). Assume that we want to calculate the corresponding norm-minimal controls using semi-discretization in space. We then obtain a system of coupled second-order-in-time ordinary differential equations with three control inputs. Controllability of the latter system can easily been checked by Kalman's rank condition on each space discretization level h. One expects, as h tends to zero, that the exact controllability of the continuous system is revealed. This expectation is frustrated, as has been shown by Infante and Zuazua (1998) for a single string and by Zuazua (1999) for a membrane. Indeed, it was shown there that uniformity of observability estimates is lost in the limit. On the other hand, spectral filtering allows to cure this pathology. We show in this paper that similar results hold for our string network. The generalization to arbitrary networks of strings in the out-of-the-plane as well as in the in-plane or 3-d-setup is then a technical matter. Therefore, this paper essentially extends the existing results to semidiscretizations of wave equations on arbitrary irregular computational grids

On Boundary Null Controllability of Strongly Degenerate Hyperbolic Systems on Star-Shaped Planar Network

In this paper we discuss the problem of boundary exact null controllability for weakly and strongly degenerate linear wave equation defined on star-shaped planar network. The network is represented by a singular measure in a bounded planar domain. The novelty of this article lies in the degeneration of the leading coefficient representative of the material properties at the common node of network. We discuss the existence of weak and strong solutions to the degenerate hyperbolic problem and establish the corresponding controllability properties.In this paper we discuss the problem of boundary exact null controllability for weakly and strongly degenerate linear wave equation defined on star-shaped planar network. The network is represented by a singular measure in a bounded planar domain. The novelty of this article lies in the degeneration of the leading coefficient representative of the material properties at the common node of network. We discuss the existence of weak and strong solutions to the degenerate hyperbolic problem and establish the corresponding controllability properties.

Topological sensitivity analysis for elliptic problems on graphs

We consider elliptic problems on graphs under given loads and bilateral contact conditions. We ask the question: which graph is best suited to sustain the loads and the constraints. More precisely, given a cost function we may look at a multiple node of the graph with edge degree q and ask as to whether that node should be resolved into a number of nodes of edge degree less than q, in order to decrease the cost. With this question in mind, we are looking into the sensitivity analysis of a graph carrying a second order elliptic equation with respect to changing its topology by releasing nodes with high edge degree or including an edge. With the machinery at hand developed here, we are in the position to define the topological gradient of an elliptic problem on a graph