On damping a control system with global aftereffect on quantum graphs. Stochastic interpretation

TL;DR

This paper models quantum graphs as temporal networks with global delay, extending control system damping with aftereffect from intervals to complex tree graphs, and establishes a variational framework with proven unique solutions.

Contribution

It introduces a novel approach to quantum graphs as temporal networks with global delay, extending damping control problems to tree structures and formulating a variational problem with proven solvability.

Findings

Extended damping control with aftereffect to tree graphs.

Established equivalence to a self-adjoint boundary value problem.

Proved unique solvability of the formulated problems.

Abstract

Quantum graphs model processes in complex systems represented as spatial networks in various fields of natural science and technology. An example is the oscillations of elastic string networks, the nodes of which, besides the continuity conditions, also obey the Kirchhoff conditions, expressing the balance of tensions. In this paper, we propose a new look at quantum graphs as {\it temporal} networks, which means that the variable parametrizing the edges of a graph is interpreted as time, while each internal vertex is a branching point giving several different scenarios for the further trajectory of a process. Then Kirchhoff-type conditions may also arise. Namely, they will be satisfied by such a trajectory of the process that is optimal with account of all the scenarios simultaneously. By employing the recent concept of global delay, we extend the problem of damping a first-order control system with aftereffect, considered earlier only on an interval, to an arbitrary tree graph. The first means that the delay, imposed starting from the initial moment of time, associated with the root of the tree, propagates through all internal vertices. Bringing the system into the equilibrium and minimizing the energy functional with account of the anticipated probability of each scenario, we come to a variational problem. Then, we establish its equivalence to a self-adjoint boundary value problem on the tree for some second-order equations involving both the global delay and the global advance. The unique solvability of both problems is proved. We also illustrate that the interval case when the coefficients of the equation are discrete stochastic processes in discrete time can be viewed as the extension to a tree.