Abstract This study investigates the finite-time output feedback stabilization problem for a class of unstable wave equations with Dirichlet boundary control, namely those with either unstable or anti-stable terms which all fall under the category of unstable wave equations. To address this problem, an auxiliary system that exhibits finite-time stability is initially proposed. We then construct a bounded and invertible backstepping transformation, through which the equivalence relationship between the original system and the auxiliary system is rigorously established. Leveraging this equivalence, we further extend the design of a finite-time state feedback controller. Thereafter, finite-time observers are developed to facilitate the derivation of an output feedback control law. Finally, numerical simulations are implemented to deliver visual verification of the theoretical conclusions.
Abstract The Ulam–Hyers–Rassias ($\mathcal{UHR}$) stability and trajectory (T-)controllability of fractional neutral stochastic integro-delay differential equations with impulses driven by time-changed Brownian motion are investigated in this study. The requirement to represent real-world systems with memory effects, hereditary characteristics, stochastic disturbances and delayed reactions is what drives this research. We use fractional calculus, stochastic analysis and M$\ddot{o}$nch’s fixed point theorem in a suitable Hilbert space scenario to prove adequate criteria for trajectory controllability. The model is robust against external disturbances since the $\mathcal{UHR}$ stability analysis guarantees that slight changes in the initial conditions result in restricted deviations in the behavior of the system over time. By improving the modeling of random time distortions, TCBM expands on conventional Brownian motion and more accurately represents stochastic influences in the real world. Additionally, we establish requirements for the presence and uniqueness of mild solutions and verify the theoretical findings using a real-world case backed by numerical simulations. With ramifications for complex dynamical networks, mathematical finance and control systems, the study’s conclusions extend stochastic control theory. The explicit formula for the option pricing model is created using the partial differential equation approach.
Abstract We study an optimal investment-consumption problem with proportional transaction costs formulated as follows: the investor allocates total wealth between a non-risky asset and a risky asset, with the possibility of consuming the non-risky asset and transferring funds between the two at any time. The central question is how the investor should optimally consume and rebalance the portfolio over time so as to maximize recursive utility, and how the corresponding value function can be characterized. Another modelling feature is the empirically realistic assumption that the risky asset is traded in a financial market with regime switches. In a continuous-time Epstein–Zin recursive utility, we formulate this optimal problem as a system of Hamilton–Jacobi–Bellman (HJB) variational inequalities. We then prove the existence and uniqueness of the value function as the viscosity solution, and our results provide further characterizations of the viscosity solution to the HJB system. We develop essential modifications of the common viscosity solution arguments and propose new analytical methods to deal with technical challenges brought by the strong non-linearity, lack of convexity and degenerate diffusion through modelling improvements.
Abstract Driven by the robust control requirements of complex time-delay systems in industrial processes, this paper investigates a model-free adaptive integral sliding mode predictive control algorithm for a class of nonlinear unknown time-delay systems affected by external disturbances. A one-step forward prediction error is utilized to obtain information on the delay present in the system, which is then used within the output predictor. Then, an output predictor based optimal extended parameter estimator is designed to establish the dynamic linearization model. An integral sliding mode predictive control algorithm is then proposed, which provides robustness to the impacts of the unknown time-delay and the external disturbances. The convergence of the estimated parameters and the stability of the closed-loop system are proved mathematically. Finally, the effectiveness of the proposed method is demonstrated via both simulation and experimental testing.
In this paper, we are interested in the dynamical analysis and boundary optimal control of counterflow heat exchanger in the case where the dynamics is described by hyperbolic partial differential equations. This topic is addressed by describing the dynamical model of the heat exchanger in an infinite-dimensional state-space, with bounded control and observation operators. First, we review the well-posedness problem and some fundamental properties relating to control theory, such as positivity, stability, reachability, stabilization and observability. These properties are complemented by spectral and pseudospectral analyses. Next, in the view of some results relating to the linear quadratic-optimal control of hyperbolic systems to which the model considered in this paper belongs, we introduce a certain state transformation that allows to put the abstract system in the lower triangular form so as to guarantee the uniqueness of solution of the operator Riccati equation. Finally, the design of an observer-based optimal control law coupled with an integral action is considered. The results are illustrated by means of numerical simulations for the set point tracking, and show the interest of the control approach proposed in this paper.
