The optimal time for the controllability of linear hyperbolic systems in one dimensional space with one-side controls has been obtained recently for time-independent coefficients in our previous works. In this paper, we consider linear hyperbolic systems with time-varying zero-order terms. We show the possibility that the optimal time for the null-controllability becomes significantly larger than the one of the time-invariant setting even when the zero-order term is indefinitely differentiable. When the analyticity with respect to time is imposed for the zero-order term, we also establish that the optimal time is the same as in the time-independent setting.
In this paper, we discuss our recent works on the null-controllability, the exact controllability, and the stabilization of linear hyperbolic systems in one dimensional space using boundary controls on one side for the optimal time. Under precise and generic assumptions on the boundary conditions on the other side, we first obtain the optimal time for the null and the exact controllability for these systems for a generic source term. We then prove the null-controllability and the exact controllability for any time greater than the optimal time and for any source term. Finally, for homogeneous …
In this paper, we study the local exact boundary controllability of entropy solutions to a class linearly degenerate hyperbolic systems of conservation laws with constant multiplicity. The authors prove the two-sided boundary controllability, one-sided boundary controllability and two-sided controllability with less controls, by applying the strategy used originally for classical solutions with essential modifications. Our constructive method is based on the well-posedness of semi-global solutions constructed by the limit of $ \e $-approximate front tracking solutions to the mixed initial-boun…
We are concerned about the controllability of a general linear hyperbolic system of the form $\partial_t w (t, x) = Σ(x) \partial_x w (t, x) + γC(x) w(t, x) $ ($γ\in \mR$) in one space dimension using boundary controls on one side. More precisely, we establish the optimal time for the null and exact controllability of the hyperbolic system for generic $γ$. We also present examples which yield that the generic requirement is necessary. In the case of constant $Σ$ and of two positive directions, we prove that the null-controllability is attained for any time greater than the optimal time for all…
This paper is devoted to the controllability of a general linear hyperbolic system in one space dimension using boundary controls on one side. Under precise and generic assumptions on the boundary conditions on the other side, we previously established the optimal time for the null and the exact controllability for this system for a generic source term. In this work, we prove the null-controllability for any time greater than the optimal time and for any source term. Similar results for the exact controllability are also discussed.
Approximation of Degenerate Hyperbolic Equations with Interior Degeneracy and Applications to Controllability
School of Mathematics and Statistics, Central South University（中南大学数学与统计学院）；Academy of Mathematics and Systems Science, Academia Sinica（中国科学院数学与系统科学研究院）
In this paper, we study the exact boundary controllability of the linear fourth-order Schrödinger equation, with variable physical parameters and clamped boundary conditions on a bounded interval. The control acts on the first spatial derivative at the left endpoint. We prove that this control system is exactly controllable at any time $T>0$. The proofs are based on a detailed spectral analysis and on the use of nonharmonic Fourier series.
In the paper, the problems of controllability and approximate controllability are studied for the control system $w_t=\frac{1}ρ\left(kw_x\right)_x+γw$, $\left.\left(\sqrt{\frac{k}ρ}w_x\right)\right|_{x=0}=u$, $x>0$, $t\in(0,T)$, where $u$ is a control, $u\in L^\infty(0,T)$. It is proved that each initial state of the control system is approximately controllable to any target state in a given time $T>0$. To obtain this result, the transformation operator generated by the equation data $ρ$, $k$, $γ$ is applied. The results are illustrated by examples.
We provide global and semi-global controllability results for hyperbolic conservation laws on a bounded domain, with a general (not necessarily convex)flux and a time-dependent source term acting as a control. The results are achieved for, possibly critical, both continuously differentiable states and BV states. The proofs are based on a combination of the return method and on the analysis of the Riccati equaiton for the space derivative of the solution.
