Method for finding a solution to a linear nonstationary interval ODE system

A. V. Fominyh

Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 17 (2021) no. 2, pp. 148-165

Résumé

The article analyses a linear nonstationary interval system of ordinary differential equations so that the elements of the matrix of the system are the intervals with the known lower and upper bounds. The system is defined on the known finite time interval. It is required to construct a trajectory, which brings this system from the given initial position to the given final state. The original problem is reduced to finding a solution of the differential inclusion of a special form with the fixed right endpoint. With the help of support functions, this problem is reduced to minimizing a functional in the space of piecewise continuous functions. Under a natural additional assumption, this functional is Gateaux differentiable. For the functional, Gateaux gradient is found, necessary and sufficient conditions for the minimum are obtained. On the basis of these conditions, the method of the steepest descent is applied to the original problem. Some numerical examples illustrate the constructed algorithm realization.

Détail

Keywords: linear nonstationary interval system of ordinary differential equations, differential inclusion, support function, the steepest descent method.

Bibliographie

[1] Durieu C., Walter E., Polyak B., “Multi-input multi-output ellipsoidal state bounding”, Journal of Optimization Theory and Applications, 111:2 (2001), 273–303 | DOI | MR | Zbl

[2] Harrison G. W., “Dynamic models with uncertain parameters”, Proceedings of the First International Conference on Mathematical Modeling, v. 1, ed. X. J. R. Avula, University of Missouri Publ, Rolla, 1977, 295–304 | MR

[3] Maksarov D. G., Norton J. P., “Computationally efficient algorithms for state estimation with ellipsoidal approximation”, International Journal of Adaptive Control and Signal Processing, 16:6 (2002), 411–434 | DOI | Zbl

[4] Jaulin L., “Nonlinear bounded-error state estimation of continuous-time systems”, Automatica, 38:6 (2002), 1079–1082 | DOI | MR | Zbl

[5] Scott J. K., Barton P. I., “Bounds on the reachable sets of nonlinear control systems”, Automatica, 49 (2013), 93–100 | DOI | MR | Zbl

[6] Singer A. B., Barton P. I., “Bounding the solutions of parameter dependent nonlinear ordinary differential equations”, SIAM J. Sci. Comput., 27:6 (2006), 2167–2182 | DOI | MR | Zbl

[7] Kieffer M., Walter E., Simeonov I., “Guaranteed nonlinear parameter estimation for continuous-time dynamical models”, Proceedings of 14th IFAC Symposium on System Identification, 2006, 843–848

[8] Dubois D., Prade H., “Towards fuzzy differential calculus. Pt 3. Differentiation”, Fuzzy Sets and Systems, 1982, no. 8, 225–233 | DOI | MR | Zbl

[9] Chang S. L., Zadeh L. A., “On fuzzy mapping and control”, IEEE Trans. Systems Man Cybernet, 2 (1972), 30–34 | DOI | MR | Zbl

[10] Goetschel R., Voxman W., “Elementary fuzzy calculus”, Fuzzy Sets and Systems, 24 (1987), 31–43 | MR

[11] Kandel A., Byatt W. J., “Fuzzy differential equations”, Proceedings of lnternational Conference on Cybernetics and Society, 1978, 1213–1216 | MR

[12] Kaleva O., “The Cauchy problem for fuzzy differential equations”, Fuzzy Sets and Systems, 35 (1990), 389–396 | DOI | MR | Zbl

[13] Stefanini L., Bede B., “Generalized Hukuhara differentiability of interval-valued functions and interval differential equations”, Nonlinear Analysis, 71 (2009), 1311–1328 | DOI | MR | Zbl

[14] Ma M., Friedman M., Kandel A., “Numerical solutions of fuzzy differential equations”, Fuzzy Sets and Systems, 105 (1990), 133–138 | MR

[15] Pederson S., Sambandham M., “Numerical solution to hybrid fuzzy systems”, Mathematical and Computer Modelling, 45 (2007), 113–144 | DOI | MR

[16] Jayakumar T., Maheskumar D., Kanagarajan K., “Numerical solution of fuzzy differential equations by Runge — Kutta method of order five”, Applied Mathematical Sciences, 6 (2012), 2989–3002 | MR | Zbl

