We develop a whole-trajectory relaxation framework for dissipative evolutions with non-convex, state- and history-dependent kinetics. The framework arises naturally from a game-theoretical formulation of energy--dissipation evolution. State--rate consistency is eliminated by causal Volterra reconstruction, reducing the problem to a diagonal equilibrium condition on complete rate trajectories. We establish a direct method based on vanishing-defect approximability, compactness, and complementary semicontinuity. The relaxation mechanism is computed exactly for a reduced simple-shear model with a non-convex kinetic potential and a local memory variable. The enriched state is a Young measure on complete local histories. We identify explicitly the sequential lower-semicontinuous envelope of the pure diagonal defect, characterize its zero set, construct pure recovery sequences for every relaxed zero, and prove existence through a frozen-best-response time discretization. The resulting coexistence of distinct rate populations with distinct memories provides a reduced mechanism for kinetic shear banding and connects naturally with negative strain-rate sensitivity and dynamic strain aging such as underlie the Portevin--Le Chatelier effect.
We present a theory for the optimal control of stochastic systems in structured environments, represented by penalty terms in the cost functional. We show that such control problems generically feature sharp finite-time transitions associated with a qualitative change in the control strategy at a critical time. Startin...
Jan Meibohm, Samuel Monter, Clemens Bechinger et al.· 1 citation
We develop a quantitative framework for the long-time behavior of McKean--Vlasov stochastic differential equations with singular coefficients and possible phase transitions. The framework separates the existence of invariant measures from their uniqueness. For existence, we introduce a generalized Lyapunov condition wi...
Metastability separates apparent equilibration from true convergence: a dynamical system can remain near an intermediate state long before reaching its stationary state. In quantum dynamics used for optimization, such a plateau can delay progress toward a target concentrated near the global minimizers of a nonconvex po...
We formulate a two-intermediate model for precursor-fed autocatalysis with a bounded phenomenological thermokinetic factor. Positivity, dissipativity, existence of a positive equilibrium, an a priori equilibrium bound, and a conditional uniqueness criterion are established analytically. Generic saddle-node and Hopf con...
Saad Jamhan Aldosari, M. S. Shabbir, A. Tassaddiq et al.· Match-communications in Math...· 0 citations
We propose and analyze a time-staggered numerical scheme for mean field game (MFG) systems with individual noise on finite graphs. Numerically solving such coupled forward--backward systems is delicate because the density evolves in the open probability simplex and the coefficients may degenerate at its boundary. The s...
Dimension reduction removes variables, but a reliable reduction should not erase their dynamical influence. We study this principle for linear block evolution systems on resolved and unresolved Hilbert spaces. Exact elimination of the unresolved component yields a Volterra equation with a memory kernel and an effective...
Hiroki Ishizaka· 0 citations
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