A Chain Rule for Reduced Functional Differential Inclusions and Stability Theorems

Öz

In order to represent real-world problems, modeling and stability concepts of a system are two essential steps, and functional differential inclusions become favorable among other methods because of their flexibility and robustness to handle those problems. Thus, functional differential inclusions (FDIs) provide a solid foundation for engineering problems, and the calculation of their derivatives becomes an important issue in checking the stability of them. Especially, to check the Lyapunov stability, various chain rules for FDIs are defined in the literature. In this work, a new chain rule is introduced in terms of the reduction procedure, a comparison with another one is represented, and the stability theorems in terms of Lyapunov are extended to the reduced functional differential inclusions.

Anahtar Kelimeler

• Functional differential inclusions
• Set-valued analysis
• Convex analysis
• Stability

Kaynakça

1. Aitalioubrahim M, Raghib T. 2023. Functional differential inclusions with maximal monotone operators and nonconvex perturbations. Filomat, 37(20): 6793-6811.
2. Aubin JP, Cellina A. 1984. Differential inclusions: Set-valued maps and viability theory. Springer, Berlin, Germany, pp: 56-59.
3. Bacciotti A, Ceragioli F. 1999. Stability and stabilization of discontinuous systems and nonsmooth Lyapunov functions. ESAIM Control Optim Calc Var, 4: 361-376.
4. Bokalo M, Skira I, Bokalo T. 2024. Strong nonlinear functional-differential variational inequalities: Problems without initial conditions. Front Appl Math Stat, 10: 54-61.
5. Clarke FH, Ledyaev YS, Stern RJ. 1998. Nonsmooth analysis and control theory. Springer, New York, USA, pp: 48-59.
6. Filippov AF. 1988. Differential equations with discontinuous right-hand sides. Kluwer Academic Publishers, Dordrecht.
7. Haddad G. 1981a. Monotone viable trajectories for functional differential inclusions. J Differ Equ, 42: 1-24.
8. Haddad G. 1981b. Topological properties of the sets of solutions for functional differential inclusions. Nonlinear Anal Theory Methods Appl, 5(12): 1349-1366.

9. Hale JK. 1977. Theory of functional differential equations. Springer, New York, USA, pp: 152-159.
10. Kamalapurkar R, Dixon W, Teel AR. 2020. On the reduction of differential inclusions and Lyapunov stability. ESAIM Control Optim Calc Var, 26: 24.
11. Kolmanovskii VB, Myshkis AD. 1992. Applied theory of functional differential equations. Kluwer Academic Publishers, Netherlands, pp: 45-49.
12. LaSalle JP. 1976. The stability of dynamical systems. SIAM, Philadelphia,USA, pp: 65-69.
13. Liu KZ, Sun XM, Wang W, Liu J. 2015. Invariance principles for delay differential inclusions. Proc 27th Chin Control Decis Conf (CCDC): 123-135.
14. Liu KZ, Sun XM, Liu J, Teel AR. 2016. Stability theorems for delay differential inclusions. IEEE Trans Autom Control, 61(10): 3215-3220.
15. Mahmudov EN, Mastaliyeva D. 2024. Optimal control of second order hereditary functional-differential inclusions with state constraints. J Ind Manag Optim, 20(11): 3562-3579.
16. Moreau JJ, Valadier M. 1987. A chain rule involving vector functions of bounded variation. J Funct Anal, 74: 333-345.
17. Paden BE, Sastry SS. 1987. A calculus for computing Filippov’s differential inclusion with application to the variable structure control of robot manipulators. IEEE Trans Circuits Syst, 34(1): 73-82.
18. Shevitz D, Paden B. 1994. Lyapunov stability theory of nonsmooth systems. IEEE Trans Autom Control, 39(9): 1910-1914.
19. Surkov AV. 2007. On the stability of functional-differential inclusions with the use of invariantly differentiable Lyapunov functionals. Differ Equ, 43(8): 1079-1087.