Rigorous criterion for unique continuation across piecewise-smooth breaking points in state-dependent DDEs

Establish a rigorous theorem proving that the condition d/dt (t − τ(t, u±(t)))|_{t=ξ_j} > 0 for both one-sided solutions u−(t) and u+(t) guarantees unique continuability of solutions for delay differential equations with state-dependent delays of the form u(t) = f(t, u(t), u(t − τ(t, u(t)))) across points t = ξ_j satisfying t − τ(t, u(t)) = ξ_i, in the presence of piecewise-smooth initial histories.

Background

The paper analyzes breaking points that arise in delay differential equations with state-dependent delays when initial functions are piecewise smooth. It argues heuristically that a ‘no-sliding’ condition, namely monotonicity of t − τ(t, u(t)) through the breaking point, should ensure unique continuation of the solution across discontinuities in the history.

While numerical examples support this condition, the authors emphasize the absence of a formal proof guaranteeing unique continuability under the stated derivative condition, highlighting a gap between practical algorithms and rigorous theory.