The goal of the present article is to study controllability properties of mixed systems of linear parabolic-transport equations, with possibly non-diagonalizable diffusion matrix, on the one-dimensional torus. The equations are coupled by zero or first order coupling terms, with constant coupling matrices, without any structure assumptions on them. The distributed control acts through a constant matrix operator on the system, so that there might be notably less controls than equations, encompassing the case of indirect and simultaneous controllability. More precisely, we prove that in small ti…
This paper is devoted to study exact controllability of two one-dimensional coupled wave equations with first-order coupling terms with coefficients depending on space and time. We give a necessary and sufficient condition for both exact controllability in high frequency in the general case and the unique continuation in the cascade case.
We consider the finite-time stabilization of homogeneous quasilinear hyperbolic systems with one side controls and with nonlinear boundary condition at the other side. We present time-independent feedbacks leading to the finite-time stabilization in any time larger than the optimal time for the null controllability of the linearized system if the initial condition is sufficiently small. One of the key technical points is to establish the local well-posedness of quasilinear hyperbolic systems with nonlinear, non-local boundary conditions.
For linear control systems in discrete time controllability properties are characterized. In particular, a unique control set with nonvoid interior exists and it is bounded in the hyperbolic case. Then a formula for the invariance pressure of this control set is proved.
Let us consider a nonlinear degenerate reaction-diffusion equation with application to climate science. After proving that the solution remains nonnegative at any time, when the initial state is nonnegative, we prove the approximate controllability between nonnegative states at any time via multiplicative controls, that is, using as control the reaction coefficient.
In this paper, we consider a linear hybrid system which is composed of $N+1$ non-homogeneous thin rods connected by $N$ interior-point masses with a Dirichlet boundary condition on the left end, and Dirichlet control on the right end. Using a detailed spectral analysis and the moment theory, we prove that this system is null controllable at any positive time $T$. To this end, firstly, we implement the Wronskian technique to obtain the characteristic equation for the eigenvalues $(λ_{n})_{n\in\N^*}$ associated with this system. Secondly, we provide that the eigenvalues $(λ_{n})_{n\in\N^*}$ inte…
We study the controllability of a coupled system of linear parabolic equations, with non-negativity constraint on the state. We establish two results of controllability to trajectories in large time: one for diagonal diffusion matrices with an "approximate" nonnegativity constraint, and a another stronger one, with "exact" nonnegativity constraint, when all the diffusion coefficients are equal and the eigenvalues of the coupling matrix have nonnegative real part. The proofs are based on a "staircase" method. Finally, we show that state-constrained controllability admits a positive minimal time…
In this paper we study the exact boundary controllability for the following Boussinesq equation with variable physical parameters: \begin{array}{lll} ρ(x)y_{tt}=-(σ(x)y_{xx})_{xx}+(q(x)y_x)_x-(y^2)_{xx},&&t>0,~x\in(0,l),\\ y(t,0)=σ(l)y_{xx}(t,0)=y(t,l)=0,~~σ(l)y_{xx}(t,l)=u(t)&&t>0, \end{array} where $l>0$, the coefficients $ρ(x)>0,σ(x)>0 $, $q(x)\geq0$ in $[0,l]$ and $u$ is the control acting at the end $x=l$. We prove that the linearized problem is exactly controllable in any time $T>0$. Our approach is essentially based on a detailed spectral analysis together with the moment method. Furthe…
In this paper, we prove the local-controllability to positive constant trajectories of a nonlinear system of two coupled ODE equations, posed in the one-dimensional spatial setting, with nonlocal spatial nonlinearites, and using only one localized control with a moving support. The model we deal with is derived from the well-known nonlinear reaction-diffusion Gray-Scott model when the diffusion coefficient of the first chemical species $d_u$ tends to $0$ and the diffusion coefficient of the second chemical species ${d_v}$ tends to $+ \infty$. The strategy of the proof consists in two main step…
We consider both the internal and boundary controllability problems for wave equations under non-negativity constraints on the controls. First, we prove the steady state controllability property with nonnegative controls for a general class of wave equations with time-independent coefficients. According to it, the system can be driven from a steady state generated by a strictly positive control to another, by means of nonnegative controls, when the time of control is long enough. Secondly, under the added assumption of conservation and coercivity of the energy, controllability is proved betwee…
