Annotation of the results obtained in 2023
The theory is developed and effective numerical methods for solving crisp and fuzzy Volterra integral equations with piecewise continuous kernels are constructed. The existence and uniqueness theorems of solutions of equations of such new classes are proved. A theoretical and empirical study of convergence of numerical methods has been carried out. Namely, the sync-collocation method for the numerical solution of fuzzy Volterra integral equations with piecewise continuous kernels is theoretically justified and implemented. To evaluate and improve the accuracy of the numerical solution of fuzzy Volterra integral equations with piecewise continuous kernels, a stochastic arithmetic methodology based on the Controle et Estimation Stochastique des Arrondis de Calculs (CESTAC) method and the Control of Accuracy and Debugging for Numerical Applications (CADNA) library is employed. An estimate of the accuracy of the approximate solution is obtained, with a stopping rule using the Hausdorff criterion. Thus, the method takes into account both the uncertainty of the initial data setting and rounding errors. The issues of construction and justification of stable numerical methods for solving crisp linear and nonlinear Volterra integral equations of the second kind with a piecewise continuous kernel are investigated. A new iterative method is developed to solve this problem. The convergence theorem is proved and error estimates of the method are obtained. A stable numerical method for solving linear and nonlinear Volterra integral equations of the second kind with a piecewise continuous kernel is developed, which allows us to achieve the optimal results. Namely, to obtain the minimum error achieved at the optimal iteration of the method and the optimal approximation. The main theorem of the CESTAC method is proved: the equality of the number of common significant digits (NCSD) of two consecutive iterations with exact and approximate solutions is established. Thus, to show the accuracy and efficiency of the method, a new stopping rule was involved. An integral dynamic fuzzy linear Volterra model of the first kind was developed and validated on real data to determine the mode parameters of energy storage devices (charge level and alternating power function of storage devices). In the field of development of theory and methods of fuzzy control of nonlinear dynamic systems with delays, Takagi-Sugeno fuzzy models capable of approximating nonlinear systems with a given accuracy by means of accessory functions have been investigated. This allows the application of classical methods of control techniques for linear systems and Lyapunov-Krasovskii functional theory for fuzzy dynamic systems.

 

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Annotation of the results obtained in 2022
1) A theory and efficient numerical methods for solving Volterra fuzzy integral equations with piecewise continuous kernels have been developed. Existence and uniqueness theorems for solutions of such a class of fuzzy integral equations are proved. The convergence of the method of successive approximations is established and an estimate of the error of the method is obtained. An efficient stable numerical method for solving such fuzzy linear Volterra integral equations based on sinc-collocation with double exponential and single exponential decay has been developed. 2) To solve crisp linear Volterra integral equations of the first kind with piecewise continuous kernels, the polynomial spline collocation method is proposed. At the same time, to approximate the integrals during discretization, the proposed projection method uses the Gaussian quadrature formula. An estimate for the accuracy of the approximate solution is obtained. Stochastic arithmetic is used based on the CESTAC method (Contrôle et Estimation STochastique des Arrondis de Calculs) and the CADNA library (Control of Accuracy and Debugging for Numerical Applications). Using this approach, one can find the optimal parameters of the proposed projection method. 3) A new inverse nonlinear problem is posed and solved for the crisp Volterra integral equations with piecewise continuous kernels. For such Volterra integral equations of the first kind, it is assumed that the desired ones are the kernel's discontinuity curves, but the rest of the information is known. The resulting integral equation is nonlinear with respect to discontinuity curves corresponding to the integration limits. For its numerical solution, a direct discretization method with a posteriori verification of calculations has been developed and tested. 4) Questions of stability and stabilization of a class of fuzzy systems of the Takagi-Sugeno type with a time-varying delay are studied. The corresponding Lyapunov-Krasovskii functional containing special integral terms is constructed. Based on the methodology of generalized free-matrix-based integral inequality, a criterion for ensuring the stability of such fuzzy systems is formulated. The validity and feasibility of the stability criterion, as well as the method of designing fuzzy controllers, are demonstrated using the model examples. 5) An efficient strategy for charging and discharging storages based on data is proposed, taking into account wind energy forecasting intervals. Two concrete scientific results can be here distinguished. First, a power interval prediction model implemented on a long-term memory network with lower and upper bound estimation (LUBE) to quantify wind energy uncertainty. Secondly, it is an energy storage management method implemented as a Markov decision process and solved using a deep reinforcement learning method. The state space, action space, and agent-environment interaction reward function are used, and the value function is approximated via a deep Q-network. Then, according to the real-time state (such as wind power, power prediction intervals, local load, dynamic electricity price, and state of charge), the proposed approach enables an automatic charge/discharge scheduling.

 

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