Targeted therapies suppress tumor growth by inhibiting specific oncogenic signaling pathways. However, sustained inhibition perturbs intracellular signaling and exerts selective stress on cancer cells, triggering compensatory responses that alter proliferation, survival, and drug sensitivity. To capture these mechanisms in a tractable, control-oriented framework, we propose a hybrid model in which stress-induced changes in cellular phenotype and treatment response are represented through state-dependent parameter switching and absorbing dynamics. Tumor growth is described by a proliferating cell population coupled to adaptive traits representing the activity and interactions of three key pathways: Wnt/β-catenin, PI3K/Akt, and MAPK/ERK. Trait evolution captures reversible stress-induced plasticity, while sustained stress triggers discrete parameter updates and, upon saturation, absorbing states associated with irreversible resistance and loss of therapeutic efficacy. Baseline parameters are calibrated using in vivo xenograft data under pathway-specific inhibition. We then propose a hybrid model predictive control strategy with a supervisory layer that adapts therapy to the current discrete mode and selectively suspends treatment on critical pathways. Numerical simulations show the dual benefit of the framework, which preserves modeling tractability while enabling policies that prevent resistance.
Hybrid model predictive control with both continuous and discrete variables is widely applicable to robotic control tasks, especially those involving contacts with the environment. Due to combinatorial complexity, the solving speed of hybrid MPC can be insufficient for real-time applications. In this paper, we propose a hybrid MPC algorithm based on Generalized Benders Decomposition. The algorithm enumerates and stores cutting planes online inside a finite buffer and transfers them across MPC iterations to provide warm-starts for new problem instances, significantly enhancing solving speed. We theoretically analyze this warm-starting performance by modeling the deviation of mode sequences through temporal shifting and stretching, deriving bounds on the dual gap between transferred optimality cuts and the true optimal costs, and utilizing these bounds to quantify the level of suboptimality guaranteed in the first solve of the Benders Master Problem. Our algorithm is validated in simulation through controlling a cart–pole system with soft contact walls, a free-flying robot navigating around obstacles, and a humanoid robot standing on one leg while pushing against walls with its hands for balance. For our benchmark problems, the algorithm enumerates cuts on the order of only tens to hundreds while reaching speeds 2–3 times faster than the off-the-shelf solver Gurobi, oftentimes exceeding 1000 Hz. The code is available at https://github.com/XuanLin/Benders-MPC.
This paper studies a hybrid discrete-time signed gradient-consensus method with heavy-ball momentum, paired communication delays, and fixed logarithmic quantization. The method preserves the aggregate supply-demand equality at every iteration. For a static connected undirected graph and bounded delays, a Lyapunov-Krasovskii analysis establishes practical stability by isolating the discontinuous quantization error as an additive disturbance and closing a computable invariant-sublevel argument. The resulting ergodic residual bound is quadratic in the quantization resolution, and exact convergence to the smooth penalized equality-constrained optimizer is recovered in the unquantized case. For delay-free unquantized quadratic costs, an exact modal analysis identifies the heavy-ball stability region and optimal parameters, with the predicted rate validated on a fixed sparse network. A reproducible delayed experiment reports all constants and a certificate whose guaranteed steady-state level lies below the reported initialization. A separate mixed generator-battery experiment exercises the signed update directly. Finite smooth penalties approximate, but do not generally enforce, the local box constraints exactly.
This paper proposes a distributed resource allocation algorithm in which each agent communicates only the sign of its local cost gradient, a single bit per directed link per iteration. The algorithm is based on Laplacian-gradient dynamics and preserves the equality (demand) constraint at every iterate, guaranteeing anytime feasibility irrespective of termination time. First, we provide a complete Lyapunov-based convergence analysis for the fixed-step-size regime: the iterates are shown to enter, in at most O(1/ε^2) iterations, a neighborhood of the optimizer whose radius scales as O(τ) in the decision variable and O(τ^2) in the cost residual, where τ is the step-size. Crucially, the step-size bound is a time-invariant network constant (depending only on the graph Laplacian spectrum and the cost curvature bounds), so no adaptive tuning is required. Second, we characterize the invariant set under B-connected, weight-balanced digraphs, strictly weaker than the all-time connectivity and doubly-stochastic weights assumed in most prior work, and show that any odd, sign-preserving nonlinearity yields the same invariant-set and descent structure, with the single-bit protocol being the extreme communication-minimizing instance of a broader nonlinear class. Third, we address local box constraints via smooth penalty relaxation and quantify explicitly how the penalty weight interacts with the step-size bound and the optimality gap. Simulations on economic dispatch (1000 generators) and CPU scheduling (500 servers) demonstrate that the single-bit protocol achieves 5-60 times reduction in total communication cost relative to multi-bit baselines, at the price of a tunable and explicitly bounded optimality gap. A diminishing step-size schedule is shown empirically to eliminate this gap; its formal convergence analysis is identified as an open problem.
This paper establishes an asymptotic ergodic classification for mass-conserving biological switching diffusions subjected to multiplicative common noise and Markovian regime-switching. The inclusion of multiplicative common noise yields a positive predictable quadratic variation density on the spatial intersection manifold whenever the biological mass is active, precluding the application of classical pathwise spatial comparison theorems. To resolve this measure-theoretic bottleneck, we impose constant stoichiometry and Beddington–DeAngelis spatial interference to guarantee the uniform fractional integrability and derive algebraic bounds for the continuous spatial variance terms. The global dynamics are classified into a trichotomy governed by the algebraic sign of the principal continuous-time hybrid Lyapunov exponent, λ. For λ<0, we bypass intersection bounds using a continuous-time logarithmic discrepancy tracking sequence and continuous local martingale limit dichotomies to establish global exponential extinction. At the critical threshold λ=0, we prove biological extinction in expectation by applying the multidimensional generator to the boundary Poisson resolvent. Because the abiotic stochastic flow is pathwise affine, it preserves spatial concavity, yielding a non-positive spatial operator difference. For λ>0, we construct a separable Meyn–Tweedie Foster–Lyapunov mapping to secure positive Harris recurrence and exponential convergence in total variation distance. The theoretical results are illustrated numerically.