In this paper we study the global approximate multiplicative controllability for nonlinear degenerate parabolic Cauchy problems. In particular, we consider a one-dimensional semilinear degenerate reaction-diffusion equation in divergence form governed via the coefficient of the \-reaction term (bilinear or multiplicative control). The above one-dimensional equation is degenerate since the diffusion coefficient is positive on the interior of the spatial domain and vanishes at the boundary points. Furthermore, two different kinds of degenerate diffusion coefficient are distinguished and studied …
In this article, we study the boundary null-controllability properties of the one-dimensional linearized (around $(Q_0,V_0)$ with constants $Q_0>0, V_0>0$) compressible Navier-Stokes equations in the interval $(0,1)$ when a control function is acting either on the density or velocity component at one end of the interval. We first prove that the linearized system, with a Dirichlet boundary control on the density component and homogeneous Dirichlet boundary conditions on the velocity component, is null-controllable in $H^s_{per}(0,1) \times L^2(0,1)$ for any $s > 1/2$ provided the time $T > 1$, …
We consider the linear degenerate wave equation, on the interval $(0, 1)$ $$ w_{tt} - (x^αw_x)_x = p(t) μ(x) w, $$ with bilinear control $p$ and Neumann boundary conditions. We study the controllability of this nonlinear control system, locally around a constant reference trajectory, the ground state. We prove that, generically with respect to $μ$, any target close to the ground state in the $H^3\times H^2$ topology (suitably adapted to the underlying degenerate operator) is reachable in time $T > \frac{4}{2-α}$, with controls in $L^2((0, T ),\mathbb R)$. Under some classical and generic assum…
We prove that the KdV equation on the circle remains exactly controllable in arbitrary time with localized control, for sufficiently small data, also in presence of quasi-linear perturbations, namely nonlinearities containing up to three space derivatives, having a Hamiltonian structure at the highest orders. We use a procedure of reduction to constant coefficients up to order zero, classical Ingham inequality and HUM method to prove the controllability of the linearized operator. Then we prove and apply a modified version of the Nash-Moser implicit function theorems by Hörmander.
We consider the observability problem for non-autonomous evolution systems (i.e., the operators governing the system depend on time). We introduce an averaged Hautus condition and prove that for skew-adjoint operators it characterizes exact observability. Next, we extend this to more general class of operators under a growth condition on the associated evolution family. We give an application to the Schrödinger equation with time dependent potential and the damped wave equation with a time dependent damping coefficient.
Motivated by the self-pursuit of controlled objects, we consider the exact controllability of a linear mean-field type game-based control system (MF-GBCS, for short) generated by a linear-quadratic (LQ, for short) Nash game. A Gram-type criterion for the general timevarying coefficients case and a Kalman-type criterion for the special time-invariant coefficients case are obtained. At the same time, the equivalence between the exact controllability of this MF-GBCS and the exact observability of a dual system is established. Moreover, an admissible control that can steer the state from any initi…
In this paper, we study the relative controllability of linear difference equations with multiple delays in the state by using a suitable formula for the solutions of such systems in terms of their initial conditions, their control inputs, and some matrix-valued coefficients obtained recursively from the matrices defining the system. Thanks to such formula, we characterize relative controllability in time $T$ in terms of an algebraic property of the matrix-valued coefficients, which reduces to the usual Kalman controllability criterion in the case of a single delay. Relative controllability is…
S. Hansen and E. Zuazua [SIAM J. Cont. Optim., 1995] studied the problem of exact controllability of two strings connected by a point mass with constant physical coefficients. In this paper we study the same problem with variable physical coefficients. This system is generated by the following equations $$ρ(x) u_{tt}=(σ(x) u_{x})_{x}-q(x)u,~~~~x\in (-1,0)\cup (0,1),~t>0,$$ $$Mu_{tt}(0,t)+σ_{1}(0)u_{x}(0^{-},t)-σ_{2}(0)u_{x}(0^{+},t)=0,~~~t>0,$$ with Dirichlet boundary condition on the left end and a control acts on the right end. We prove that this system is exactly controllable in an asymmetr…