In this paper, we consider the stability and discretization of a tree-shaped string network, which models flexible space structures like large deployable antennas. We first show that the continuous system, described by coupled wave equations with boundary feedback, is exponentially stable. To achieve this, we design frequency-domain multipliers and use only two controllers-fewer than in previous studies. Next, we discretize the system using a reduced-order finite difference scheme and construct discrete multipliers to prove that the semi-discrete system preserves exponential stability. Numerical simulations support the theoretical findings. This work extends the frequency-domain multiplier method to discrete partial differential equation systems with network structures and demonstrates that exponential stability can be maintained after discretization.
In this paper, adaptive continuous-time algorithms with event-triggered mechanism are studied to solve the optimization problem. First, an event-triggered adaptive algorithm is introduced, and it is proven that this algorithm can effectively solve the optimization problem. Second, to solve the optimization problem, two event-triggered algorithms are proposed, one considering uniform quantization information and the other addressing external disturbances. It is demonstrated that the states of the multi-agent systems practically converge to the global optimal point. Three numerical cases demonstrate the effectiveness of the relevant results.
This paper addresses the stabilization problem of a star-shaped network of strings with joint anti-damping, where each string may have different wave speeds and lengths. By designing invertible transformations, the wave system is converted into an equivalent system with conservative linkage. We design feedback controllers with collocated and non-collocated observations to cancel the instabilities at the control ends and obtain the target system. Through the Riesz basis method and the PDE approach, the well-posedness and exponential stability of the non-dissipative closed-loop system are established. The effectiveness of the proposed feedback control law is verified by numerical simulations.
In this work, the semigroup operator for a two-dimensional transport-reaction model that is described by a first-order hyperbolic system is derived. Two different cases of one spatially varying velocity and two spatially varying velocities are considered. A discrete in-time model setting is formulated without any model reduction or approximation, typically present in controller designs of distributed parameter systems. This setting is obtained by an exact time discretization that makes use of the semigroup operator. Additionally, adjoint operators, which are utilized in the design of the controller, are derived. The system is assumed to have a point observation and a distributed actuation function. A model predictive controller has been designed for this system to account for the constraints present in the input and output. The controller's efficiency in achieving convergence and satisfying both input and output constraints is demonstrated through the use of numerical simulations.
This paper studies the cooperative tracking consensus problem of open multi-agent systems (MASs), where agents may dynamically enter or leave the system as the topology switches. A novel method of extending the structure matrix is proposed to deal with the unmatched dimensions of dynamic topologies. The switching topology is only required to be uniformly connected on average. A cooperative tracking consensus control strategy is proposed on the basis of the agent's own information and dynamically increasing or decreasing neighbours' information. The sufficient conditions based on the uniformly exponential stability of the Lyapunov function and a sequence of differential inequalities, as well as an average dwell time condition, are derived to guarantee the tracking consensus of open MASs. Moreover, this result is applied to the tracking consensus problem of the virtually coupled train set under dynamic coupling/decoupling. Finally, simulations are provided to show the effectiveness and superiority of the proposed tracking control scheme.
Extremum seeking (ES) is an effective real-time optimization method for partial differential equation (PDE) systems in cascade with non-linear quadratic maps. To address PDEs in the feedback loop, a boundary control law and a re-design of the additive probing signal are mandatory. The latter, commonly called 'trajectory generation' or 'motion planning', involves designing perturbation signals that anticipate their propagation through PDEs. Specifically, this requires solving motion planning problems for systems governed by parabolic and hyperbolic PDEs. Physics-informed neural networks (PINNs) is a powerful tool for solving PDEs by embedding physical laws as constraints in the neural network's loss function, enabling efficient solutions for high-dimensional, non-linear and complex problems. This paper proposes a novel construction integrating PINN and ES, automating the motion planning process for specific PDE systems and eliminating the need for case-by-case analytical derivations. The proposed strategy efficiently extracts perturbation signals, optimizing the PDE system.
In this paper, we consider the problem of state and parameter estimation problem for an Ordinary differential equation-Partial differential equation (ODE-PDE) cascade system, in which the output of the ODE dynamics serves as the driving force for the PDE dynamics, governed by a wave equation. We develop an adaptive observer that combines a state observer with a least-squares parameter adaptation law and six auxiliary filters. A rigorous analysis of well-posedness and convergence properties is subsequently conducted. The estimated parameter value converged to the actual value exponentially under a specific persistent excitation condition. Numerical results demonstrate the effectiveness of the estimation scheme.
