Approximation of exact controls for semilinear wave and heat equations through space-time methods

The initial value problems for some semilinear wave and heat equations are investigated in two space dimensions. By proving the existence of ground state, strong instability of standing waves for the associated wave and heat equations are obtained.

The numerous examples of Chapters 2 and 3 demonstrate how the classical method of separation of variables is used to generate solutions of Laplace’s and the heat equation over specified finite domains. Utilizing the modern methods inherent in Green’s functions and Green’s Theorem, solutions of Poisson’s equation over an arbitrary finite domain are established. Can such modern methods be applied to the heat equation? Unlike Laplace’s or Poisson’s equations which are static in time, the heat equation evolves in time. Indeed, the heat equation is the prototype for what are commonly called evolution equations. Nonlinear evolution equations are examined in Chapter 6 and a special nonlinear system is detailed in Chapter 7. For now, the focus will be on the linear heat and wave equations. Instead of Green’s Theorem, one of the most powerful ideas in modern mathematics is applied: The Fourier transform. The program of study for this chapter then is to define the Fourier transform, develop its properties, apply it to the heat and wave equations, and derive analytic solutions.

We report the problem of feedback stabilization along a path of steady-states, and of exact boundary controllability of semilinear one-dimensional heat and wave equations, investigated in [5], [6]. The main result is that it is possible to move from any steady-state to any other one by means of a boundary control, provided that they are in the same connected component of the set of steady-states. The proof is based on an effective feedback stabilization procedure which is efficiently implementable.

In this paper we address the problem of null-controllability of heat equations in two different cases: (a) The semilinear heat equation in bounded domains and (b) The linear heat equation in the half line. Concerning the first problem (a) we show that a number of systems in which blow-up arises may be controlled by means of external forces which are localized in an arbitrarily small open set. In the frame of problem (b) we prove that compactly supported initial data may not be driven to zero if the control is supported in a bounded set. This shows that although the velocity of propagation in the heat equation is infinite, this is not sufficient to guarantee null-controllability properties.We also include a list of open problems.KeywordsHeat EquationMoment ProblemApproximate ControllabilityCarleman EstimateOpen Nonempty SubsetThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We consider the linear heat equation with potential in a n‐dimensional thin cilinder Ωε=Ω×(0,ε) where Ω is a bounded open smooth set of $\mathbb{R}^{n-1}$ with n≥2 and ε is a small parameter. We study the null controllability problem when the control acts in a cylindrical region ωε=ω×(0,ε), where ω⊂Ω is an open and non‐empty subset of Ω. We prove that, under appropriate boundary conditions, for a suitable class of potentials the heat equation is uniformly null controllable as ε→0. We also prove the convergence of the controls to a null control for the n−1‐dimensional heat equation in Ω. Similar results are proved for the semilinear heat equation with globally Lipschitz nonlinearities.

The aim of this paper is threefold. The first and also main purpose is to provide numerical evidence for the conjecture proposed by Bizoń et al. [“Dynamics near the threshold for blowup in the one-dimensional focusing nonlinear Klein-Gordon equation,” J. Math. Phys. 52, 103703 (2011)]10.1063/1.3645363 that the blowup evolution of spherically symmetric semilinear Klein-Gordon equations is similar to the evolution of spherically symmetric semilinear wave equations, i.e., the mass term can be neglected when the amplitude of a solution grows. The second aim is to describe the relationship between different types of blowup for energy critical semilinear wave equations. The third goal is to present numerical evidence for the fact that the special class of self-similar profiles of semilinear wave equations found by Kycia [“On self-similar solutions of semilinear wave equations in higher space dimensions,” Appl. Math Comput. 217, 9451–9466 (2011)]10.1016/j.amc.2011.04.039 play the same role in the evolution of semilinear wave and Klein-Gordon equations as the previously known ordinary profiles. All the results are presented in spherical symmetry.

