# zbMATH — the first resource for mathematics

A class of fractional delay nonlinear integrodifferential controlled systems in Banach spaces. (English) Zbl 1223.45007
This work deals with the fractional delay nonlinear integrodifferential controlled system
$\begin{cases}\text{}^C\!D_t^qx(t)+Ax(t)=f\left(t,x_t,\displaystyle\int_0^tg(t,s,x_s)ds\right)+B(t)u(t),\,\,\,0<t\leq T,\\ x(t)=\varphi(t),\,\,\,-r\leq t\leq 0,\end{cases}.\tag{1}$
where $$\text{}^C\!D_t^q$$ denotes the Caputo fractional derivative of order $$q\in (0,1)$$, $$-A:D(A)\to X$$ is the infinitesimal generator of an analytic semigroup of uniformly bounded linear operators $$\{S(t),\,\,t\geq 0\}$$ on a separable reflexive Banach space $$X$$, $$f$$ is $$X$$-value function and $$g$$ is $$X_{\alpha}$$-value function. Here $$X_{\alpha}=D(A^{\alpha})$$ is a Banach space with the norm $$\|x\|_{\alpha}=\|A^{\alpha}x\|$$ for $$x\in X_{\alpha}$$, $$u$$ takes values from another separable reflexive Banach space $$Y$$, $$B$$ is a linear operator from $$Y$$ into $$X$$, and $$x_t:[-r,0]\to X_{\alpha},\,\,t\geq 0$$ represents the history of the state from time $$t-r$$ up to the present time $$t$$, defined by $$x_t=\{x(t+s),\,\,\,s\in [-r,0]\}$$. The authors prove the existence and uniqueness of $$\alpha$$-mild solutions for $$(1)$$, and the continuous dependence result of these solutions. The Lagrange problem of system $$(1)$$ is also formulated and an existence result of optimal controls is presented. To illustrate the obtained results, an example is finally addressed.

##### MSC:
 45J05 Integro-ordinary differential equations 26A33 Fractional derivatives and integrals 49J21 Existence theories for optimal control problems involving relations other than differential equations 93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) 45G10 Other nonlinear integral equations
Full Text:
##### References:
  Agarwal, R.P.; Benchohra, M.; Hamani, S., A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta appl math, 109, 973-1033, (2010) · Zbl 1198.26004  Agarwal, R.P.; Belmekki, M.; Benchohra, M., A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative, Adv diff equ, 47, (2009), Article ID 981728 · Zbl 1182.34103  Agarwal, R.P.; Zhou, Yong; He, Yunyun, Existence of fractional neutral functional differential equations, Comp math appl, 59, 1095-1100, (2010) · Zbl 1189.34152  Amann, H., Invariant sets and existence for semilinear parabolic and elliptic systems, J math anal appl, 65, 432-469, (1978) · Zbl 0387.35038  Balder, E., Necessary and sufficient conditions for L1-strong-weak lower semicontinuity of integral functional, Nonlinear anal, 11, 1399-1404, (1987) · Zbl 0638.49004  Benchohra, M.; Henderson, J.; Ntouyas, S.K.; Ouahab, A., Existence results for fractional order functional differential equations with infinite delay, J math anal appl, 338, 1340-1350, (2008) · Zbl 1209.34096  Benchohra, M.; Henderson, J.; Ntouyas, S.K.; Ouahab, A., Existence results for fractional functional differential inclusions with infinite delay and application to control theory, Fract calc appl anal, 11, 35-56, (2008) · Zbl 1149.26010  Belmekki, M.; Benchohra, M., Existence results for fractional order semilinear functional differential with nondense domain, Nonlinear anal, 72, 925-932, (2010) · Zbl 1179.26018  Chang, Y.K.; Kavitha, V.; Arjunan, M.M., Existence and uniqueness of mild solutions to a semilinear integrodifferential equation of fractional order, Nonlinear anal, 71, 5551-5559, (2009) · Zbl 1179.45010  El-Borai, M.M., Some probability densities and fundamental solutions of