Semilinear Hyperbolic Systems Research Articles

Event-Triggered Boundary Control of Semilinear Hyperbolic Systems

We present an event-triggered boundary control scheme for hyperbolic systems. The trigger condition is based on predictions of the state on determinate sets, and the control input is updated only when the predictions deviate from the reference by a given margin. Nominal closed-loop stability, the absence of Zeno behaviour, and robustness to uncertainty and disturbances, are all established analytically. For the special case of linear systems, the trigger condition can be expressed in closed-form as an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L_{2}$</tex-math></inline-formula> -scalar product of kernels with the distributed state. The presented controller can also be combined with existing observers to solve the event-triggered output-feedback control problem. A numerical simulation demonstrates the effectiveness of the proposed approach.

Wong–Zakai approximations for quasilinear systems of Itô's-type stochastic differential equations driven by fBm with H &gt; 1 2

In a recent paper, Lanconelli and Scorolli 10 extended to the multidimensional case a Wong–Zakai-type approximation for Itô stochastic differential equations (SDEs) proposed by Øksendal and Hu 7 . The aim of this paper is to extend the latter result to system of SDEs of Itô type driven by fractional Brownian motion (fBm) like those considered by Hu 6 . This extension is not trivial since the covariance structure of the fBm precludes us from using the same approach as that used by Lanconelli and Scorolli. Instead we employ a truncated Cameron–Martin expansion as the approximation for the fBm. We are naturally led to the investigation of a semilinear hyperbolic system of evolution equations in several space variables that we utilize for constructing a solution of the Wong–Zakai approximated systems. We show that the law of each element of the approximating sequence solves in the sense of distribution a Fokker–Planck equation and that the sequence converges to the solution of the Itô equation, as the number of terms in the expansion goes to infinite.

Unknown Input Observer Design for a Class of Semilinear Hyperbolic Systems With Dynamic Boundary Conditions

The problem of unknown input observer design is considered for coupled PDE/ODE systems subject to incremental sector bounded nonlinearities and unknown boundary inputs. Assuming available measurements at the boundary of the distributed domain, the synthesis of the unknown input observer is based on Lyapunov methods and recast in terms of some matrix inequalities. Numerical simulations support and confirm the theoretical findings, illustrating the robust estimation performances of the proposed nonlinear unknown input observer.

Solvability of One-Dimensional Semilinear Hyperbolic Systems and Sine-Gordon Equation

We study semilinear hyperbolic Eq. (1.1). We derive an explicit solution representation for some nonlinear terms F and G. For other nonlinear terms, it is shown that the solutions of the equations are related with the variable coefficient sine-Gordon equation.

Averaged Control Problems Governed by a Semilinear Distributed Systems

In this work, we consider the regional averaged controllability (RAC) problem governed by a class of semilinear hyperbolic systems. We start by giving the definitions of the exact and approximate RAC systems. After that, we state the problem of RAC for semilinear systems. We propose two methods of solution: using a condition of the analytical operator to the nonlinear part of the system to characterize the optimal control via the fixed point theorem and the Hilbert Uniqueness Method (HUM) with an asymptotic condition on the nonlinear part to find the optimal control of the considered problem. Finally, we present a numerical example to show the effectiveness of the main results.

Solvability of certain one-dimensional semilinear hyperbolic systems and hyperbolic Liouville equation

We study the initial-value problem for semilinear hyperbolic systems in one space dimension. When the power of nonlinearity is even , we have global existence of solution which is related with the repulsive Liouville equation. When n = 2k−1, we may have a finite time blow-up solution which is related with the attractive Liouville equation.

Leaderless Consensus of Semilinear Hyperbolic Multiagent Systems with Semipositive or Seminegative Definite Convection

This paper deals with a leaderless consensus of semilinear first-order hyperbolic partial differential equation-based multiagent systems (HPDEMASs). A consensus controller under an undirected graph is designed. Dealing with different convection assumptions, two different boundary conditions are presented, one right endpoint and the other left endpoint. Two sufficient conditions for leaderless consensus of HPDEMAS are presented by giving the gain range in the case of the symmetric seminegative definite convection coefficient and the semipositive definite convection coefficient, respectively. Two examples are presented to show the effectiveness of the control methods.