Shooting methods for numerical solutions of exact controllability problems constrained by linear and semilinear wave equations with local distributed controls

The solutions to certain stochastic partial differential equations with linear Gaussian noise constitute interesting examples of self-similar processes. In this chapter we analyze these classes of self-similar processes. We focus on the solution to the linear heat and wave equation driven by a Gaussian noise which behaves as a Brownian motion or fractional Brownian motion with respect to the time variable and is white or colored with respect to the space variable. We consider various aspects of these self-similar processes. In particular we present the conditions for the existence of the solution, the sharp regularity of their trajectories, we study the law of the solution to the linear heat equation and its connection with the bifractional Brownian motion.

The use of the local truncation error to improve arbitrary-order finite elements for the linear wave and heat equations

Existence and nonexistence of time-global solutions to damped wave equation on half-line

The exact controllability of the semilinear wave equation $$y_{tt}-y_{xx}+ f(y)=0$$ , $$x\in (0,1)$$ assuming that f is locally Lipschitz continuous and satisfies the growth condition $$\limsup _{\vert r\vert \rightarrow \infty } \vert f(r)\vert /(\vert r\vert \ln ^{p}\vert r\vert )\leqslant \beta $$ for some $$\beta $$ small enough and $$p=2$$ has been obtained by Zuazua (Ann Inst H Poincaré Anal Non Linéaire 10(1):109–129, 1993). The proof based on a non-constructive fixed point arguments makes use of precise estimates of the observability constant for a linearized wave equation. Under the above asymptotic assumption with $$p=3/2$$ , by introducing a different fixed point application, we present a simpler proof of the exact boundary controllability which is not based on the cost of observability of the wave equation with respect to potentials. Then, assuming that f is locally Lipschitz continuous and satisfies the growth condition $$\limsup _{\vert r\vert \rightarrow \infty } \vert f^\prime (r)\vert /\ln ^{3/2}\vert r\vert \leqslant \beta $$ for some $$\beta $$ small enough, we show that the above fixed point application is contracting yielding a constructive method to approximate the controls for the semilinear equation. Numerical experiments illustrate the results. The results can be extended to the multi-dimensional case and for nonlinearities involving the gradient of the solution.

In this paper, we study the null controllability of linear heat and wave equations with spatial nonlocal integral terms. Under some analyticity assumptions on the corresponding kernel, we show that the equations are controllable. We employ compactness-uniqueness arguments in a suitable functional setting, an argument that is harder to apply for heat equations because of its very strong time irreversibility. Some possible extensions and open problems concerning other nonlocal systems are presented.

In this paper, several abstract results concerning the controllability of semilinear evolution systems are obtained. First, approximate controllability conditions for semilinear systems are obtained by means of a fixed-point theorem of the Rothe type; in this case, the compactness of the linear operator is assumed. Next, the exact controllability of semilinear systems with nonlinearities having small Lipschitz constants is derived by means of the Banach fixed-point theorem; in this case, the compactness of the operators is not assumed. In both cases, it is proven that the controllability of the linear system implies the controllability of the associated semilinear system. Finally, these abstract results are applied to the controllability of the semilinear wave and heat equations.

With the rise of distributed and connected modern engineering systems, the need for safe and reliable distributed parameter systems is becoming increasingly prominent. However, fault detection of partial differential equation systems remains relatively unexplored compared to its ordinary differential equation counterpart. In this letter, we attempt to advance such research by addressing the fault detectability problem for linear deterministic heat equations with additive faults. Fault detectability examines whether a fault is detectable given a system model and measured outputs. Essentially, similar to controllability and observability, fault detectability is an intrinsic property of the system and has to be verified before designing diagnostic algorithms. In this context, we derive fault detectability conditions for heat equations by utilizing the singular values of fault-to-output transfer function. We illustrate the proposed detectability conditions on battery thermal fault diagnosis case study.

Stability analysis of a linear system coupling wave and heat equations with different time scales