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Approximate controllability for abstract measure differential systems. (English) Zbl 1250.93035
Summary: In this paper, we investigate approximate controllability for abstract measure differential systems based on generalizing knowledge for ordinary differential systems. We first introduce new concepts of the reachable set and approximate controllability for abstract measure differential systems. Then, based on the nonlinear alternative for $\alpha$-condensing mapping in Banach space, we present sufficient conditions of approximate controllability for a class of abstract measure differential systems. Our results in dealing with approximate controllability are less conservative than those in the previous literature. Finally, an example is given to illustrate the availability of our results for approximate controllability.
##### MSC:
 93B05 Controllability 93B03 Attainable sets 93C25 Control systems in abstract spaces
##### References:
 [1] Sharma, R. R.: An abstract measure differential equation, Proc. am. Math. soc. 32, 503-510 (1972) · Zbl 0229.34054 · doi:10.2307/2037847 [2] Dhage, B. C.; Chate, D. N.; Ntouyas, S. K.: Abstract measure differential equations, Dyn. syst. Appl. 13, No. 1, 105-117 (2004) · Zbl 1064.34038 [3] Joshi, S. R.; Kasaralikar, S. N.: Differential inequalities for a system of abstract measure delay differential equations, J. math. Phys. sci. 16, No. 6, 515-523 (1982) · Zbl 0518.34054 [4] Dhage, B. C.; Bellale, S. S.: Existence theorems for perturbed abstract measure differential equations, Nonlinear anal. 71, No. 12, e319-e328 (2009) [5] Shendge, G. R.; Joshi, S. R.: Abstract measure differential inequalities and applications, Acta math. Hung. 41, No. 1–2, 53-59 (1983) · Zbl 0536.34040 · doi:10.1007/BF01994061 [6] Dhage, B. C.: On abstract measure integro–differential equations, J. math. Phys. sci. 20, No. 5, 367-380 (1986) · Zbl 0619.45005 [7] Dhage, B. C.: Existence theory for quadratic perturbations of abstract measure integro–differential equations, Differ. equ. Appl. 1, No. 3, 307-323 (2009) · Zbl 1179.34062 · doi:http://files.ele-math.com/abstracts/dea-01-16-abs.pdf [8] Respondek, J. S.: Numerical analysis of controllability of diffusive–convective system with limited manipulating variables, Int. comm. Heat. mass. Transfer 34, No. 8, 934-944 (2007) [9] Garcia, V. M.; Serra, M.; Llorca, J.: Controllability study of an ethanol steam reforming process for hydrogen production, J. power sources 196, No. 9, 4411-4417 (2011) [10] Respondek, J. S.: Controllability of dynamical systems with constraints, Systems control lett. 54, No. 4, 293-314 (2005) · Zbl 1129.93326 · doi:10.1016/j.sysconle.2004.09.001 [11] Schmitendorf, W. E.; Barmish, B. R.: Controlling a constrained linear system to an affine target, IEEE trans. Automat. control 26, No. 3, 761-763 (1981) · Zbl 0484.93014 · doi:10.1109/TAC.1981.1102689 [12] Respondek, J. S.: Numerical simulation in the partial differential equation controllability analysis with physically meaningful constraints, Math. comput. Simul. 81, No. 1, 120-132 (2010) · Zbl 1203.65100 · doi:10.1016/j.matcom.2010.07.016 [13] Paige, C. C.: Properties of numerical algorithms related to computing controllability, IEEE trans. Automat. control 26, No. 1, 130-138 (1981) · Zbl 0463.93024 · doi:10.1109/TAC.1981.1102563 [14] Respondek, J. S.: Numerical approach to the non-linear diofantic equations with applications to the controllability of infinite dimensional dynamical systems, Internat. J. Control 78, No. 13, 1017-1030 (2005) · Zbl 1108.93021 · doi:10.1080/00207170500197605 [15] Klamka, J.: Controllability of dynamical systems, (1991) [16] Klamka, J.; Wyrwal, J.: Controllability of second-order infinite-dimensional systems, Systems control lett. 57, No. 5, 386-391 (2008) · Zbl 1139.93004 · doi:10.1016/j.sysconle.2007.10.002 [17] Kalogeropoulos, G.; Psarrakos, P.: A note on the controllability of higher order linear systems, Appl. math. Lett. 17, No. 12, 1375-1380 (2004) · Zbl 1073.93005 · doi:10.1016/j.am1.2003.12.008 [18] Respondek, J. S.: Approximate controllability of the n-th order infinite dimensional systems with controls delayed by the control devices, Int. J. Syst. sci. 39, No. 8, 765-782 (2008) [19] Respondek, J. S.: Approximate controllability of infinite dimensional systems of the n-th order, Int. J. Appl. math. Comput. sci. 18, No. 2, 199-212 (2008) [20] Fu, X. L.; Mei, K. D.: Approximate controllability of semilinear partial functional differential equations, J. dyn. Control syst. 15, No. 3, 425-443 (2009) · Zbl 1203.93022 · doi:10.1007/s10883-009-9068-x [21] Muthukumara, P.; Balasubramaniam, P.: Approximate controllability of second-order damped mckean–Vlasov stochastic evolution equations, Comput. math. Appl. 60, No. 10, 2788-2796 (2010) · Zbl 1207.93012 · doi:10.1016/j.camwa.2010.09.032 [22] Dhage, B. C.: Periodic boundary value problems of first order ordinary Carathéodory and discontinuous differential equations, Nonlinear funct. Anal. appl. 13, No. 2, 323-352 (2008) · Zbl 1171.34005