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Causal state feedback representation for linear quadratic optimal control problems of singular Volterra integral equations

This work was partially supported by the National Natural Science Foundation of China under grant 12071067, National Key R & D Program of China under grant 2020YFA0714102, and NSF grant DMS–1812921

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  • This paper is concerned with a linear quadratic optimal control for a class of singular Volterra integral equations. Our framework covers the problems for fractional differential equations. Under some necessary convexity conditions, an optimal control exists, and can be characterized via Fréchet derivative of the quadratic functional in a Hilbert space or via maximum principle type necessary conditions. However, these (equivalent) characterizations are not causal, meaning that the current value of the optimal control depends on the future values of the optimal state. Practically, this is not feasible. We obtain a causal state feedback representation of the optimal control via a Fredholm integral equation. Finally, a concrete form of our results for fractional differential equations is presented.

    Mathematics Subject Classification: Primary: 45D05, 45F15, 49N10, 49N35, 93B52.


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