This work tackles the output regulation problem for a reaction-diffusion system subject to input delay and unknown multi-channel disturbances with unknown frequencies and amplitudes. A control framework is developed by combining modal decomposition with a dual-observer scheme, which enables real-time estimation of states and disturbances through a state observer and an adaptive disturbance estimator. Unlike existing methods that address delay compensation or disturbance rejection in isolation, the proposed tracking-error-based control law achieves both simultaneously, thereby guaranteeing exponential convergence of the output to the reference signal. Numerical simulations confirm the effectiveness of the output-feedback strategy.
In this paper, we study the well-posedness and the input-to-state type stability of a one-dimensional fluid-particle interaction system. A distinctive feature, not yet considered in the ISS literature, is that our system involves a free boundary. More precisely, the fluid is described by the viscous Burgers equation, and the motion of the particle obeys Newton second law. The point mass is subject to both a feedback control and an open-loop control. We first establish the well-posedness of the system for any open-loop input in the L2(0, infinity) space. Assuming the input also belongs to the L1(0,infinity) space, we prove that the particle's position remains uniformly bounded and that the system is input-to-state type stable. The proof is based on the construction of a Lyapunov functional derived from a special test function.
Sufficient conditions for the existence and non-existence of a fixed-order controller stabilizing a polytope of systems are addressed in this paper. To achieve this goal, a solution based on the Hermite-Biehler theorem and an adaptation of the Edge theorem allows us to treat this issue as a simple interpolation problem of stable polynomials. Then, feasibility and infeasibility certificates are established from a reformulation of the interpolation conditions, which enable us to conclude the existence or non-existence of a compensator directly from the points to interpolate.
This paper models and simulates conservative linear acoustic propagation in axisymmetric pipes with time-space-dependent cross-sections. It extends the horn equation to moving-wall bores. The physical equations (partial differential equations) satisfy a power balance, allowing their formulation as a Port-Hamiltonian system. A two-step numerical method is proposed, which preserves mass, momentum and power conservation in discrete domains. Spatial discretization yields a system of Ordinary Differential Equations (ODE) that satisfies known acoustic characteristics for static walls and ensures power-balanced propagation for controlled dynamic walls. Time discretization via the discrete-gradient method results in a discrete time-space model. Simulations validate conservative propagation and resonances for static walls and capture dynamic vocal tract acoustics during articulation.
This paper focuses on stabilizing the unstable semi-linear parabolic stochastic partial differential system (SPDS) driven by L & eacute;vy noise. The boundary control for SPDS is designed using backstepping method through Volterra integral transformation. With the help of linear matrix inequality and Lyapunov analysis, the stability of the transformed system has been presented. Since the selected transformation is invertible, the stability of the transformed system implies the stability of the original system with designed control. The result guarantees the system's asymptotic stability under Neumann boundary conditions. Finally, the effectiveness of the proposed results are validated through numerical examples by demonstrating the stabilizability of the system under designed control.
We develop switched predictor-feedback control laws to achieve global asymptotic stabilization for both finite-dimensional non-linear systems and infinite-dimensional reaction-diffusion partial differential equations (PDEs) subject to input delay and almost state quantization. The proposed approach generalizes the predictor-feedback framework, for non-linear systems and reaction diffusion PDEs, introduced by Krstic in the seminal papers Krstic (2009a, Systems Control Lett., 58, 773-82) and Krstic (2009b, IEEE Trans. Autom. Control, 55, 287-303), by incorporating quantized measurements of the plant and actuator states into the predictor state formulation. To address constraints imposed by quantization, we introduce a dynamic switching strategy that adjusts the quantizer's range in a piecewise constant manner-initially capturing large states and subsequently refining precision to reduce quantization error. Stability of the closed-loop systems is established using backstepping transformations combined with small-gain and input-to-state stability arguments. The developed results are further extended to account for almost input quantization, for both non-linear ordinary differential equation and reaction-diffusion PDE systems with input delay.
This article discusses the sliding mode control problem for a kind of impulsive reaction-diffusion systems (RDSs) with state delay and time-varying disturbance. A novel sliding surface containing regulation terms is constructed. The designed sliding mode controller can remove the impulsive effects on RDSs and drive the closed-loop state trajectories onto the specified sliding surface in finite time. By means of Linear matrix inequalities, the robust exponential stability of sliding motion is studied. Finally, a numerical example is provided to verify the availability of our findings.
