تقرير
On damping a control system with global aftereffect on quantum graphs. Stochastic interpretation
| العنوان: | On damping a control system with global aftereffect on quantum graphs. Stochastic interpretation |
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| المؤلفون: | Buterin, Sergey |
| سنة النشر: | 2023 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Optimization and Control, 34K35 34K10 93C23 49J55 49K45 93E20 |
| الوصف: | 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. In particular, the interval case when the coefficients of the equation are assumed to be discrete-time stochastic processes can be viewed as the extension to a tree. Comment: 24 pages, 5 figures |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2308.00496 |
| رقم الأكسشن: | edsarx.2308.00496 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |