Günter Leugering

In this article, we study optimal control problems driven by space-time fractional parabolic equations (STFPEs) of Sturm-Liouville type on metric graphs, considering both distributed and boundary controls. The considered problem involves the fractional time derivative in the Caputo sense, and the spatial fractional derivative is of the Sturm-Liouville type which is the composition of the right-sided Caputo fractional derivative and left-sided Riemann-Liouville fractional derivative. By introducing the appropriate function spaces for the involved fractional operators in both the time and spatial variables, we first establish well-posedness results for the weak solution for the governing STFPEs using the Galerkin approximation method. Then, we prove that the associated optimal control problem possesses a unique optimal solution. Moreover, to characterize the control variables, we develop an adjoint problem involving a right-sided Caputo fractional derivative in time and derive optimality conditions for the considered problem, using the Lagrange multiplier method. Finally, we propose a difference scheme to find the numerical solution of the considered optimal control problem, particularly on a metric star graph by approximating the Caputo time derivative using the L1 method and spatial fractional derivative with the Grünwald-Letnikov formula. The performance and the accuracy of the proposed difference scheme have been illustrated via an example.

We consider networks of elastic strings with end masses, where the coupling is modeled via elastic springs. The model is representative of a network of nonlinear strings, where the strings are coupled to elastic bodies. The coupled system converges to the classical string network model with Kirchhoff and continuity transmission conditions as the spring stiffness terms approach infinity and the masses at the nodes vanish. Due to the presence of point masses at the nodes, the boundary conditions become dynamic, and consequently, the corresponding first-order system of quasilinear balance laws exhibits nonlocal boundary conditions. We demonstrate well-posedness in the sense of semi-global classical solutions \cite{li} (i.e., for arbitrarily large time intervals provided that the initial and boundary data are small enough) and observe extra regularity at the masses as in \cite{WangLeugeringLi2017,WangLeugeringLi2019}. We prove local and global-local exact boundary controllability of a star-like network when control is active at the endpoints of the string-spring-mass system, except for one clamped end. In this case, at multiple nodes, a complex smoothing pattern appears, leading to asymmetric control spaces when the springs and masses are present. Furthermore, the rank of the Laplacian matrix at the junction is crucial for the controllability property, particularly in models containing wave equations with degeneration at dynamic boundaries, which can be interpreted as damage in mechanical vibration systems where parts of the springs are missing.

We consider the in-plane motion of elastic strings on tree-like network, observed from the 'leaves'. We investigate the inverse problem of recovering not only the physical properties i.e. the 'optical lengths' of each string, but also the topology of the tree which is represented by the edge degrees and the angles between branching edges. To this end use the boundary control method for wave equations established in~\cite{AK,B}. It is shown that under generic assumptions the inverse problem can be solved by applying measurements at all leaves, the root of the tree being fixed.

We consider a non-overlapping domain decomposition method for optimal control problems of the tracking type governed by time-fractional diffusion equations in one space dimension, where the fractional time derivative is considered in the Caputo sense. We concentrate on a transmission problem defined on two adjacent intervals, where at the interface we introduce an iterative non-overlapping domain decomposition in the spirit of P.L. Lions for the corresponding first-order optimality system, such that the optimality system corresponding to the optimal control problem on the entire domain is iteratively decomposed into two systems on the respective sub-domains; this approach can be framed as first optimize, then decompose. We show that the iteration involving the states and adjoint states converges in the appropriate spaces. Moreover, we show that the decomposed systems on the sub-domain can in turn be interpreted as optimality systems of so-called virtual control problems on the sub-domains. Using this property, we are able to solve the original optimal control problem by an iterative solution of optimal control problems on the sub-domains. This approach can be framed as first decompose, then optimize. We provide a mathematical analysis of the problems as well as a numerical finite difference discretization using the L1-method with respect to the Caputo derivative, along with two examples in order to verify the method.

Exact controllability for the wave equation on a metric graph consisting of a cycle and two attached edges is proven. One boundary and one internal control are used. At the internal vertices, delta-prime conditions are satisfied. As a second example, we examine a tripod controlled at the root and the junction, while the leaves are fixed. These examples are key to understanding controllability properties in general metric graphs.

In the present paper we deal with parabolic fractional initial-boundary value problems of Sturm–Liouville type in an interval and in a general star graph. We first give several existence, uniqueness and regularity results of weak and very-weak solutions. We prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal contol via the Euler–Lagrange first order optimality conditions. We then investigate the analogous problems for a fractional Sturm–Liouville problem in a general star graph with mixed Dirichlet and Neumann boundary controls. The existence and uniqueness of minimizers, and the characterization of the first order optimality conditions are obtained in a general star graph by using the method of Lagrange multipliers.