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Optimal control problem for a degenerate fractional differential equation. (English) Zbl 1468.49020

The classical Riemann-Liouville left fractional derivative is \[ ^{RL} D^\alpha_{0+} f(t) = \frac{1}{\Gamma(n-\alpha)} \Big(\frac{d}{dt} \Big)^n \int_0^t (t - \tau)^{n - \alpha - 1} f(\tau) d\tau \qquad (n-1 < \alpha \le n) \, , \] its right counterpart based on the integration interval \(t \le \tau \le T.\) The Caputo left fractional derivative uses instead the definition \[ ^{C} D^\alpha_{0+} f(t) = \frac{1}{ \Gamma(n-\alpha)} \int_0^t (t - \tau)^{n - \alpha - 1} f^{(n)}(\tau) d\tau \qquad (n-1 < \alpha \le n) \, , \] its right counterpart based on \(t \le \tau \le T.\) Both the \(RL\) and the \(C\) derivatives reduce to the ordinary derivative when \(\alpha = n.\) One advantage of the second definion is, the \(C\) derivative of any order \(\alpha\) of a constant is zero, while we have \(^{RL} D^\alpha_{0-}c = (c t^{n-\alpha}/\Gamma(n - \alpha + 1))^{(n)},\) which is zero only for integer values of \(\alpha.\) Another is, the Laplace transform of Caputo \(\alpha^{th}\) derivatives involves \(f(0), \dots, f^{(n-1)}(0)\) rather than fractional derivatives of \(f(\cdot)\) at \(t = 0\), thus allowing pointwise initial conditions in fractional differential equations.
The control systems in this paper are of the form \[ v(t) ^{C} D_{0+}^\alpha u(t) + a(t)u(t) = \varphi(t, \nu(t)) \, \quad (0 \le t \le T), \qquad u(0) = \varphi_0 \] where \(0 < \alpha \le 1,\) \(\nu(t) = (\nu_1(t), \dots \nu_m(t))\) is an \(m\)-dimensional control function in a control set \(\Omega\) and \(v(t)\) is positive and locally integrable. The problem is to minimize the cost functional \[ F(\nu) = \sum_{k=1}^N \alpha_k u(t^{(k)}) \] where the \(\{t^{(k)}\}\) are given in \([0, T].\) The authors derive the adjoint equation and obtain Pontryagin’s maximum principle for the optimal control \(\bar \nu\) in the usual maximization-of-Hamiltonian form \[ H(t, f_\alpha(t), \bar \nu(t)) = \max_{\nu \in \Omega}H(t, f_\alpha(t), \nu) \] where \(f_\alpha(t)\) is the solution of the adjoint equation and \(H(t, f_\alpha(t), \nu) = f_\alpha (t) \cdot \varphi(t, \nu)\) is the Hamiltonian.

MSC:

