For Caputo fractional-order systems, the optimal value must depend on the entire trajectory history; with that history dependence, the dynamic programming principle and a Hamilton-Jacobi-Bellman equation hold, and smooth solutions yield optimal feedback controls.
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Dynamic programming principle and Hamilton-Jacobi-Bellman equations for fractional-order systems
For Caputo fractional-order systems, the optimal value must depend on the entire trajectory history; with that history dependence, the dynamic programming principle and a Hamilton-Jacobi-Bellman equation hold, and smooth solutions yield optimal feedback controls.