Time-optimal control with bang-bang property for strongly coupled nonlinear microwave heating systems

Dongsheng Luo; Lianying Zhang; Peiyong Zhang; Ailiang Zhao

doi:10.59400/adecp3873

Time-optimal control with bang-bang property for strongly coupled nonlinear microwave heating systems

Dongsheng Luo
Department of Mathematics, Zunyi Normal University, Zunyi 563006, China
Lianying Zhang
Department of Foundational Education, Zunyi Medical and Pharmaceutical College, Zunyi 563006, China
Peiyong Zhang
School of Medical Information Engineering, Zunyi Medical University, Zunyi 563006, China
Ailiang Zhao
Department of Mathematics, Zunyi Normal University, Zunyi 563006, China

Article ID: 3873

Keywords: nonlinear microwave heating system; strongly coupled PDE system; time-optimal control; controllability; bang-bang property

Abstract

This paper makes a rigorous mathematical analysis of the time-optimal control problem for a nonlinear microwave heating system described by coupled partial differential equations. The work extends the established theory for linear models to a physically realistic nonlinear regime, where magnetic field permeability exhibits a nonlinear dependence on the material’s evolving temperature field. The first result establishes the exact controllability of this nonlinear distributed parameter system. This is achieved by applying the Kakutani Fixed-Point Theorem to an appropriately defined solution operator, proving that the system state can be driven from any admissible initial temperature distribution to a specified target profile within a finite time horizon using suitable control inputs. Leveraging this controllability foundation and employing crucial a priori energy estimates, we subsequently prove the existence of at least one time-optimal control via minimizing sequences and weak compactness arguments. The central contribution is the rigorous analytic proof of the bang-bang property for these time-optimal controls. This structural property is demonstrated by contradiction, using a pivotal quantitative relation—derived from the controllability analysis—that links the minimum achievable control time to the L 2 -norm of the control force. The proof conclusively shows that any time-optimal control must saturate the prescribed control constraints almost everywhere in the time-space domain, taking values only at the extremes of the admissible set. These results lay a firm theoretical foundation for optimal control protocol design in nonlinear microwave heating, confirming that efficient strategies are inherently of switching type and offering a benchmark for future numerical and experimental work as well.

Published

2026-03-12

How to Cite

Luo, D., Zhang, L., Zhang, P., & Zhao, A. (2026). Time-optimal control with bang-bang property for strongly coupled nonlinear microwave heating systems. Advances in Differential Equations and Control Processes, 33(1). https://doi.org/10.59400/adecp3873

Download Citation

Issue

Vol. 33 No. 1 (2026)

Section

Article

This work is licensed under a Creative Commons Attribution 4.0 International License.

References

[1]Metaxas AC. Foundations of Electroheat: A Unified Approach. John Wiley and Sons; 1996.

[2]Zhu NK, Wang S, He D, et al. Study on Improving Microwave Heating Uniformity Based on Phase-Frequency Simultaneous Modulation Technique. IEEE Microwave and Wireless Technology Letters. 2025; 35(11): 1871–1874.

[3]Wei W, Yin HM, Tang J. An Optimal Control Problem for Microwave Heating. Nonlinear Analysis. 2012; 75: 2024–2036.

[4]Yin HM. Regularity of Solutions of Maxwell’s Equations in Quasi-Stationary Electromagnetic Field and Applications. Partial Differential Equations. 1997; 22: 1029–1053.

[5]Xu J, Hao K, Meng C, et al. Dynamic Synthesis Augmented Time GAN and Adaptive Temperature Control for Microwave Heating. Journal of Manufacturing Systems. 2025; 80: 723–733.

[6]Kime KA, Russell DL. Boundary Controllability of Maxwell’s Equations in a Spherical Region. SIAM Journal on Control and Optimization. 2006; 28(2): 259–262.

[7]Krigman SS, Wayne CE. Boundary Controllability of Maxwell’s Equations with Nonzero Conductivity Inside a Cube, I: Spectral Controllability. Journal of Mathematical Analysis and Application. 2007; 329(2): 1375–1396.

[8]Luo D, Wei W, Liao Y, et al. Bang-Bang Property of Time Optimal Control for a Kind of Microwave Heating Problem. Journal of Optimization Theory and Applications. 2019; 183(1): 317–331.

[9]Tröltzsch F. Optimal Control of Partial Differential Equations, Theory, Methods and Applications: Graduate Studies in Mathematics Volume 112. American Mathematical Society; 2010.

[10]Prastiyanto D, Apriaskar E, Amelia I, et al. A Novel Approach to Microwave Heating Control System: Crow Search Algorithm-Based PID Controllers. Journal of Advanced Research in Applied Mechanics. 2024; 128(1): 1–10

[11]Wang GS. L∞-nullcontrollability for the Heat Equation and Its Consequence for the Time Optimal Control Problem. SIAM Journal on Control and Optimization. 2008; 47: 1701–1720.

[12]Wang G, Wang L, Xu Y, et al. Time Optimal Control of Evolution Equations. Birkhäser Springer International Publishing AG; 2018.

[13]Yang B, Zhang MZ, Weng ZG. Time-optimal consensus control for enhanced thermal uniformity in multi-source microwave heating systems. International Journal of Heat and Mass Transfer. 2025; 252: 127426.

[14]Ning C, Wang Y, Yang DH. Time-Varying Bang-Bang Property of Time Optimal Controls for Heat Equation and Its Application. Systems and Control Letters. 2018; 112: 18–23.

[15]Wang GS, Xu Y, Zhang Y. Attainable Subspaces and the Bang-bang Property of Time Optimal Controls for Heat Equations. Siam Journal on Control and Optimization. 2014; 53(2): 592–621.

