Günter Leugering

Domain Decomposition of Optimal Control Problems for Dynamic Networks of Elastic Strings

We consider optimal control problems related to exact- and approximate controllability of dynamic networks of elastic strings. In this note we concentrate on problems with linear dynamics, no state and no control constraints. The emphasis is on approximating target states and velocities in part of the network using a dynamic domain decomposition method (d3m) for the optimality system on the network. The decomposition is established via a Uzawa-type saddle-point iteration associated with an augmented Lagrangian relaxation of the transmission conditions at multiple joints. We consider various cost functions and prove convergence of the infinite dimensional scheme for an exemplaric choice of the cost. We also give numerical evidence in the case of simple exemplaric networks.

On the semi-discretization of optimal control problems for networks of elastic strings: global optimality systems and domain decomposition

We consider a star-graph as an examplary network, with elastic strings stretched along the edges. The network is allowed to perform out-of-the plane displacements. We consider such networks as being controlled at its simple nodes via Dirichlet conditions. The objective is to steer given initial data to final target data in a given time T with minimal control costs. This problem is discussed in the continuous as well as in the discrete case. We discuss an iterative domain decomposition technique and its discrete analogue. We prove convergence and show some numerical results.

ON FEEDBACK CONTROLS FOR DYNAMIC NETWORKS OF STRINGS AND BEAMS AND THEIR NUMERICAL SIMULATION

In these notes we want to present some control strategIes for dynamIc nctworks of strmgs and beams m those SItuatIOns where the classIcal concepts of exact or approxImate controllabIlIty faIl ThIS IS, m partIcular, the case for networks contammg CIrCUIts GenerIcally, a reSIdual motIOn settles m such CIrCUIts, even If all nodes are subject to controls In those SItuatIOns we resort to controls whIch dIrect the flux of energy USIng such controls m a network, we are able to steer the entIre energy to preassIgned parts of the structure In practIce these parts are more massIve and can absorb energy more eaSIly than the fragIle elements We also provIde numerIcal eVIdence for the control strategIes dIscussed m these notes The materIal IS related to JOInt work WIth J E Lagnese and E G P G SchmIdt, and IS essentIally mcluded (14) 1991 Mathemat%cs Subject Class%ficahon 39C20, 39B52, 65M06, 65M60

Modeling and Controllability of Interconnected Membranes

We shall consider in this Chapter the motion of an elastic body consisting of several interconnected elastic membranes. Our goals are to obtain the dynamic equations of motion of such a configuration, especially to elucidate the geometric and dynamic conditions which must hold in junction regions where two or more such elastic elements are joined together, and to study the controllability properites of the resulting (linearized) system of equations. The modeling proceeds through two stages. In Section 1 the equations of motion and boundary conditions of a general nonlinear membrane and the corresponding Hamilton’s principle will be derived, assuming a linear stress-strain relation. The dynamic equations of a system of interconnected membranes are derived in Section 2 by minimizing an appropriate Lagrangian over the class of deformations which satisfies certain geometric constraints. In particular, the geometric constraint imposed in the junction regions is the very natural one that the initially connected system of membranes remains connected throughout the deformation process. When this constraint is imposed on the admissible deformations, the optimal motion is forced to satisfy a second, dynamic constaint in the junction regions that has the interpretation of a balance of forces law there. Both the equations of motion and the dynamic junction conditions involve nonlinear couplings among the components of the displacement vector. When these are linearized around the trivial equilibrium the familiar equations of motion of a single elastic membrane, in which in-plane displacements are not coupled to transverse motion, axe obtained for the displacement of each individual element. However, all displacements remain coupled through the geometric and (linearized) dynamic junction conditions. The question of exact controllability of this linearized coupled system is then considered in Section 3.

Modeling of Networks of Thermoelastic Beams

We consider the deformation of a thin beam of length l with a given initial curvature and torsion, and with variable doubly symmetric cross section. To be precise, the undeformed beam, in its initial reference configuration, occupies the region

Modeling and Controllability of Systems of Linked Plates

\Omega :\left\{ {r: = {{r}_{0}}\left( {{{x}_{1}}} \right) + {{x}_{2}}{{e}_{2}}\left( {{{x}_{1}}} \right) + {{x}_{3}}\left( {{{x}_{1}}} \right)\left. {{{e}_{3}}} \right|{\text{ }}{{x}_{1}} \in \left[ {0,\ell } \right],\left( {{{x}_{2}},{{x}_{3}}} \right): = {{x}_{2}}{{e}_{2}}\left( {{{x}_{1}}} \right) + {{x}_{3}}{{e}_{3}}\left( {{{x}_{1}}} \right) \in A\left( {{{x}_{1}}} \right)} \right\},

Modeling and Controllability of Coupled Plate-Beam Systems

where r0: [0,l] ↦ ℝ3 is a smooth function representing the centerline of the beam at rest, and the orthonormal triads e1(•), e2(•), e3(•) are chosen to be smooth functions of x1 such that e1 is the direction of the tangent of the centerline with respect to the variable x1, i.e. \(\frac{d}{{d{{x}_{1}}}}{{r}_{0}}\left( {{{x}_{1}}} \right) = {{e}_{1}}\left( {{{x}_{1}}} \right)\), and such that e2(x1), e3(x1) span the orthogonal cross section at x1. The meaning of the variables x i are as follows: x1 denotes the length along the undeformed centerline, x2 and x3 denote the lengths along lines orthogonal to the reference line. The set Ω can then be viewed as obtained by translating the reference curve r0(x1) to the position x2e2 + x3e3 within the cross section vertical to the tangent of r0. The cross section at x1 is defined as