On the solvability of one boundary value problem for one class of higher-order semilinear hyperbolic systems

On the solvability of one boundary value problem for one class of higher-order semilinear hyperbolic systems

Blow‐up and lifespan estimates for Nakao's type problem with nonlinearities of derivative type

In the present paper, we investigate blow‐up and lifespan estimates for a class of semilinear hyperbolic coupled system in with , which is a part of the so‐called Nakao's type problem weakly coupled a semilinear damped wave equation with a semilinear wave equation with nonlinearities of derivative type. By constructing two time‐dependent functionals and employing an iteration method for unbounded multiplier with slicing procedure, the results of blow‐up and upper bound estimates for the lifespan of energy solutions are derived. The model seems to be hyperbolic‐like instead of parabolic‐like. Particularly, the blow‐up result for one‐dimensional case is optimal.

Prescribed-time control for a class of semilinear hyperbolic PDE-ODE systems

A prediction-based controller is shown to achieve prescribed-time stabilization of a nonlinear infinite-dimensional system, which consists of a general boundary controlled first-order semilinear hyperbolic PDE that is bidirectionally interconnected with nonlinear ODEs at its unactuated boundary. The approach uses a coordinate transformation to map the nonlinear system into a form suitable for control. In particular, this transformation is based on predictions of system trajectories, which can be obtained by solving a general nonlinear Volterra integro-differential equation. Then, a prediction-based controller is designed to stabilize the system in prescribed-time. Numerical simulations illustrate the performance of both the prescribed-time controller and an asymptotically stabilizing one, which follows as a special case.

Numerical boundary control for semilinear hyperbolic systems

&lt;p style='text-indent:20px;'&gt;This work is devoted to the design of boundary controls of physical systems that are described by semilinear hyperbolic balance laws. A computational framework is presented that yields sufficient conditions for a boundary control to steer the system towards a desired state. The presented approach is based on a Lyapunov stability analysis and a CWENO-type reconstruction.&lt;/p&gt;

Time-domain decomposition for optimal control problems governed by semilinear hyperbolic systems with mixed two-point boundary conditions

Abstract In this article, we study the time-domain decomposition of optimal control problems for systems of semilinear hyperbolic equations and provide an in-depth well-posedness analysis. This is a continuation of our work, Krug et al. (2021) in that we now consider mixed two-point boundary value problems. The more general boundary conditions significantly enlarge the scope of applications, e.g., to hyperbolic problems on metric graphs with cycles. We design an iterative method based on the optimality systems that can be interpreted as a decomposition method for the original optimal control problem into virtual control problems on smaller time domains.

Sampled-data in space nonlinear control of scalar semilinear parabolic and hyperbolic systems

The paper describes a novel method of sampled-data in space (spatial variable) nonlinear control of scalar semilinear parabolic and hyperbolic systems with unknown parameters, distributed disturbances and finite number of measurements along the spatial variable. Differently from recent results based on piecewise constant control laws, the proposed one is used piecewise nonlinear functions choosing by designer for providing some properties in the closed-loop system. In particular, we propose several types of functions providing reduced control. The gain design in the control law is found as a solution of linear matrix inequalities with minimum ultimate bound guarantee. The simulations confirm theoretical results and show the efficiency of the proposed control scheme compared with some existing ones.

Transient gas pipeline flow: analytical examples, numerical simulation and a comparison to the quasi-static approach

The operation of gas pipeline flow with high pressure and small Mach numbers allows to model the flow by a semilinear hyperbolic system of partial differential equations. In this paper we present a number of transient and stationary analytical solutions of this model. They are used to discuss and clarify why a PDE model is necessary to handle certain dynamic situations in the operation of gas transportation networks. We show that adequate numerical discretizations can capture the dynamical behavior sufficiently accurate. We also present examples that show that in certain cases an optimization approach that is based on multi-period optimization of steady states does not lead to approximations that converge to the optimal state.

