arXiv subjects

Özkan Karabacak

Publications and source records attributed to Özkan Karabacak.

12 recordsLinked to original sources

arXiv2026

Construction of Lyapunov density for nonautonomous dynamical systems on hypertorus

We present a semidefinite programming framework for constructing time-varying Lyapunov densities for nonautonomous dynamical systems on a hypertorus. The formulation leverages Gram matrix representations of hybrid (real-trigonometric) polynomials. In addition, we introduce a novel block decomposition of these Gram representations to confine the blow-up of the resulting density to a prescribed set. The results are then applied to establish the almost global synchronization of a time-varying Kuramoto model and the robust almost-global stability of a parameter-varying nonautonomous system. These examples demonstrate the applicability of the proposed method and validate the theoretical results. All computational results are obtained using an open-source MATLAB implementation, as referenced in the text, thereby facilitating reproducibility of the reported examples.

math.DS↗

arXiv2026

Dual Lyapunov-based Synchronization Control of Rössler System

This paper proposes a novel approach for the synchronization problem of nonlinear dynamical systems, integrating dual Lyapunov stability analysis with polynomial optimization. A comprehensive review of the relevant scientific literature on synchronization methods is conducted, with a particular focus on classical Lyapunov-based methods for chaotic systems. In this study, the Rössler system is synchronized by employing dual Lyapunov-based closed-loop synchronization method. This method uses semidefinite programming and sum-of-squares polynomials to compute a nonlinear state feedback function which synchronize a chaotic system to a selected reference model. It is aimed that chaotic behavior is destroyed and, instead, a limit cycle becomes attracting. Simulation works are performed for randomly selected 100 different initial conditions to show that synchronization process is successfully performed. Furthermore, bifurcation diagrams and phase portraits are evaluated to analyze the system dynamics. The paper discusses results and how new constraints should be employed and adapted to more complex systems.

math.DS↗

arXiv2026

Semidefinite Programming Certificates for Synchronization of Kuramoto Oscillators on Arcs

A class of Kuramoto models with a general coupling function that can be expressed in terms of a finite number of harmonics, each comprising sinusoidal terms, is studied. We propose a novel approach for certifying local phase synchronization in this class for all initial conditions lying on an arc. The trace parametrization property and Gram matrix representation of a trigonometric polynomial are utilized along with Putinar's Positivstellensatz to obtain semidefinite programming certificates for the stability of the phase-difference system, which in turn implies synchronization of the original system. The results can be extended to any system of coupled oscillators where the forward-invariance on arcs can be established.

math.DS↗

arXiv2026

Feedback-Based Quantum Algorithm for Excited States Calculation

Recently, feedback-based quantum algorithms have been introduced to calculate the ground states of Hamiltonians, inspired by quantum Lyapunov control theory. This paper aims to generalize these algorithms to the problem of calculating an eigenstate of a given Hamiltonian, assuming that the lower energy eigenstates are known. To this aim, we propose a new design methodology that combines the layer-wise construction of the quantum circuit in feedback-based quantum algorithms with a new feedback law based on a new Lyapunov function to assign the quantum circuit parameters. We present two approaches for evaluating the circuit parameters: one based on the expectation and overlap estimation of the terms in the feedback law and another based on the gradient of the Lyapunov function. We demonstrate the algorithm through an illustrative example and through an application in quantum chemistry. To assess its performance, we conduct numerical simulations and execution on IBM's superconducting quantum computer.

quant-ph↗

arXiv2025

Feedback-Based Quantum Strategies for Constrained Combinatorial Optimization Problems

Feedback-based quantum algorithms have recently emerged as potential methods for approximating the ground states of Hamiltonians. One such algorithm, the feedback-based algorithm for quantum optimization (FALQON), is specifically designed to solve quadratic unconstrained binary optimization problems. Its extension, the feedback-based algorithm for quantum optimization with constraints (FALQON-C), was introduced to handle constrained optimization problems with equality and inequality constraints. In this work, we extend the feedback-based quantum algorithms framework to address a broader class of constraints known as invalid configuration (IC) constraints, which explicitly prohibit specific configurations of decision variables. We first present a transformation technique that converts the constrained optimization problem with invalid configuration constraints into an equivalent unconstrained problem by incorporating a penalizing term into the cost function. Then, leaning upon control theory, we propose an alternative method tailored for feedback-based quantum algorithms that directly tackles IC constraints without requiring slack variables. Our approach introduces a new operator that encodes the optimal feasible solution of the constrained optimization problem as its ground state. Then, a controlled quantum system based on the Lyapunov control technique is designed to ensure convergence to the ground state of this operator. Two approaches are introduced in the design of this operator to address IC constraints: the folded spectrum approach and the deflation approach. These methods eliminate the need for slack variables, significantly reducing the quantum circuit depth and the number of qubits required. We show the effectiveness of our proposed algorithms through numerical simulations.

