PARTIAL CONTROLLABILITY OF SEMILINEAR SYSTEMS

Year 2026, Volume: 16 Issue: 3, 319 - 330, 17.03.2026

Agamirza Bashirov

https://izlik.org/JA95JY92BD

Abstract

The paper considers semilinear control system in the product of Hilbert spaces $X=H\times G$ driven by densely defined closed linear operator $A$ generating a strongly continuous semigroup. For the linear operator $L$, projecting $X$ to $H$, it is proved a sufficient condition for $L$-partially exact controllability to $L(D(A))$ which means that for every initial state $\xi \in X$ and every $\eta \in D(A)$ there exists a control $u$ such that $Lx^{\xi ,u}(T)=L\eta $, where $x^{\xi ,u}$ is the state process corresponding the initial state $\xi $ and the control $u$. The result is demonstrated on examples.

Keywords

Exact controllability , partial controllability , deterministic system , semilinear system , heat equation

References

• Kalman, R. E., (1960), Contributions to the theory of optimal control, Boletin de la Sociedad Matematica Mexicana, 5, pp. 102–119.
• Fattorini, H. O., (1966), Some remarks on complete controllability, SIAM Journal on Control, 4(4), pp. 686–694.
• Russel, D. L., (1967), Nonharmonic Fourier series in the control theory of distributed parameter systems, Journal of Mathematical Analysis and Applications, 18(3), pp. 542–560.
• Curtain, R. F. and Zwart, H. J., (1995), An Introduction to Infinite Dimensional Linear Systems Theory, Springer-Verlag, Berlin.
• Bensoussan, A., (1992), Stochastic Control of Partially Observable Systems, Cambridge University Press, London.
• Zabczyk, J., (1995), Mathematical Control Theory: An Introduction, Ser. Systems & Control: Foundations & Applications, Birkhauser, Berlin.
• Klamka, J., (1991), Controllability of Dynamical Systems, Kluwer Academic, Dordrecht.
• Balachandran, K. and Dauer, J., (2002), Controllability of nonlinear systems in Banach spaces: a survey, Journal of Optimization Theory and Applications, 115(1), pp. 7–28.
• Carrasco, A., Leiva, H., and Merentes, N., (2015), Controllability of semilinear systems of parabolic equations with delay on the state, Asian Journal of Control, 17(6), pp. 2105–2114.
• Leiva, H., (2015), Controllability of the semilinear nonaotonomous systems, International Journal of Control, 88(3), pp. 585–592.
• McKibben, N. A., (2003), Approximate controllability for a class of abstract second-order functional evolution equations, Journal of Optimization Theory and Applications, 117(2), pp. 397–414.
• Muslim, M., Kumar, A., and Sakthivel, R., (2018), Exact and trajectory controllability of second-order evolution systems with impulses and deviated arguments, Mathematical Methods in the Applied Sciences, 41(11), pp. 4259–4272.
• Kumar, A., Muslim, M., and Sakthivel, R., (2018), Controllability of the second-order nonlinear differential equations with non-instantaneous impulses, Journal of Dynamical and Control Systems, 24(2), pp. 325–342.
• Guevara, C. and Leiva, H., (2018), Controllability of the impulsive semilinear heat equation with memory and delay, Journal of Dynamical and Control Systems, 24(1), pp. 1–11.
• Acosta, A. and Leiva, H., (2018), Robustness of the controllability for the heat equation under the influence of multiple impulses and delays, Quaestiones Mathematicae, 41(6), pp. 761–772.
• Pighin D. and Zuazua, E., (2018), Controllability under positivity constrains of semilinear heat equations, Mathematical Control and Related Fields, 8(3-4), pp. 935–964.
• Chaves-Silva, F. W., Zhang, X., and Zuazua, E., (2017), Controllability of evolution equations with memory, SIAM Journal on Control and Optimization, 55(4), pp. 2437–2459.
• Klamka, J., Babiarz, A., and Niezabitowski, M., (2016), Banach fixed-point theorem in semilinear controllability problems - a survey, Bulletin of the Polish Academy of Sciences: Technical Sciences, 64(1), pp. 21–35.
