On the Convergence of the Approximate Solution to the Optimization Problem for Oscillatory Processes
DOI:
https://doi.org/10.31489/2025m3/22-33Keywords:
optimal control, optimal process, minimal value of functional, non-linear optimization problem, approximations of complete solution, resolvent approximation, finite-dimensional approximation, convergenceAbstract
This article addresses the non-linear optimization problem of oscillatory processes governed by partial integro-differential equations involving a Fredholm integral operator. A distinctive feature of the problem is that both the objective functional and the functions describing external and boundary influences are non-linear with respect to the vector controls. The integro-differential equation describing the state of the oscillatory process includes Fredholm integral operator, which has a significant impact on the structure and properties of the solutions. The algorithm for constructing the complete solution to this problem, as well as the effect of the Fredholm integral operator on the solution of the corresponding boundary value problem, has been published in previous studies. This article is dedicated to the investigation of the convergence of approximate solutions to the exact solution of the considered non-linear optimization problem. The influence of the Fredholm integral operator on the convergence behavior of the approximations is examined. It is demonstrated that the presence of the integral operator necessitates the construction of three distinct types of approximations of the optimal process: “Resolvent” approximations, based on the resolvent of the kernel of the integral operator; Approximations by optimal controls, constructed through the approximation of control functions; Finite-dimensional approximations.
References
Butkovskii, A.G. (1965). Teoriia optimalnogo upravleniia sistemami s raspredelennymi parametrami [Theory of optimal control of systems with distributed parameters]. Moscow: Nauka [in Russian].
Egorov, A.I. (1978). Optimalnoe upravlenie teplovymi i diffuzionnymi protsessami [Optimal control of thermal and diffusion processes]. Moscow: Nauka [in Russian].
Sirazetdinov, T.K. (1987). Ustoichivost sistemy s raspredelennymi parametrami [Stability of systems with distributed parameters]. Novosibirsk: Nauka [in Russian].
Li, J., Xu, Z., Zhu, D., Dong, K., Yan, T., Zeng, Z., & Yang, S.X. (2021). Bio-inspired intelligence with applications to robotics: a survey. Intelligence and Robotics, 1(1), 58–83. https://doi.org/10.20517/ir.2021.08
Saveriano, M., Abu-Dakka, F.J., Kramberger, A., & Peternel, L. (2023). Dynamic movement primitives in robotics: A tutorial survey. The International Journal of Robotics Research, 42(13), 1133–1184. https://doi.org/10.1177/02783649231201196
Plotnikov, V.I. (1968). An energy inequality and the overdeterminacy property of a system of eigenfunctions. Mathematics of the USSR-Izvestiya, 2(4), 695–707. https://doi.org/10.1070/IM1968v002n04ABEH000656
Raskin, L., & Sira, O. (2019). Construction of the fractional-non-linear optimization method. Eastern-European Journal of Enterprise Technologies, 4(4(100)), 37–43. https://doi.org/10.15587/1729-4061.2019.174079
Krasnichenko, L.S. (2012). Reshenie zadachi nelineinoi optimizatsii teplovykh protsessov pri granichnom upravlenii [Solution of the problem of nonlinear optimization of thermal processes under boundary control. Candidate’s thesis. Bishkek [in Russian].
Asanova, Zh.K. (2012). Tochechnoe podvizhnoe nelineinoe optimalnoe upravlenie protsessom teploperedachi [Pointwise moving nonlinear optimal control of heat transfer process]. Candidate’s thesis. Bishkek [in Russian].
Ermekbaeva, A. (2022). Optimalnoe tochnoe upravlenie teplovymi protsessami, opisyvaemymi fredgolmovo integro-differentsialnymi uravneniiami [Optimal exact control of thermal processes described by Fredholm integro-differential equations]. Candidate’s thesis. Bishkek [in Russian].
Baetov, A.K. (2014). Priblizhennye resheniia zadach nelineinoi optimizatsii kolebatelnykh protsessov [Approximate solutions of problems of nonlinear optimization of oscillatory processes]. Candidate’s thesis. Bishkek [in Russian].
Uryvskaya, T.Yu. (2010). Priblizhennoe reshenie zadachi nelineinoi optimizatsii teplovykh protsessov [Approximate solution of the problem of nonlinear optimization of thermal processes]. Candidate’s thesis. Bishkek [in Russian].
Kerimbekov, A., & Abdyldaeva, E. (2016). On the Solvability of a Nonlinear Tracking Problem Under Boundary Control for the Elastic Oscillations Described by Fredholm Integro-Differential Equations. In: Bociu, L., Desideri, JA., Habbal, A. (Eds.). System Modeling and Optimization. 27th IFIP TC 7 Conference. CSMO 2015, 312–321. Springer. https://doi.org/10.1007/978-3-319-55795-3_29
Abdyldaeva, E.F., & Kerimbekov, A. (2023). On features of the nonlinear optimization problem of oscillation processes under distributed and boundary vector controls. Journal of the Institute of Automation and Information Technologies of the National Academy of Sciences of the Kyrgyz Republic: Problems of Automation and Control, 14–25.