[16]Fattorini HO. The Time-Optimal Control Problem in Banach Spaces. Applied Mathematics and Optimization. 1974; 1(2): 163–188.

[17]Bonifacius L, Pieper K. Strong Stability of Linear Parabolic Time-optimal Control Problems. ESIAM Control Optimisation and Calculus of Variations. 2019; 25: 20152–20170.

[18]Phung KD, Wang G, Zhang X. On the Existence of Time Optimal Controls for Linear Evolution Equations. Discrete and Continuous Dynamical Systems Series B. 2007; 8: 925–941.

[19]Wang GS, Wang LJ. The Bang-bang Principle of Time Optimal Controls for the Heat Equation with Internal Controls. Systems Control Letter. 2007; 56: 709–713.

[20]Karl K, Wang JL. Time Optimal Control of the Heat Equation with Pointwise Control Constraints. ESAIM Control Optimisation and Calculus of Variations. 2013; 19(2): 460–485.

[21]Yao Q, Zheng X, Wang R, et al. Control of Thermal Uniformity in Microwave Heating Process by BPNN and Adaptive Particle Swarm Optimization. Heliyon. 2024; 10(21): e37971.

[22]Kunisch K, Wang LJ. Bang-bang Property of Time Optimal Control of Semilinear Parabolic Equation. Discrete Continuous Dynamic Systems. 2016; 36: 279–302.

[23]Anh CT, Nguyet TM. Time Optimal Control of the Unsteady 3D Navier-Stokes-Voigt Equations. Applied Mathematics and Optimization. 2019; 79(2): 397–426.

[24]Phung KD, Wang LJ, Zhang C. Bang-bang Property for Time Optimal Control of Semilinear Heat Equation. Annales de l’Institut Henri Poincaré (C), Analyse non linéaire. 2014; 31: 477–499.

[25]Khaneja N. Time Optimal Control of Coupled Spin Dynamics: A Global Analysis. Automatica. 2020; 111: 108639–108659.

[26]Gonalves SN, Albuquerque DMS, Pereira JCF. Modelling and Energy Efficiency Analysis of the Microwave Continuous Processing of Limestone. Journal of Cleaner Production. 2024; 5: 37–53.

[27]Wu S, Wang M, Jin M, et al. Uniform L1 Stability of the inelastic Bolzman Equation with Large External Force for Hard Potential. Discrete and Continuous Dynamical Systems. 2019; 12(4): 1005–1013.

[28]Rousseau JL, Lebeau G, Robbiano L. Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume I: Dirichlet Boundary Conditions on Euclidean Space. Birkhauser; 2022.

[29]Corella AD, Wachsmuth G. Stability and Genericity of Bang-Bang Controls in Affine Problems. SIAM Journal on Control and Optimization. 2024; 62(3): 1669–1689.

[30]Temporelli Q, Micheau P, Boisvert M. Control of an Electromechanical Clutch Actuator by a Parallel Adaptive Feedforward and Bang-Bang Controller: Simulation and Experimental Results. International Federation of Automatic Control. 2017; 50(1): 4787–4793.

[31]Jork NA, Simon JS. Analysis of Unregularized Optimal Control Problems Constrained by the Two-Dimensional Boussinesq System. SIAM Journal on Mathematical Analysis. 2025; 57(4): 3910–3952.

Early fault signals of the rolling bearing in the rotor are weak and present the characteristics of non-periodic and non-stationary; it is more difficult to carry out fault diagnosis on it. In this regard, this paper proposes a weak rolling bearing fault diagnosis algorithm based on whale optimization algorithm, simplistic geometry mode decomposition, and maximum correlated kurtosis deconvolution (WOA-SGMD-MCKD). Firstly, the vibration signal of the rotor platform is obtained, and the Symmetric Geometric Mode Decomposition (SGMD) is used to reconstruct the vibration signal. To obtain the best decomposition effect of the SGMD and overcome modal aliasing, the Whale Optimization Algorithm (WOA) is used to optimize the embedding dimension. Secondly, for the reconstructed vibration signal, the Maximum Correlated Kurtosis Deconvolution (MCKD) is used to extract its impulse component, and the WOA is used to optimize the filter length and deconvolution period of the MCKD so that the frequency envelope spectrum of the vibration signal can be obtained, which can provide the basis for the fault diagnosis of rolling bearings. Finally, the effectiveness and feasibility of the algorithm proposed are verified by a non-periodic and non-stationary simulation platform and rotor maneuvering platform in this paper.

Numerical solution of a 3D mathematical model for the progression of tumor angiogenic factor in a tissue

In this work, the movement of tumor angiogenic factor in a three-dimensional tissue is obtained by the Method of Lines. This method transforms a partial differential equation into a system of ordinary differential equations together with the initial and boundary conditions. The more the number of lines is increased, the more the accuracy of the method increases. This method results in very accurate numerical solutions for linear and non-linear problems in contrast with other existing methods. We present Matlab-generated figures, which are the movement of tumor angiogenic factor in porous medium and explain the biological importance of this progression. The computer codes are also provided.

Mathematical modelling and controllability analysis of fractional order coal mill pulverizer model

This paper investigates the controllability of nonlinear dynamical systems and their applications, with a focus on fractional-order systems and coal mill models. A novel theorem is proposed, providing sufficient conditions for controllability, including constraints on the steering operator and nonlinear perturbation bounds. The theorem establishes the existence of a contraction mapping for the nonlinear operator, enabling effective control strategies for fractional systems. The methodology is demonstrated through rigorous proof and supported by an iterative algorithm for controller design. Additionally, the controllability of a coal mill system represented as a nonlinear differential system, is analyzed. The findings present new insights into the interplay of fractional dynamics and nonlinear systems, offering practical solutions for real-world control problems.