Wong–Zakai approximations for quasilinear systems of Itô’s type stochastic differential equations

We extend to the multidimensional case a Wong–Zakai-type theorem proved by Hu and Øksendal (1996) for scalar quasi-linear Itô stochastic differential equations (SDEs). More precisely, with the aim of approximating the solution of a quasilinear system of Itô’s SDEs, we consider for any finite partition of the time interval [0,T] a system of differential equations, where the multidimensional Brownian motion is replaced by its polygonal approximation and the product between diffusion coefficients and smoothed white noise is interpreted as a Wick product. We remark that in the one dimensional case this type of equations can be reduced, by means of a transformation related to the method of characteristics, to the study of a random ordinary differential equation. Here, instead, one is naturally led to the investigation of a semilinear hyperbolic system of partial differential equations that we utilize for constructing a solution of the Wong–Zakai approximated systems. We show that the law of each element of the approximating sequence solves in the sense of distribution a Fokker–Planck equation and that the sequence converges to the solution of the Itô equation, as the mesh of the partition tends to zero.

An optimal control problem by a first-order hyperbolic system with delay and integral control constraints

A non-classic optimality condition and numerical algorithm for smooth boundary control in semi-linear first-order hyperbolic system with delay are presented. An additional integral control constraint is considered. The suggested approach is based on special variations of admissible continuously differentiable controls. These variations lead to a necessary optimality condition which is a base for a numerical method. An illustrative example demonstrates the effectiveness of the proposed approach.

Time-Domain Decomposition for Optimal Control Problems Governed by Semilinear Hyperbolic Systems

In this article, we extend the time-domain decomposition method described by Lagnese and Leugering [Systems Control Lett., 48 (2003), pp. 229--242] to semilinear optimal control problems for hyperb...

Stability for semilinear hyperbolic coupled system with frictional and viscoelastic localized damping

In this paper we prove stability results for a semilinear hyperbolic coupled system subject to a viscoelastic localized damping acting in the first equation and a frictional localized one acting in the second equation of the system. We divide the proof into two parts. In the first part the equations are posed in a homogeneous medium Ω with the damping acting in a boundary neighbourhood. In the second part the equations posed in an inhomogeneous medium Ω with the damping acting in a collar of the boundary and in an appropriate mesh in the interior. Due to the assumptions on the nonlinear function involving the frictional damping, general decay rates are obtained. To prove the results we used the tools of the Microlocal Analysis Theory.

Output feedback boundary control of heterodirectional semilinear hyperbolic systems

We solve the problem of stabilizing a general class of 1-d semilinear hyperbolic systems with an arbitrary number of states convecting in each direction and with the actuation and sensing restricted to one boundary. The control design is based on the dynamics on the characteristic lines along which the inputs propagate through the domain and the predictability of states in the interior of the domain up to the time they are affected by the inputs. In the context of broad solutions, the state-feedback controller drives systems with globally Lipschitz nonlinearities from an arbitrary initial condition to the origin in minimum time. Alternatively, it is possible to satisfy a tracking objective at the uncontrolled boundary or, for systems with C1-coefficients and initial conditions, to design the control inputs to obtain classical C1-solutions that also reach the origin in finite time. Further, we design an observer that estimates the distributed state from boundary measurements only. The observer combined with the state-feedback controller solves the output-feedback control problem.

On L∞ stabilization of diagonal semilinear hyperbolic systems by saturated boundary control

This paper considers a diagonal semilinear system of hyperbolic partial differential equations with positive and constant velocities. The boundary condition is composed of an unstable linear term and a saturated feedback control. Weak solutions with initial data in L2([0, 1]) are considered and well-posedness of the system is proven using nonlinear semigroup techniques. Local L∞ exponential stability is tackled by a Lyapunov analysis and convergence of semigroups. Moreover, an explicit estimation of the region of attraction is given.