State-Dependent Delays in Optimal Control and Hamilton-Jacobi Equations
Abstract
We develop a Hamilton-Jacobi theory for finite-horizon optimal control problems governed by state-dependent delay equations. In this setting, the delayed-time map depends on the controlled trajectory itself, so that both the state and the point at which the past state is evaluated vary simultaneously. The natural state variable is therefore the entire history, and the value functional is defined on a space of Lipschitz histories. We first establish the dynamic programming principle and characterize the value functional as the unique viscosity solution of the associated Hamilton-Jacobi equation with co-invariant derivatives. To accommodate Lipschitz histories, we introduce a notion of viscosity solution based on finite-dimensional projections generated by polygonal extensions, and prove a comparison principle in a suitable class of functionals. Then we derive a Pontryagin minimum principle whose adjoint equation contains advanced terms induced by the state dependence of the delay. We also obtain a generalized transversality relation formulated in terms of the delay superdifferential of the value functional. Finally, we establish the semiconcavity of the value functional in the history variable, as well as its joint semiconcavity in time and history on an appropriate solution manifold.
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BibTeX
@misc{jiang2026statedependent,
title = {{State-Dependent Delays in Optimal Control and Hamilton-Jacobi Equations}},
author = {Yiming Jiang and Cristian Mendico and Yawei Wei and Fei Zeng and Zimeng Zheng},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27776},
eprint = {2609.27776},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05745399v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - State-Dependent Delays in Optimal Control and Hamilton-Jacobi Equations
AU - Yiming Jiang
AU - Cristian Mendico
AU - Yawei Wei
AU - Fei Zeng
AU - Zimeng Zheng
PY - 2026
DA - 2026/09/01
PB - HAL
DO - 10.48550/arxiv.2609.27776
UR - https://hal.science/hal-05745399v1
AB - We develop a Hamilton-Jacobi theory for finite-horizon optimal control problems governed by state-dependent delay equations. In this setting, the delayed-time map depends on the controlled trajectory itself, so that both the state and the point at which the past state is evaluated vary simultaneously. The natural state variable is therefore the entire history, and the value functional is defined on a space of Lipschitz histories. We first establish the dynamic programming principle and characterize the value functional as the unique viscosity solution of the associated Hamilton-Jacobi equation with co-invariant derivatives. To accommodate Lipschitz histories, we introduce a notion of viscosity solution based on finite-dimensional projections generated by polygonal extensions, and prove a comparison principle in a suitable class of functionals. Then we derive a Pontryagin minimum principle whose adjoint equation contains advanced terms induced by the state dependence of the delay. We also obtain a generalized transversality relation formulated in terms of the delay superdifferential of the value functional. Finally, we establish the semiconcavity of the value functional in the history variable, as well as its joint semiconcavity in time and history on an appropriate solution manifold.
ER -