quant-ph↗

arXiv2024

Weighted Feedback-Based Quantum Algorithm for Excited States Calculation

Drawing inspiration from the Lyapunov control technique for quantum systems, feedback-based quantum algorithms have been proposed for calculating the ground states of Hamiltonians. In this work, we consider extending these algorithms to tackle calculating excited states. Inspired by the weighted subspace-search variational quantum eigensolver algorithm, we propose a novel weighted feedback-based quantum algorithm for excited state calculation. We show that depending on how we design the weights and the feedback law, we can prepare the $p$th excited state or lowest energy states up to the $p$th excited state. Through an application in quantum chemistry, we show the effectiveness of the proposed algorithm, evaluating its efficacy via numerical simulations.

quant-ph↗

arXiv2024

Feedback-Based Quantum Algorithm for Constrained Optimization Problems

The feedback-based algorithm for quantum optimization (FALQON) has recently been proposed to solve quadratic unconstrained binary optimization problems. This paper efficiently generalizes FALQON to tackle quadratic constrained binary optimization (QCBO) problems. For this purpose, we introduce a new operator that encodes the problem's solution as its ground state. Using Lyapunov control theory, we design a quantum control system such that the state converges to the ground state of this operator. When applied to the QCBO problem, we show that our proposed algorithm saves computational resources by reducing the depth of the quantum circuit and can perform better than FALQON. The effectiveness of our proposed algorithm is further illustrated through numerical simulations.

quant-ph↗

arXiv2024

Lyapunov Densities For Markov Processes: An Application To Quantum Systems With Non-Demolition Measurements

Stochastic convergence of discrete time Markov processes has been analysed based on a dual Lyapunov approach. Using some existing results on ergodic theory of Markov processes, it has been shown that existence of a properly subinvariant function (counterpart of the Lyapunov density in deterministic systems) implies sweeping of a Markov process out of the sets where this function is integrable. Such a function can be used as a certificate of convergence in probability of a stochastic system. We apply this technique to Markov processes induced by a quantum system with non-demolition measurement and propose dual Lyapunov certificates to certify sweeping.

math.DS↗

arXiv2023

Measurement-Based Control for Minimizing Energy Functions in Quantum Systems

In variational quantum algorithms (VQAs), the most common objective is to find the minimum energy eigenstate of a given energy Hamiltonian. In this paper, we consider the general problem of finding a sufficient control Hamiltonian structure that, under a given feedback control law, ensures convergence to the minimum energy eigenstate of a given energy function. By including quantum non-demolition (QND) measurements in the loop, convergence to a pure state can be ensured from an arbitrary mixed initial state. Based on existing results on strict control Lyapunov functions, we formulate a semidefinite optimization problem, whose solution defines a non-unique control Hamiltonian, which is sufficient to ensure almost sure convergence to the minimum energy eigenstate under the given feedback law and the action of QND measurements. A numerical example is provided to showcase the proposed methodology.

quant-ph↗

arXiv2017

Graph-Based Minimum Dwell Time and Average Dwell Time Computations for Discrete-Time Switched Linear Systems

Discrete-time switched linear systems where switchings are governed by a digraph are considered. The minimum (or average) dwell time that guarantees the asymptotic stability can be computed by calculating the maximum cycle ratio (or maximum cycle mean) of a doubly weighted digraph where weights depend on the eigenvalues and eigenvectors of subsystem matrices. The graph-based method is applied to systems with defective subsystem matrices using Jordan decomposition. In the case of bimodal switched systems scaling algorithms that minimizes the condition number can be used to give a better minimum (or average) dwell time estimates.

eess.SY↗

arXiv2015

A computational model revealing the effect of dopamine on action selection

In order to reveal the effect of nigrostriatal dopamine system on action selection, first a computational model of the cortex-basal ganglia-thalamus loop is proposed and based on this model a simple compound model realizing the Stroop effect is established. Even though Stroop task is mostly used to examine selective attention, the main objective of this work is to investigate the effect of action selection on Stroop task. The computational model of the cortex-basal ganglia-thalamus loop is a non-linear dynamical system which is not only capable of revealing the action selection property of basal ganglia but also capable of modelling the effect of dopamine on action selection. While the interpretation of action selection is based on the solutions of the non-linear dynamical system, the effect of dopamine is modelled by a parameter of the model. The inhibiting effect of dopamine on the habitual behaviour which corresponds to word reading in Stroop task and letting the novel one occur corresponding to colour naming is investigated using the compound computational model established in this work.

q-bio.NC↗

arXiv2015

Uni-directional synchronization and frequency difference locking induced by a heteroclinic ratchet

A system of four coupled oscillators that exhibits unusual synchronization phenomena has been ana- lyzed. Existence of a one-way heteroclinic network, called heteroclinic ratchet, gives rise to uni-directional (de)synchronization between certain groups of cells. Moreover, we show that locking in frequency differ- ences occur when a small white noise is added to the dynamics of oscillators.

nlin.AO↗