A conditional gradient-flow framework for edge-length optimization of quasilinear hyperbolic networks
Abstract
We develop a conditional gradient-flow dynamical system (GFDS) framework for finite-dimensional edge-length optimization of fixed-topology networks governed by quasilinear hyperbolic balance laws. Each physical edge is mapped to the unit interval, and one prescribed parameter-dependent family specifies the coefficients, sources, initial data, node maps, controls, targets, and regularizer. Geometric dependence acts through the metric length vector, whereas every additional explicit parameter dependence is retained separately. Under a parameter-uniform classical regime---including a Lipschitz estimate for the nonlinear state and identified continuity of the linear broad-solution operators---the design-to-state map is continuously Fr\'echet differentiable as a map into X0, the product C0 topology. The reduced differential is represented by transposition and, under stronger regularity and terminal compatibility, by a classical backward network. Under the stated additional finite-dimensional regularity, compactness, and retraction assumptions, the intrinsic equality-constrained GFDS satisfies energy dissipation, global continuation on compact interior sublevels, stationarity of accumulation points, convergence to isolated stationary points, conditional local exponential stability, and an analogous exact retraction--Armijo stationarity result. A scalar affine two-edge model verifies the structural hypotheses and displayed direct-differentiation and Armijo identities in that special class. The general result remains conditional, concerns classical pre-shock solutions on a fixed graph, and does not infer parameter-uniform quasilinear estimates from fixed-parameter well-posedness.
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BibTeX
@misc{sokolowski2026conditional,
title = {{A conditional gradient-flow framework for edge-length optimization of quasilinear hyperbolic networks}},
author = {Jan Sokolowski},
howpublished = {HAL},
year = {2026},
month = aug,
url = {https://hal.science/hal-05731944v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A conditional gradient-flow framework for edge-length optimization of quasilinear hyperbolic networks
AU - Jan Sokolowski
PY - 2026
DA - 2026/08/31
PB - HAL
UR - https://hal.science/hal-05731944v1
AB - We develop a conditional gradient-flow dynamical system (GFDS) framework for finite-dimensional edge-length optimization of fixed-topology networks governed by quasilinear hyperbolic balance laws. Each physical edge is mapped to the unit interval, and one prescribed parameter-dependent family specifies the coefficients, sources, initial data, node maps, controls, targets, and regularizer. Geometric dependence acts through the metric length vector, whereas every additional explicit parameter dependence is retained separately. Under a parameter-uniform classical regime---including a Lipschitz estimate for the nonlinear state and identified continuity of the linear broad-solution operators---the design-to-state map is continuously Fr\'echet differentiable as a map into X_0, the product C^0 topology. The reduced differential is represented by transposition and, under stronger regularity and terminal compatibility, by a classical backward network. Under the stated additional finite-dimensional regularity, compactness, and retraction assumptions, the intrinsic equality-constrained GFDS satisfies energy dissipation, global continuation on compact interior sublevels, stationarity of accumulation points, convergence to isolated stationary points, conditional local exponential stability, and an analogous exact retraction--Armijo stationarity result. A scalar affine two-edge model verifies the structural hypotheses and displayed direct-differentiation and Armijo identities in that special class. The general result remains conditional, concerns classical pre-shock solutions on a fixed graph, and does not infer parameter-uniform quasilinear estimates from fixed-parameter well-posedness.
ER -