49K15 Optimality conditions for problems involving ordinary differential equations
93C10 Nonlinear systems in control theory
93C15 Control/observation systems governed by ordinary differential equations
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[1] Hilfer, R., Applications of Fractional Calculus in Physics (2000), Singapore: World Scientific, Singapore · Zbl 0998.26002
[2] Podlubny, I., Fractional Differential Equations (1999), San Diego: Academic, San Diego · Zbl 0924.34008
[3] Samko, S. G.; Kilbas, A. A.; Marichev, D. I., Fractional Integrals and Derivatives: Theory and Applications (1993), Philadelphia, PA: Gordon and Breach Science, Philadelphia, PA · Zbl 0818.26003
[4] A. A. Kilbas, H. Srivastava, and J. Trujillo, Theory and Applications of Fractional Differential Equations (North-Holland Mathematics Studies, Elsevier, Amsterdam, 2006). · Zbl 1092.45003
[5] Machado, J. T.; Kiryakova, V.; Mainardi, F., Recent history of fractional calculus, Commun. Nonlin. Sci. Numer. Simul., 16, 1140-1153 (2011) · Zbl 1221.26002 · doi:10.1016/j.cnsns.2010.05.027
[6] Mainardi, F., Fractional Calculus and Waves in Linear Viscoelaticity (2010), London: Imperial College Press, London · Zbl 1210.26004 · doi:10.1142/p614
[7] Malinowska, A. B.; Torres, D. F. M., Introduction to the Fractional Calculus of Variations (2012), London: Imperial College Press, London · Zbl 1258.49001 · doi:10.1142/p871
[8] Alsaedi, A.; Alghamdi, N.; Agrawal, R. P.; Ntouyas, S. K.; Ahmad, B., Multi-term fractional-order boundary-value problems with nonlocal integral boundary conditions, Electron. J. Differ. Equat., 2018, 1-16 (2018) · Zbl 1445.34030 · doi:10.1186/s13662-017-1452-3
[9] Bachar, I.; Mâagli, H.; Rădulescu, V. D., Positive solutions for superlinear Riemann-Liouville fractional boundary-value problems, Electron. J. Differ. Equat., 2017, 1-16 (2017) · Zbl 1422.35169 · doi:10.1186/s13662-016-1057-2
[10] Tarasov, V. E., Fractional Dynamics: Fractional Dynamics Applications of Fractional Calculus to Dynamics of Particles, Fields and Media (2011), New York: Springer, New York · Zbl 1214.81004
[11] T. K. Yuldashev and E. T. Karimov, ‘‘Inverse problem for a mixed type integro-differential equation with fractional order Caputo operators and spectral parameters,’’ Axioms 9 (4), 121-1-24 (2020).
[12] Yuldashev, T. K.; Kadirkulov, B. J., Nonlocal problem for a mixed type fourth-order differential equation with Hilfer fractional operator, Ural Math. J., 6, 153-167 (2020) · Zbl 1448.35341 · doi:10.15826/umj.2020.1.013
[13] T. K. Yuldashev and B. J. Kadirkulov, ‘‘Boundary value problem for weak nonlinear partial differential equations of mixed type with fractional Hilfer operator,’’ Axioms 9 (2), 68-1-19 (2020).
[14] Damag, F. H.; Kiliçman, A.; Ibrahim, R. W., Monotone solutions of monotone iterative technique for hybrid fractional differential equations, Lobachevskii J. Math., 40, 156-165 (2019) · Zbl 1419.34168 · doi:10.1134/S1995080219020069
[15] Houas, M.; Dahmani, Z., On existence of solutions for fractional differential equations with nonlocal multi-point boundary conditions, Lobachevskii J. Math., 37, 120-127 (2016) · Zbl 1344.34015 · doi:10.1134/S1995080216020050
[16] Baleanu, D.; Machado, J. T.; Luo, A. C. J., Fractional Dynamics and Control (2012), New York: Springer, New York · Zbl 1231.93003 · doi:10.1007/978-1-4614-0457-6
[17] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishenko, Mathematical Theory of Optimal Processes (Nauka, Moscow, 1969; Wiley, New York, 1962). · Zbl 0102.32001
[18] Kamocki, R., Pontryagin maximum principle for fractional ordinary optimal control problems, Math. Methods Appl. Sci., 37, 1668-1686 (2014) · Zbl 1298.26023 · doi:10.1002/mma.2928
[19] Ali, H. M.; Lobo Pereira, F.; Gama, S. M. A., A new approach to the Pontryagin maximum principle for nonlinear fractional optimal control problems, Math. Methods Appl. Sci., 39, 3640-3649 (2016) · Zbl 1344.49032 · doi:10.1002/mma.3811
[20] Agrawal, O. P., A general formulation and solution scheme for fractional optimal control problems, Nonlin. Dynam., 38, 323-337 (2004) · Zbl 1121.70019 · doi:10.1007/s11071-004-3764-6
[21] Agrawal, O. P.; Defterli, O.; Baleanu, D., Fractional optimal control problems with several state and control variables, J. Vibr. Control, 16, 1967-1976 (2012) · Zbl 1269.49002 · doi:10.1177/1077546309353361
[22] Kamocki, R., On the existence of optimal solutions to fractional optimal control problems, Appl. Math. Comput., 35, 94-104 (2014) · Zbl 1334.49010
[23] Bandaliyev, R. A.; Guliyev, V. S.; Mamedov, I. G.; Sadigov, A. B., The optimal control problem in the processes described by the Goursat problem for a hyperbolic equation in variable exponent Sobolev spaces with dominating mixed derivatives, J. Comput. Appl. Math., 305, 11-17 (2016) · Zbl 1359.37073 · doi:10.1016/j.cam.2016.03.024
[24] Bandaliyev, R. A.; Guliyev, V. S.; Mamedov, I. G.; Rustamov, Y. I., Optimal control problem for Bianchi equation in variable exponent Sobolev spaces, J. Optim. Theory Appl., 180, 303-320 (2019) · Zbl 1410.49027 · doi:10.1007/s10957-018-1290-9
[25] Mardanov, M. J.; Sharifov, Y. A., Pontryagin’s maximum principle for the optimal control problems with multipoint boundary conditions, Abstr. Appl. Anal., 2015, 428042-1-6 (2015) · Zbl 1345.49024 · doi:10.1155/2015/428042
[26] Pooseh, S.; Almeida, R.; Torres, D. F. M., Fractional order optimal control problems with free terminal time, J. Ind. Manag. Optim., 10, 363-381 (2014) · Zbl 1278.26013
[27] Bandaliyev, R. A.; Mamedov, I. G.; Mardanov, M. J.; Melikov, T. K., Fractional optimal control problem for ordinary differential in weighted Lebesgue spaces, Optim. Lett., 14, 1519-1532 (2020) · Zbl 1448.49026 · doi:10.1007/s11590-019-01518-6
[28] Maz’ya, V. G., Sobolev Spaces (1985), Berlin: Springer, Berlin · Zbl 0692.46023 · doi:10.1007/978-3-662-09922-3
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