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Approximation theorems for controllability problem governed by fractional differential equation

  • * Corresponding author: Rathinasamy Sakthivel

    * Corresponding author: Rathinasamy Sakthivel
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  • In this manuscript, we discuss the optimal control problem for a nonlinear system governed by the fractional differential equation in a separable Hilbert space $ X $. We utilize the fixed point technique and $ \eta $-resolvent family to present the existence of control for the fractional system. The optimal pair is obtained as the limit of the optimal pair sequence of the unconstrained problem. Further, we derive some approximation results, which guarantee the convergence of the numerical method to optimal pair sequence. Finally, the main results are validated with the aid of an example.

    Mathematics Subject Classification: Primary: 34A08; Secondary: 93B05, 49J20, 93C25.


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  • Figure 1.  Comparison between $ z_{b} $ and $ \overline{z}(b) $

    Figure 2.  Approximated optimal control $ \overline{u} $

    Figure 3.  Numerical solution $ \overline{z} $ corresponding to optimal control $ \overline{u} $ with $ \eta = 0.75 $

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