fractional evolution equations, Chaos soliton fract, 14, 433-440, (2002) · Zbl 1005.34051  El-Borai, M.M., The fundamental solutions for fractional evolution equations of parabolic type, J appl math stoch anal, 3, 197-211, (2004) · Zbl 1081.34053  Henderson, J.; Ouahab, A., Fractional functional differential inclusions with finite delay, Nonlinear anal, 70, 2091-2105, (2009) · Zbl 1159.34010  Hu, L.; Ren, Y.; Sakthivel, R., Existence and uniqueness of mild solutions for semilinear integro-differential equations of fractional order with nonlocal initial conditions and delays, Semigroup forum, 79, 507-514, (2009) · Zbl 1184.45006  Hernández, E.; O’Regan, D.; Balachandran, K., On recent developments in the theory of abstract differential equations with fractional derivatives, Nonlinear anal, 73, 3462-3471, (2010) · Zbl 1229.34004  Jaradat, O.K.; Al-Omari, A.; Momani, S., Existence of the mild solution for fractional semilinear initial value problems, Nonlinear anal, 69, 3153-3159, (2008) · Zbl 1160.34300  Miller, K.S.; Ross, B., An introduction to the fractional calculus and differential equations, (1993), John Wiley New York · Zbl 0789.26002  Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., Theory and applications of fractional differential equations, () · Zbl 1092.45003  Lakshmikantham, V.; Leela, S.; Devi, J.V., Theory of fractional dynamic systems, (2009), Cambridge Scientific Publishers · Zbl 1188.37002  Mophou, G.M.; N’Guérékata, G.M., Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay, Appl math comput, 216, 61-69, (2010) · Zbl 1191.34098  Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010  Pazy, A., Semigroup of linear operators and applications to partial differential equations, (1983), Springer-Verlag New York · Zbl 0516.47023  Ren, Y.; Qin, Y.; Sakthivel, R., Existence results for fractional order semilinear integro-differential evolution equations with infinite delay, Integral equ oper theory, 67, 33-49, (2010) · Zbl 1198.45009  Xiang, X.; Kuang, H., Delay systems and optimal controls, Acta math appl sin, 16, 27-35, (2000) · Zbl 1005.49017  Wang, JinRong; Zhou, Yong, Time optimal control problem of a class of fractional distributed systems, Int J dyn diff eq, 3, 363-382, (2010) · Zbl 1245.49010  Wang, JinRong; Zhou, Yong, A class of fractional evolution equations and optimal controls, Nonlinear anal, 12, 262-272, (2011) · Zbl 1214.34010  Wang, JinRong; Zhou, Yong, Study of an approximation process of time optimal control for fractional evolution systems in Banach spaces, Adv diff equ, 2011, 1-16, (2011), Article ID 385324 · Zbl 1222.49006  Ye, Q.; Li, Z., Introductory to reaction-diffusion equations, (1999), Science Publishing Society China  Zhou, Yong, Existence and uniqueness of fractional functional differential equations with unbounded delay, Int J dyn diff eq, 1, 239-244, (2008) · Zbl 1175.34081  Zhou, Yong; Jiao, Feng; Li, Jing, Existence and uniqueness for fractional neutral differential equations with infinite delay, Nonlinear anal, 71, 3249-3256, (2009) · Zbl 1177.34084  Zhou, Yong; Jiao, Feng, Existence of extremal solutions for discontinuous fractional functional differential equations, Int J dyn diff eq, 2, 237-252, (2009) · Zbl 1188.34108  Zhou, Yong; Jiao, Feng, Existence of mild solutions for fractional neutral evolution equations, Comput math appl, 59, 1063-1077, (2010) · Zbl 1189.34154  Zhou, Yong; Jiao, Feng, Nonlocal Cauchy problem for fractional evolution equations, Nonlinear anal, 11, 4465-4475, (2010) · Zbl 1260.34017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.