# Distributed control for semilinear equations with Gerasimov-Caputo derivatives

### Abstract

We consider the optimal control problem for semilinear evolution equations with lower fractional derivatives, resolved with respect to the higher fractional derivative, as well as having a degenerate linear operator at it. The nonlinear operator depends on the Gerasimov–Caputo fractional derivatives of lower orders. For the degenerate equation, a nonlinear operator is considered in two cases: if its image lies in the subspace without degeneration and if this operator depends only on the elements of the subspace without degeneration. It is shown that in the case when the solvability of the initial problem, for at least one admissible control, is obvious or can be shown directly, it is possible to prove the existence of an optimal control under a weaker condition of uniform in time local Lipschitz continuity with respect to the phase variables of the nonlinear operator, instead of the condition of its Lipschitz continuity. The theoretical results are applied to an optimal control problem for a system of partial differential equations with fractional time derivatives.

### References

[1]

Mainardi F. and Paradisi F., “Fractional diffusive waves,” J. Comput. Acoustics, 9, No. 4, 1417–1436 (2001).

[2]

Mainardi F. and Spada G., “Creep, relaxation and viscosity properties for basic fractional models in rheology,” Eur. Phys. J. Spec. Top., 193, 133–160 (2011).

[3]

Uchaikin V. V., “Fractional phenomenology of cosmic ray anomalous diffusion,” Usp. Phys., 56, No. 11, 1074–1119 (2013).

[4]

Bajlekova E. G., Fractional Evolution Equations in Banach Spaces, PhD Thesis, Eindhoven Univ. Technol., Univ. Press Facilities, Eindhoven (2001).

[5]

Pruss J., Evolutionary Integral Equations and Applications, Birkh¨auser-Verl., Basel (1993).

[6]

Glushak A. V. and Avad H. K., “On the solvability of an abstract differential equation of fractional order with a variable operator,” J. Math. Phys., 202, No. 5, 637–652 (2014).

[7]

Kostić M., Abstract Volterra Integro-Differential Equations, CRC Press, Boca Raton, FL (2015).

[8]

Debbouche A. and Torres D. F. M., “Sobolev type fractional dynamic equations and optimal multi-integral controls with fractional nonlocal conditions,” Fract. Calc. Appl. Anal., 18, 95–121 (2015).

[9]

Baleanu D., Machado J. A. T., and Luo A. C. J., Fractional Dynamics and Control, Springer-Verl., New York; Dordrecht; Heidelberg; London (2012).

[10]

Wang J. R. and Zhou Y., “A class of fractional evolution equations and optimal controls,” Nonlinear Anal.: Real World Appl., 12, No. 1, 262–272 (2011).

[11]

Romanova E. A. and Fedorov V. E., “Resolving operators of a linear degenerate evolution equation with Caputo derivative. The sectorial case [in Russian],” Mat. Zamet. SVFU, 23, No. 4, 58–72 (2016).

[12]

Streletskaya E. M., Fedorov V. E., and Debbouche A., “The Cauchy problem for distributed order equations in Banach spaces [in Russian],” Mat. Zamet. SVFU, 25, No. 1, 63–72 (2018).

[13]

Fedorov V. E. and Gordievskikh D. M., “Resolving operators of degenerate evolution equations with fractional derivative with respect to time,” Russ. Math., 59, 60–70 (2015).

[14]

Fedorov V. E., Gordievskikh D. M., and Plekhanova M. V., “Equations in Banach spaces with a degenerate operator under a fractional derivative,” Differ. Equ., 51, 1360–1368 (2015).

[15]

Fedorov V. E., Plekhanova M. V., and Nazhimov R. R., “Degenerate linear evolution equations with the Riemann–Liouville fractional derivative,” Sib. Math. J., 59, No. 1, 136–146 (2018).

[16]

Plekhanova M. V., “Nonlinear equations with degenerate operator at fractional Caputo derivative,” Math. Methods Appl. Sci., 40, 41–44 (2016).

[17]

Plekhanova M. V. and Baybulatova G. D., “Strong solutions of semilinear equations with lower fractional derivatives,” in: Transmutation operators and applications (V. Kravchenko, S. M. Sitnik, eds.) (Birkh¨auser Trends Math.), pp. 573–585, Birkh¨auser, Basel (2020).

[18]

Plekhanova M. V. and Baybulatova G. D., “Semilinear equations in Banach spaces with lower fractional derivatives,” in: Nonlinear Analysis and Boundary Value Problems (NABVP 2018) (Santiago de Compostela, Spain, Sep. 4–7, 2019) (I. Area, A. Cabada, J. A. Cid et al., eds.) pp. 81–93, Springer, Cham (2019) (Springer Proc. Math. Stat.; vol. 292).

[19]

Plekhanova M. V. and Baybulatova G. D., “A class of semilinear degenerate equations with fractional lower order derivatives,” in: Stability, Control, Differential Games (SCDG 2019) (T. F. Filippova, V. T. Maksimov, and A. M. Tarasyev, eds.), Proc. Int. Conf. Devoted to 95th Anniv. Acad. N. N. Krasovskii (Yekaterinburg, Russia, Sep. 16–20, 2019), pp. 444–448, Yekaterinburg (2019).

[20]

Plekhanova M. V., “Start control problems for fractional order evolution equations [in Russian],” Chelyab. Fiz.-Mat. Zh., 1, No. 3, 15–36 (2016).

[21]

Plekhanova M. and Baybulatova G., “On strong solutions for a class of semilinear fractional degenerate evolution equations with lower fractional derivatives,” Math. Methods Appl. Sci. (2020).

[22]

Plekhanova M. V., “ Distributed control problems for a class of degenerate semilinear evolution equations,” J. Comput. Appl. Math., 312, 39–46 (2017).

[23]

Baybulatova G. D., “Start control problem for a class of degenerate equations with lower order fractional derivatives [in Russian],” Chelyab. Fiz.-Mat. Zh., 5, No. 3, 271–284 (2020).

[24]

Plekhanova M. V. and Baybulatova G. D., “Problems of hard control for a class of degenerate fractional order evolution equations,” in: Mathematical Optimization Theory and Operations Research, Proc. 18th Int. Conf. (MOTOR 2019) (Ekaterinburg, Russia, July 8–12, 2019), pp. 501–512, Springer, Cham (2019) (Lect. Notes Comput. Sci.; vol. 11548).

[25]

Plekhanova M. V., “Solvability of control problems for degenerate evolution equations of fractional order [in Russian],” Chelyab. Fiz.-Mat. Zh., 2, No. 1, 53–65 (2017).

[26]

Fursikov A. V., Optimal Control of Distributed Systems, Theory and Applications, AMS, Providence, RI (1999) (Transl. Math. Monogr.; vol. 187).

[27]

Sviridyuk G. A. and Fedorov V. E., Linear Sobolev Type Equations and Degenerate Semi-groups of Operators, VSP, Utrecht; Boston (2003).

[28]

Oskolkov A. P., “Initial-boundary value problems for equations of motion of Kelvin–Voight fluids and Oldroyd fluids,” Proc. Steklov Inst. Math., 179, 137–182 (1989).

[29]

Ladyzhenskaya O. A., The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach Publ., New York; London; Paris; Montreux; Tokyo; Melbourne (1961).

*Mathematical notes of NEFU*, 28(2), pp. 47-67. doi: https://doi.org/10.25587/SVFU.2021.16.62.004.

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