• Babiarz, A., Klamka, J., and Niezabitowski, M., (2016), Shauder’s fixed point theorem in approximate controllability problems, International Journal of Applied Mathematics and Computer Science, 26(2), pp. 263–275.
• Leiva, H., (2014), Rothe’s fixed point theorem and controllability of semilinear non-autonomous systems, Systems and Control Letters, 67(1), pp. 14–18.
• Bashirov, A. E., (1996), On weakening of the controllability concepts, Proceedings of the 35th IEEE Conference on Decision and Control, Kobe, Japan, 1996, 640–645.
• Bashirov, A. E. and Jneid, M., (2014), Partial complete controllability of deterministic semilinear systems, TWMS Journal of Applied and Engineering Mathematics, 4(2), pp. 216-225.
• Bashirov, A.E., (2021), On exact controllability of semilinear systems, Mathematical Methods in the Applied Sciences, 44(2021), 7455-7462, doi: 10.1002/mma.6265.
• Eden, J., Tan, Y., Lau, D., and Oetomo, D., (2016), On the positive output-controllability of linear time invariant systems, Automatica, 71, pp. 202–209.
• Morse, A. S., (1971), Output controllability and system synthesis, SIAM Journal on Control and Optimization, 9(2), pp. 143–148.
• Sarachik, P. E. and Kreindler, E., (1065), Controllability and observability of linear discrete-time systems, International Journal of Control, 1(5), pp. 419–432.
• Garcia-Planas, M. I. and Dominguez-Garcia, J. L., (2013), Alternative tests for functional and point-wise output-controllability of linear-invariant systems, Systems and Control Letters, 62(5), pp. 382–387.
• Tie, L., (2020), Output-controllability and output-near-controllability of drift-less discrete-time bilinear systems, SIAM Journal on Control and Optimization, 58(4), pp. 2114–2142.
• Saldi N., (2024),, Common Information Approach for Static Team Problems with Polish Spaces and Existence of Optimal Policies, Applied and Computational Mathematics, 23(3), pp.307-324.
• Aliev, F. A., Hajiyeva, N. S., Velieva, N. I., Mutallimov, M. M., and Namazov, A. A., (2024), Algorithm for Solution of Linear Quadratic Optimization Problem with Constraint in the Form of Equalities on Control, Applied and Computational Mathematica, 23(3), pp.404-414.
• Bashirov, A. E., Etikan, H., and S¸emi, N., (2010), Partial controllability of stochastic linear systems, International Journal of Control, 83(12), pp. 2564–2572.
• Bashirov, A. E. and Ghahramanlou, N., (2015), On partial S-controllability of semilinear partially observable systems, International Journal of Control, 88(5), pp. 969-982.
• Li, X. and Yong, L., (1995), Optimal Control for Infinite Dimensional Systems: Ser. Systems & Control: Foundations & Applications, Birkhauser, Boston.
• Bucy, R. S. and Joseph, P. D., (1968), Filtering for Stochastic Processes with Application to Guidance, Interscience, New York.
• Bashirov, A. E., Eppelbaum, L. V., and Mishne, L. R., (1992), Improving Eötvös corrections by wide band noise Kalman filtering, Geophysical Journal International, 108(5), pp. 193–197.
• Bashirov, A. E., Etikan, H., and S¸emi, N., (1997), Filtering, smoothing and prediction for wide band noise driven systems, Journal of the Franklin Institute, 334(4), pp. 667–683.
• Bashirov, A. E. and Uğural, S., (2002), Analysing wide-band noise processes with application to control and filtering, IEEE Transactions on Automatic Control, 47(2), pp. 323–327.
• Bashirov, A. E. and Uğural, S., (2002), Representation of systems disturbed by wide band noise, Applied Mathematics Letters, 15(5), pp. 607–613.
• Bashirov, A. E., (2005), Filtering for linear systems with shifted noises, International Journal of Control, 78(7), pp. 521-529.
• Abuassbeh, K. and Bashirov, A. E., (2022), Derivation of Kalman-type filter for linear systems with pointwise delay in signal noise, Boundary Value Problems, 2022:64, https://doi.org/10.1186/s13661-022-01646-6.
• Zuazua, E., (1997), Finite dimensional null controllability for the semilinear heat equation, Journal of Math. Pure Appl., 76, pp. 237-264.