[17] Bandyopadhyay A., Kar K., “Fuzzy continuous dynamical system: a multivariate optimization technique”, Annual meeting of the North American Fuzzy Information Processing Society (Berkeley, CA, USA, 2007), 1–7 | MR

[18] Abbasbandy S., Viranloo T. A., “Numerical solutions of fuzzy differential equations by Taylor method”, Computational Methods in Applied Mathematics, 2:2 (2002), 113–124 | DOI | MR

[19] Morozov A. Yu., Reviznikov D. L., “Modification of methods for solving the Cauchy problem for systems of ordinary differential equations with interval parameters”, Proceedings of MAI Institute, 89 (2016), 1–17 (In Russian)

[20] Hashemi M. S., Malekinagad J., Marasi H. R., “Series solution of the system of fuzzy differential equations”, Advances in Fuzzy Systems, 2012, 1–16 | DOI | MR

[21] Liao S., “Homotopy analysis method: a new analytical technique for nonlinear problems”, Communications in Nonlinear Science and Numerical Simulation, 1997, no. 2, 95–100 | DOI | MR

[22] Mosleh M., Otadi M., “Minimal solution of fuzzy linear system of differential equations”, Neural Computing $\$ Applications, 21 (2012), 329–336 | DOI | MR

[23] Wu J., Zhang Yu., Chen L., Luo Zh., “A Chebyshev interval method for nonlinear dynamic systems under uncertainty”, Applied Mathematical Modelling, 37 (2013), 4578–4591 | DOI | MR | Zbl

[24] Chen M., Wu C., Xue X., Liu G., “On fuzzy boundary value problems under uncertainty”, Information Sciences, 178 (2008), 1877–1892 | DOI | MR | Zbl

[25] Sokolova S. P., Kuzmina E. A., “Dynamic properties of interval systems”, SPIIRAS Proceedings, 7 (2008), 215–221 (In Russian) | MR

[26] Rogoza A. A., “On an approach to constructing two-sided estimates of solution sets of non-linear differential equations with interval parameters on the basis of projective methods”, Differential Equations and Control Processes, 2 (2015), 19–58 (In Russian) | MR | Zbl

[27] Dobronec B. S., Strelnikova E. A., “Solutions of boundary value problems for ordinary differential equations with interval uncertainties”, Computing Technologies, 10 (2005), 16–21 (In Russian)

[28] Gasilov N. A., Fatullayev A. G., Amrahovb S. E., “Solution method for a non-homogeneous fuzzy linear system of differential equations”, Applied Soft Computing, 70 (2018), 225–237 | DOI

[29] Hullermeier E., “An approach to modelling and simulation of uncertain dynamical systems”, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 5:2 (1997), 117–137 | DOI | MR | Zbl

[30] Hanss M., “The transformation method for the simulation and analysis of systems with uncertain parameters”, Fuzzy Sets and Systems, 130 (2002), 277–289 | DOI | MR | Zbl

[31] Fominyh A. V., “A numerical method for finding the optimal solution of a differential inclusion”, Vestnik St. Petersburg University. Mathematics, 51 (2018), 397–406 | DOI | MR | Zbl

[32] Blagodatskih V. I., Filippov A. F., “Differential inclusions and optimal control”, Proceedings of the Steklov Institute of Mathematics, 169, 1985, 194–252 (In Russian) | MR | Zbl

[33] Blagodatskih V. I., Introduction to optimal control, Vysshaya shkola Publ, M., 2001, 239 pp. (In Russian)

[34] Bonnans J. F., Shapiro A., Perturbation analysis of optimization problems, Springer Science+Business Media Press, New York, 2000, 601 pp. | MR | Zbl

[35] Kantorovich L. V., Akilov G. P., Functional analysis, Nauka Publ, M., 1977, 752 pp. (In Russian) | MR

[36] Penot J. P., “On the convergence of descent algorithms”, Computational Optimization and Applications, 23:3 (2002), 279–284 | DOI | MR | Zbl

[37] Dolgopolik M. V., Fominyh A. V., “Exact penalty functions for optimal control problems. I: Main theorem and free-endpoint problems”, Optimal Control Applications $\$ Methods, 40:6 (2019), 1018–1044 | DOI | MR | Zbl

[38] Vasil'ev F. P., Optimization methods, Factorial Press, M., 2002, 620 pp. (In Russian)